This entry, , was chosen because it comprehensively contains every OEIS field, filled. A046970 Dirichlet inverse of the Jordan function J_2 (A007434). 1, -3, -8, -3, -24, 24, -48, -3, -8, 72, -120, 24, -168, 144, 192, -3, -288, 24, -360, 72, 384, 360, -528, 24, -24, 504, -8, 144, -840, -576, -960, -3, 960, 864, 1152, 24, -1368, 1080, 1344, 72, -1680, -1152, -1848, 360, 192, 1584, -2208, 24, -48, 72, 2304, 504, -2808, 24, 2880, 144, 2880, 2520, -3480, -576 OFFSET 1,2 COMMENTS B(n+2) = -B(n)*((n+2)*(n+1)/(4*Pi^2))*z(n+2)/z(n) = -B(n)*((n+2)*(n+1)/(4*Pi^2)) * Sum_{j>=1} a(j)/j^(n+2). Apart from signs also Sum_{d|n} core(d)^2*mu(n/d) where core(x) is the squarefree part of x. - Benoit Cloitre, May 31 2002 REFERENCES M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1965, pp. 805-811. T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, p. 48. LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. P. G. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408. Paul W. Oxby, A Function Based on Chebyshev Polynomials as an Alternative to the Sinc Function in FIR Filter Design, arXiv:2011.10546 [eess.SP], 2020. Wikipedia, Riemann zeta function. FORMULA Multiplicative with a(p^e) = 1 - p^2. a(n) = Sum_{d|n} mu(d)*d^2. abs(a(n)) = Product_{p prime divides n} (p^2 - 1). - Jon Perry, Aug 24 2010 From Wolfdieter Lang, Jun 16 2011: (Start) Dirichlet g.f.: zeta(s)/zeta(s-2). a(n) = J_{-2}(n)*n^2, with the Jordan function J_k(n), with J_k(1):=1. See the Apostol reference, p. 48. exercise 17. (End) a(prime(n)) = -A084920(n). - R. J. Mathar, Aug 28 2011 G.f.: Sum_{k>=1} mu(k)*k^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 15 2017 EXAMPLE a(3) = -8 because the divisors of 3 are {1, 3} and mu(1)*1^2 + mu(3)*3^2 = -8. a(4) = -3 because the divisors of 4 are {1, 2, 4} and mu(1)*1^2 + mu(2)*2^2 + mu(4)*4^2 = -3. E.g., a(15) = (3^2 - 1) * (5^2 - 1) = 8*24 = 192. - Jon Perry, Aug 24 2010 G.f. = x - 3*x^2 - 8*x^3 - 3*x^4 - 24*x^5 + 24*x^6 - 48*x^7 - 3*x^8 - 8*x^9 + ... MAPLE Jinvk := proc(n, k) local a, f, p ; a := 1 ; for f in ifactors(n)[2] do p := op(1, f) ; a := a*(1-p^k) ; end do: a ; end proc: A046970 := proc(n) Jinvk(n, 2) ; end proc: # R. J. Mathar, Jul 04 2011 MATHEMATICA muDD[d_] := MoebiusMu[d]*d^2; Table[Plus @@ muDD[Divisors[n, {n, 60}] (Lopez) Flatten[Table[{ x = FactorInteger[n]; p = 1; For[i = 1, i
Entry fields ; ID number : Every sequence in the OEIS has a
serial number, a six-digit positive
integer, prefixed by A (and zero-padded on the left prior to November 2004). The letter "A" stands for "absolute". Numbers are either assigned by the editor(s) or by an A number dispenser, which is handy for when contributors wish to send in multiple related sequences at once and be able to create cross-references. An A number from the dispenser expires a month from issue if not used. But as the following table of arbitrarily selected sequences shows, the rough correspondence holds. : Even for sequences in the book predecessors to the OEIS, the ID numbers are not the same. The 1973
Handbook of Integer Sequences contained about 2400 sequences, which were numbered by lexicographic order (the letter N plus four digits, zero-padded where necessary), and the 1995
Encyclopedia of Integer Sequences contained 5487 sequences, also numbered by lexicographic order (the letter M plus 4 digits, zero-padded where necessary). These old M and N numbers, as applicable, are contained in the ID number field in parentheses after the modern A number. ; Sequence data : The sequence field lists the numbers themselves, to about 260 characters. More terms of the sequences can be provided in so-called B-files. The sequence field makes no distinction between sequences that are finite but still too long to display and sequences that are infinite; instead, the keywords "fini", "full", and "more" are used to distinguish such sequences. To determine to which
n the values given correspond, see the offset field, which gives the
n for the first term given. ; Name : The name field usually contains the most common name for the sequence, and sometimes also the formula. For example, 1, 8, 27, 64, 125, 216, 343, 512, () is named "The
cubes: a(n) = n^3.". ; Comments : The comments field is for information about the sequence that does not quite fit in any of the other fields. The comments field often points out interesting relationships between different sequences and less obvious applications for a sequence. For example, Lekraj Beedassy in a comment to A000578 notes that the cube numbers also count the "total number of
triangles resulting from criss-crossing
cevians within a triangle so that two of its sides are each
n-partitioned", and Neil Sloane points out an unexpected relationship between
centered hexagonal numbers () and second
Bessel polynomials () in a comment to A003215. ; References : References to printed documents (books, papers, ...). ; Links : Links, i.e.
URLs, to online resources. These may be: :# references to applicable articles in journals :# links to the index :# links to text files which hold the sequence terms (in a two column format) over a wider range of indices than held by the main database lines :# links to images in the local database directories which often provide combinatorial background related to
graph theory :# others related to computer codes, more extensive tabulations in specific research areas provided by individuals or research groups ; Formula : Formulas,
recurrences,
generating functions, etc. for the sequence. ; Example : Some examples of sequence member values. ; Maple :
Maple code. ; Mathematica :
Wolfram Language code. ; Program : Originally
Maple and
Mathematica were the preferred programs for calculating sequences in the OEIS, each with their own field labels. , Mathematica was the most popular choice with 100,000 Mathematica programs followed by 50,000
PARI/GP programs, 35,000 Maple programs, and 45,000 in other languages. : As for any other part of the record, if there is no name given, the contribution (here: program) was written by the original submitter of the sequence. ; Crossrefs : Sequence cross-references originated by the original submitter are usually denoted by "
Cf." : Except for new sequences, the "see also" field also includes information on the lexicographic order of the sequence (its "context") and provides links to sequences with close A numbers (A046967, A046968, A046969, A046971, A046972, A046973, in our example). The following table shows the context of our example sequence, A046970: ; Keyword : The OEIS has its own
lexicon: a standard set of mostly four-letter keywords which
characterizes each sequence: :*
allocated – An A-number which has been set aside for a user but for which the entry has not yet been approved (and perhaps not yet written). :*
base – The results of the calculation depend on a specific
positional base. For example, 2, 3, 5, 7, 11, 101, 131, 151, 181 ... are prime numbers regardless of base, but they are
palindromic specifically in base 10. Most of them are not palindromic in binary. Some sequences rate this keyword depending on how they are defined. For example, the
Mersenne primes 3, 7, 31, 127, 8191, 131071, ... does not rate "base" if defined as "primes of the form 2^n − 1". However, defined as "
repunit primes in binary," the sequence would rate the keyword "base". :*
bref – "sequence is too short to do any analysis with", for example, , the number of
isomorphism classes of
associative non-
commutative non-anti-associative
anti-commutative closed
binary operations on a
set of order
n. :*
changed The sequence is changed in the last two weeks. :*
cofr – The sequence represents a
continued fraction, for example the continued fraction expansion of
e () or π (). :*
cons – The sequence is a decimal expansion of a
mathematical constant, such as
e () or π (). :*
core – A sequence that is of foundational importance to a branch of mathematics, such as the prime numbers (), the Fibonacci sequence (), etc. :*
dead – This keyword used for erroneous sequences that have appeared in papers or books, or for duplicates of existing sequences. For example, is the same as . :*
dumb – One of the more subjective keywords, for "unimportant sequences," which may or may not directly relate to mathematics, such as
popular culture references, arbitrary sequences from Internet puzzles, and sequences related to
numeric keypad entries. , "Mix digits of pi and e" is one example of lack of importance, and , "Price is Right wheel" (the sequence of numbers on the
Showcase Showdown wheel used in the U.S. game show
The Price Is Right) is an example of a non-mathematics-related sequence, kept mainly for trivia purposes. :*
easy – The terms of the sequence can be easily calculated. Perhaps the sequence most deserving of this keyword is 1, 2, 3, 4, 5, 6, 7, ... , where each term is 1 more than the previous term. The keyword "easy" is sometimes given to sequences "primes of the form
f(
m)" where
f(
m) is an easily calculated function. (Though even if
f(
m) is easy to calculate for large
m, it might be very difficult to determine if
f(
m) is prime). :*
eigen – A sequence of
eigenvalues. :*
fini – The sequence is finite, although it might still contain more terms than can be displayed. For example, the sequence field of shows only about a quarter of all the terms, but a comment notes that the last term is 3888. :*
frac – A sequence of either numerators or denominators of a sequence of fractions representing
rational numbers. Any sequence with this keyword ought to be cross-referenced to its matching sequence of numerators or denominators, though this may be dispensed with for sequences of
Egyptian fractions, such as , where the sequence of numerators would be . This keyword should not be used for sequences of continued fractions; cofr should be used instead for that purpose. :*
full – The sequence field displays the complete sequence. If a sequence has the keyword "full", it should also have the keyword "fini". One example of a finite sequence given in full is that of the
supersingular primes , of which there are precisely fifteen. :*
hard – The terms of the sequence cannot be easily calculated, even with raw number crunching power. This keyword is most often used for sequences corresponding to unsolved problems, such as "How many
n-spheres can touch another
n-sphere of the same size?" lists the first ten known solutions. :*
hear – A sequence with a graph audio deemed to be "particularly interesting and/or beautiful", some examples are collected at the OEIS site. :*
less – A "less interesting sequence". :*
look – A sequence with a graph visual deemed to be "particularly interesting and/or beautiful". Two examples out of several thousands are A331124 A347347. :*
more – More terms of the sequence are wanted. Readers can submit an extension. :*
mult – The sequence corresponds to a
multiplicative function. Term
a(1) should be 1, and term
a(
mn) can be calculated by multiplying
a(
m) by
a(
n) if
m and
n are
coprime. For example, in ,
a(12) =
a(3)
a(4) = −8 × −3. :*
new – For sequences that were added in the last couple of weeks, or had a major extension recently. This keyword is not given a checkbox in the Web form for submitting new sequences; Sloane's program adds it by default where applicable. :*
nice – Perhaps the most subjective keyword of all, for "
exceptionally nice sequences." :*
nonn – The sequence consists of nonnegative integers (it may include zeroes). No distinction is made between sequences that consist of nonnegative numbers only because of the chosen offset (e.g.,
n3, the cubes, which are all nonnegative from
n = 0 forwards) and those that by definition are completely nonnegative (e.g.,
n2, the squares). :*
obsc – The sequence is considered obscure and needs a better definition. :*
recycled – When the editors agree that a new proposed sequence is not worth adding to the OEIS, an editor blanks the entry leaving only the keyword line with keyword:recycled. The A-number then becomes available for allocation for another new sequence. :*
sign – Some (or all) of the values of the sequence are negative. The entry includes both a Signed field with the signs and a Sequence field consisting of all the values passed through the
absolute value function. :*
tabf – "An irregular (or funny-shaped) array of numbers made into a sequence by reading it row by row." For example, , "Triangle read by rows giving successive states of
cellular automaton generated by "rule 62." :*
tabl – A sequence obtained by reading a geometric arrangement of numbers, such as a triangle or square, row by row. The quintessential example is
Pascal's triangle read by rows, . :*
uned – The sequence has not been edited but it could be worth including in the OEIS. The sequence may contain computational or typographical errors. Contributors are encouraged to edit these sequences. :*
unkn – "Little is known" about the sequence, not even the formula that produces it. For example, , which was presented to the
Internet Oracle to ponder. :*
walk – "Counts walks (or
self-avoiding paths)." :*
word – Depends on the words of a specific language. For example, zero, one, two, three, four, five, etc. For example, 4, 3, 3, 5, 4, 4, 3, 5, 5, 4, 3, 6, 6, 8, 8, 7, 7, 9, 8, 8 ... , "Number of letters in the English name of
n, excluding spaces and hyphens." : Some keywords are mutually exclusive, namely: core and dumb, easy and hard, full and more, less and nice, and nonn and sign. ; Offset : The offset is the index of the first term given. For some sequences, the offset is obvious. For example, if we list the sequence of square numbers as 0, 1, 4, 9, 16, 25 ..., the offset is 0; while if we list it as 1, 4, 9, 16, 25 ..., the offset is 1. The default offset is 0, and most sequences in the OEIS have offset of either 0 or 1. Sequence , the
magic constant for
n ×
n magic square with prime entries (regarding 1 as a prime) with smallest row sums, is an example of a sequence with offset 3, and , "Number of stars of visual magnitude
n." is an example of a sequence with offset −1. Sometimes there can be disagreement over what the initial terms of the sequence are, and correspondingly what the offset should be. In the case of the
lazy caterer's sequence, the maximum number of pieces you can cut a pancake into with
n cuts, the OEIS gives the sequence as 1, 2, 4, 7, 11, 16, 22, 29, 37, ... , with offset 0, while
Mathworld gives the sequence as 2, 4, 7, 11, 16, 22, 29, 37, ... (implied offset 1). It can be argued that making no cuts to the pancake is technically a number of cuts, namely
n = 0, but it can also be argued that an uncut pancake is irrelevant to the problem. Although the offset is a required field, some contributors do not bother to check if the default offset of 0 is appropriate to the sequence they are sending in. The internal format actually shows two numbers for the offset. The first is the number described above, while the second represents the index of the first entry (counting from 1) that has an absolute value greater than 1. This second value is used to speed up the process of searching for a sequence. Thus , which starts 1, 1, 1, 2 with the first entry representing
a(1) has
1, 4 as the internal value of the offset field. ; Author(s) : The author(s) of the sequence is (are) the person(s) who submitted the sequence, even if the sequence has been known since ancient times. The name of the submitter(s) is given first name (spelled out in full), middle initial(s) (if applicable) and last name; this in contrast to the way names are written in the reference fields. The e-mail address of the submitter is also given before 2011, with the @ character replaced by "(AT)" with some exceptions such as for associate editors or if an e-mail address does not exist. Now it has been the policy for OEIS not to display e-mail addresses in sequences. For most sequences after A055000, the author field also includes the date the submitter sent in the sequence. ; Extension : Names of people who extended (added more terms to) the sequence or corrected terms of a sequence, followed by date of extension. ==Sloane's gap==