A091202 -id:A091202 - cp's OEIS frontend

A000005 d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 7, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4, 4, 8, 2, 12, 4, 6, 4, 4, 4, 12, 2, 6, 6, 9, 2, 8, 2, 8

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is Product p^e(p) then d(n) = Product (e(p) + 1). More generally, for k > 0, sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1) is the sum of the k-th powers of the divisors of n.
Number of ways to write n as n = x*y, 1 <= x <= n, 1 <= y <= n. For number of unordered solutions to x*y=n, see A038548.
Note that d(n) is not the number of Pythagorean triangles with radius of the inscribed circle equal to n (that is A078644). For number of primitive Pythagorean triangles having inradius n, see A068068(n).
Number of factors in the factorization of the polynomial x^n-1 over the integers. - T. D. Noe, Apr 16 2003
Also equal to the number of partitions p of n such that all the parts have the same cardinality, i.e., max(p)=min(p). - Giovanni Resta, Feb 06 2006
Equals A127093 as an infinite lower triangular matrix * the harmonic series, [1/1, 1/2, 1/3, ...]. - Gary W. Adamson, May 10 2007
For odd n, this is the number of partitions of n into consecutive integers. Proof: For n = 1, clearly true. For n = 2k + 1, k >= 1, map each (necessarily odd) divisor to such a partition as follows: For 1 and n, map k + (k+1) and n, respectively. For any remaining divisor d <= sqrt(n), map (n/d - (d-1)/2) + ... + (n/d - 1) + (n/d) + (n/d + 1) + ... + (n/d + (d-1)/2) {i.e., n/d plus (d-1)/2 pairs each summing to 2n/d}. For any remaining divisor d > sqrt(n), map ((d-1)/2 - (n/d - 1)) + ... + ((d-1)/2 - 1) + (d-1)/2 + (d+1)/2 + ((d+1)/2 + 1) + ... + ((d+1)/2 + (n/d - 1)) {i.e., n/d pairs each summing to d}. As all such partitions must be of one of the above forms, the 1-to-1 correspondence and proof is complete. - Rick L. Shepherd, Apr 20 2008
Number of subgroups of the cyclic group of order n. - Benoit Jubin, Apr 29 2008
Equals row sums of triangle A143319. - Gary W. Adamson, Aug 07 2008
Equals row sums of triangle A159934, equivalent to generating a(n) by convolving A000005 prefaced with a 1; (1, 1, 2, 2, 3, 2, ...) with the INVERTi transform of A000005, (A159933): (1, 1,-1, 0, -1, 2, ...). Example: a(6) = 4 = (1, 1, 2, 2, 3, 2) dot (2, -1, 0, -1, 1, 1) = (2, -1, 0, -2, 3, 2) = 4. - Gary W. Adamson, Apr 26 2009
Number of times n appears in an n X n multiplication table. - Dominick Cancilla, Aug 02 2010
Number of k >= 0 such that (k^2 + k*n + k)/(k + 1) is an integer. - Juri-Stepan Gerasimov, Oct 25 2015
The only numbers k such that tau(k) >= k/2 are 1,2,3,4,6,8,12. - Michael De Vlieger, Dec 14 2016
a(n) is also the number of partitions of 2*n into equal parts, minus the number of partitions of 2*n into consecutive parts. - Omar E. Pol, May 03 2017
From Tomohiro Yamada, Oct 27 2020: (Start)
Let k(n) = log d(n)*log log n/(log 2 * log n), then lim sup k(n) = 1 (Hardy and Wright, Chapter 18, Theorem 317) and k(n) <= k(6983776800) = 1.537939... (the constant A280235) for every n (Nicolas and Robin, 1983).
There exist infinitely many n such that d(n) = d(n+1) (Heath-Brown, 1984). The number of such integers n <= x is at least c*x/(log log x)^3 (Hildebrand, 1987) but at most O(x/sqrt(log log x)) (Erdős, Carl Pomerance and Sárközy, 1987). (End)
Number of 2D grids of n congruent rectangles with two different side lengths, in a rectangle, modulo rotation (cf. A038548 for squares instead of rectangles). Also number of ways to arrange n identical objects in a rectangle (NOT modulo rotation, cf. A038548 for modulo rotation); cf. A007425 and A140773 for the 3D case. - Manfred Boergens, Jun 08 2021
The constant quoted above from Nicolas and Robin, 6983776800 = 2^5 * 3^3 * 5^2 * 7 * 11 * 13 * 17 * 19, appears arbitrary, but interestingly equals 2 * A095849(36). That second factor is highly composite and deeply composite. - Hal M. Switkay, Aug 08 2025

			G.f. = x + 2*x^2 + 2*x^3 + 3*x^4 + 2*x^5 + 4*x^6 + 2*x^7 + 4*x^8 + 3*x^9 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
G. Chrystal, Algebra: An elementary text-book for the higher classes of secondary schools and for colleges, 6th ed, Chelsea Publishing Co., New York 1959 Part II, p. 345, Exercise XXI(16). MR0121327 (22 #12066)
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 55.
G. H. Hardy and E. M. Wright, revised by D. R. Heath-Brown and J. H. Silverman, An Introduction to the Theory of Numbers, 6th ed., Oxford Univ. Press, 2008.
K. Knopp, Theory and Application of Infinite Series, Blackie, London, 1951, p. 451.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Chap. II. (For inequalities, etc.)
S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962. Has many references to this sequence. - N. J. A. Sloane, Jun 02 2014
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
B. Spearman and K. S. Williams, Handbook of Estimates in the Theory of Numbers, Carleton Math. Lecture Note Series No. 14, 1975; see p. 2.1.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 285.
E. C. Titchmarsh, The Theory of Functions, Oxford, 1938, p. 160.
Terence Tao, Poincaré's Legacies, Part I, Amer. Math. Soc., 2009, see pp. 31ff for upper bounds on d(n).

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from N. J. A. Sloane)
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, requires Flash plugin].
G. E. Andrews, Some debts I owe, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000; see (7.1).
J. Bell, Lambert series in analytic number theory
R. Bellman and H. N. Shapiro, On a problem in additive number theory, Annals Math., 49 (1948), 333-340. [From _N. J. A. Sloane_, Mar 12 2009]
Henry Bottomley, Illustration of initial terms
D. M. Bressoud and M. V. Subbarao, On Uchimura's connection between partitions and the number of divisors, Can. Math. Bull. 27, 143-145 (1984). Zbl 0536.10013.
C. K. Caldwell, The Prime Glossary, Number of divisors
Imanuel Chen and Michael Z. Spivey, Integral Generalized Binomial Coefficients of Multiplicative Functions, Preprint 2015; Summer Research Paper 238, Univ. Puget Sound.
Jimmy Devillet and Gergely Kiss, Characterizations of biselective operations, arXiv:1806.02073 [math.RA], 2018.
P. Erdős and L. Mirsky, The distribution of values of the divisor function d(n), Proc. London Math. Soc. 2 (1952), pp. 257-271.
Paul Erdős, Carl Pomerance and András Sárközy, On locally repeated values of certain arithmetic functions, III, Proc. Amer. Math. Soc. 101 (1987), 1-7.
C. R. Fletcher, Rings of small order, Math. Gaz. vol. 64, p. 13, 1980.
Robbert Fokkink and Jan van Neerven, Problemen/UWC (in Dutch)
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors ..., J. Integer Seqs., Vol. 6, 2003.
D. R. Heath-Brown, The divisor function at consecutive integers, Mathematika 31 (1984), 141-149.
Adolf Hildebrand, The divisor function at consecutive integers, Pacific J. Math. 129 (1987), 307-319.
J. J. Holt and J. W. Jones, Counting Divisors, Discovering Number Theory, Section 1.4.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113.
M. Maia and M. Mendez, On the arithmetic product of combinatorial species, arXiv:math/0503436 [math.CO], 2005.
R. G. Martinez, Jr., The Factor Zone, Number of Factors for 1 through 600.
Math Forum, Divisor Counting.
Mathematics Stack Exchange, A question on discrete Fourier Transform of some function
K. Matthews, Factorizing n and calculating phi(n), omega(n), d(n), sigma(n) and mu(n).
Mircea Merca, A new look on the generating function for the number of divisors, Journal of Number Theory, Volume 149, April 2015, Pages 57-69.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, corollary 2.1.
J. L. Nicolas and G. Robin, Majorations Explicites Pour le Nombre de Diviseurs de N, Canadian Mathematical Bulletin, Volume 26, Issue 4, 01 December 1983, pp. 485-492.
Matthew Parker, The first 25 million terms (7-Zip compressed file).
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003.
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003. [Cached copy, with permission (pdf only)]
Omar E. Pol, Illustration of initial terms: figure 1, figure 2, figure 3, figure 4, figure 5, (2009), figure 6 (a, b, c), (2013)
S. Ramanujan, On The Number Of Divisors Of A Number.
H. B. Reiter, Counting Divisors.
W. Sierpiński, Number Of Divisors And Their Sum.
Terence Tao, Poincaré's Legacies: pages from year two of a mathematical blog, see page 59.
E. C. Titchmarsh, On a series of Lambert type, J. London Math. Soc., 13 (1938), 248-253.
Keisuke Uchimura, An identity for the divisor generating function arising from sorting theory, J. Combin. Theory Ser. A 31 (1981), no. 2, 131--135. MR0629588 (82k:05015)
Wang Zheng Bing, Robert Fokkink and Wan Fokkink, A Relation Between Partitions and the Number of Divisors, Am. Math. Monthly, 102 (Apr., 1995), no. 4, 345-347.
Eric Weisstein's World of Mathematics, Binomial Number, Dirichlet Series Generating Function, Divisor Function, and Moebius Transform.
Wikipedia, Table of divisors.
Wolfram Research, Divisors of first 50 numbers
Index entries for "core" sequences
Index entries for sequences computed from exponents in factorization of n

See A002183, A002182 for records. See A000203 for the sum-of-divisors function sigma(n).
For partial sums see A006218.
Cf. A007427 (Dirichlet Inverse), A001227, A001826, A001842, A005237, A005238, A006558, A006601, A019273, A027750, A034296, A039665, A049051, A049820, A051731, A061017, A066446, A091202, A091220, A106737, A115361, A127093, A129372, A129510, A143319, A156552, A159933, A159934, A163280, A183063, A237665, A263730.
Factorizations into given number of factors: writing n = x*y (A038548, unordered, A000005, ordered), n = x*y*z (A034836, unordered, A007425, ordered), n = w*x*y*z (A007426, ordered).
Cf. A000010, A095849.
Cf. A098198 (Dgf at s=2), A183030 (Dgf at s=3), A183031 (Dgf at s=3).

List([1..150],n->Tau(n)); # Muniru A Asiru, Mar 05 2019

divisors 1 = [1]
divisors n = (1:filter ((==0) . rem n)
               [2..n `div` 2]) ++ [n]
a = length . divisors
-- James Spahlinger, Oct 07 2012

a000005 = product . map (+ 1) . a124010_row  -- Reinhard Zumkeller, Jul 12 2013

function tau(n)
    i = 2; num = 1
    while i * i <= n
        if rem(n, i) == 0
            e = 0
            while rem(n, i) == 0
                e += 1
                n = div(n, i)
            end
            num *= e + 1
        end
        i += 1
    end
    return n > 1 ? num + num : num
end
println([tau(n) for n in 1:104])  # Peter Luschny, Sep 03 2023

[ NumberOfDivisors(n) : n in [1..100] ]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

with(numtheory): A000005 := tau; [ seq(tau(n), n=1..100) ];

Table[DivisorSigma[0, n], {n, 100}] (* Enrique Pérez Herrero, Aug 27 2009 *)
CoefficientList[Series[(Log[1 - q] + QPolyGamma[1, q])/(q Log[q]), {q, 0, 100}], q] (* Vladimir Reshetnikov, Apr 23 2013 *)
a[ n_] := SeriesCoefficient[ (QPolyGamma[ 1, q] + Log[1 - q]) / Log[q], {q, 0, Abs@n}]; (* Michael Somos, Apr 25 2013 *)
a[ n_] := SeriesCoefficient[ q/(1 - q)^2 QHypergeometricPFQ[ {q, q}, {q^2, q^2}, q, q^2], {q, 0, Abs@n}]; (* Michael Somos, Mar 05 2014 *)
a[n_] := SeriesCoefficient[q/(1 - q) QHypergeometricPFQ[{q, q}, {q^2}, q, q], {q, 0, Abs@n}] (* Mats Granvik, Apr 15 2015 *)
With[{M=500},CoefficientList[Series[(2x)/(1-x)-Sum[x^k (1-2x^k)/(1-x^k),{k,M}],{x,0,M}],x]] (* Mamuka Jibladze, Aug 31 2018 *)

numlib::tau (n)$ n=1..90 // Zerinvary Lajos, May 13 2008

{a(n) = if( n==0, 0, numdiv(n))}; /* Michael Somos, Apr 27 2003 */

{a(n) = n=abs(n); if( n<1, 0, direuler( p=2, n, 1 / (1 - X)^2)[n])}; /* Michael Somos, Apr 27 2003 */

{a(n)=polcoeff(sum(m=1, n+1, sumdiv(m, d, (-log(1-x^(m/d) +x*O(x^n) ))^d/d!)), n)} \\ Paul D. Hanna, Aug 21 2014

from sympy import divisor_count
for n in range(1, 20): print(divisor_count(n), end=', ') # Stefano Spezia, Nov 05 2018

[sigma(n, 0) for n in range(1, 105)]  # Zerinvary Lajos, Jun 04 2009

If n is written as 2^z*3^y*5^x*7^w*11^v*... then a(n)=(z+1)*(y+1)*(x+1)*(w+1)*(v+1)*...
a(n) = 2 iff n is prime.
G.f.: Sum_{n >= 1} a(n) x^n = Sum_{k>0} x^k/(1-x^k). This is usually called THE Lambert series (see Knopp, Titchmarsh).
a(n) = A083888(n) + A083889(n) + A083890(n) + A083891(n) + A083892(n) + A083893(n) + A083894(n) + A083895(n) + A083896(n).
a(n) = A083910(n) + A083911(n) + A083912(n) + A083913(n) + A083914(n) + A083915(n) + A083916(n) + A083917(n) + A083918(n) + A083919(n).
Multiplicative with a(p^e) = e+1. - David W. Wilson, Aug 01 2001
a(n) <= 2 sqrt(n) [see Mitrinovich, p. 39, also A046522].
a(n) is odd iff n is a square. - Reinhard Zumkeller, Dec 29 2001
a(n) = Sum_{k=1..n} f(k, n) where f(k, n) = 1 if k divides n, 0 otherwise (Mobius transform of A000012). Equivalently, f(k, n) = (1/k)*Sum_{l=1..k} z(k, l)^n with z(k, l) the k-th roots of unity. - Ralf Stephan, Dec 25 2002
G.f.: Sum_{k>0} ((-1)^(k+1) * x^(k * (k + 1)/2) / ((1 - x^k) * Product_{i=1..k} (1 - x^i))). - Michael Somos, Apr 27 2003
a(n) = n - Sum_{k=1..n} (ceiling(n/k) - floor(n/k)). - Benoit Cloitre, May 11 2003
a(n) = A032741(n) + 1 = A062011(n)/2 = A054519(n) - A054519(n-1) = A006218(n) - A006218(n-1) = 1 + Sum_{k=1..n-1} A051950(k+1). - Ralf Stephan, Mar 26 2004
G.f.: Sum_{k>0} x^(k^2)*(1+x^k)/(1-x^k). Dirichlet g.f.: zeta(s)^2. - Michael Somos, Apr 05 2003
Sequence = M*V where M = A129372 as an infinite lower triangular matrix and V = ruler sequence A001511 as a vector: [1, 2, 1, 3, 1, 2, 1, 4, ...]. - Gary W. Adamson, Apr 15 2007
Sequence = M*V, where M = A115361 is an infinite lower triangular matrix and V = A001227, the number of odd divisors of n, is a vector: [1, 1, 2, 1, 2, 2, 2, ...]. - Gary W. Adamson, Apr 15 2007
Row sums of triangle A051731. - Gary W. Adamson, Nov 02 2007
Sum_{n>0} a(n)/(n^n) = Sum_{n>0, m>0} 1/(n*m). - Gerald McGarvey, Dec 15 2007
Logarithmic g.f.: Sum_{n>=1} a(n)/n * x^n = -log( Product_{n>=1} (1-x^n)^(1/n) ). - Joerg Arndt, May 03 2008
a(n) = Sum_{k=1..n} (floor(n/k) - floor((n-1)/k)). - Enrique Pérez Herrero, Aug 27 2009
a(s) = 2^omega(s), if s > 1 is a squarefree number (A005117) and omega(s) is: A001221. - Enrique Pérez Herrero, Sep 08 2009
a(n) = A048691(n) - A055205(n). - Reinhard Zumkeller, Dec 08 2009
For n > 1, a(n) = 2 + Sum_{k=2..n-1} floor((cos(Pi*n/k))^2). And floor((cos(Pi*n/k))^2) = floor(1/4 * e^(-(2*i*Pi*n)/k) + 1/4 * e^((2*i*Pi*n)/k) + 1/2). - Eric Desbiaux, Mar 09 2010, corrected Apr 16 2011
a(n) = 1 + Sum_{k=1..n} (floor(2^n/(2^k-1)) mod 2) for every n. - Fabio Civolani (civox(AT)tiscali.it), Mar 12 2010
From Vladimir Shevelev, May 22 2010: (Start)
(Sum_{d|n} a(d))^2 = Sum_{d|n} a(d)^3 (J. Liouville).
Sum_{d|n} A008836(d)*a(d)^2 = A008836(n)*Sum_{d|n} a(d). (End)
a(n) = sigma_0(n) = 1 + Sum_{m>=2} Sum_{r>=1} (1/m^(r+1))*Sum_{j=1..m-1} Sum_{k=0..m^(r+1)-1} e^(2*k*Pi*i*(n+(m-j)*m^r)/m^(r+1)). - A. Neves, Oct 04 2010
a(n) = 2*A038548(n) - A010052(n). - Reinhard Zumkeller, Mar 08 2013
Sum_{n>=1} a(n)*q^n = (log(1-q) + psi_q(1)) / log(q), where psi_q(z) is the q-digamma function. - Vladimir Reshetnikov, Apr 23 2013
a(n) = Product_{k = 1..A001221(n)} (A124010(n,k) + 1). - Reinhard Zumkeller, Jul 12 2013
a(n) = Sum_{k=1..n} A238133(k)*A000041(n-k). - Mircea Merca, Feb 18 2013
G.f.: Sum_{k>=1} Sum_{j>=1} x^(j*k). - Mats Granvik, Jun 15 2013
The formula above is obtained by expanding the Lambert series Sum_{k>=1} x^k/(1-x^k). - Joerg Arndt, Mar 12 2014
G.f.: Sum_{n>=1} Sum_{d|n} ( -log(1 - x^(n/d)) )^d / d!. - Paul D. Hanna, Aug 21 2014
2*Pi*a(n) = Sum_{m=1..n} Integral_{x=0..2*Pi} r^(m-n)( cos((m-n)*x)-r^m cos(n*x) )/( 1+r^(2*m)-2r^m cos(m*x) )dx, 0 < r < 1 a free parameter. This formula is obtained as the sum of the residues of the Lambert series Sum_{k>=1} x^k/(1-x^k). - Seiichi Kirikami, Oct 22 2015
a(n) = A091220(A091202(n)) = A106737(A156552(n)). - Antti Karttunen, circa 2004 & Mar 06 2017
a(n) = A034296(n) - A237665(n+1) [Wang, Fokkink, Fokkink]. - George Beck, May 06 2017
G.f.: 2*x/(1-x) - Sum_{k>0} x^k*(1-2*x^k)/(1-x^k). - Mamuka Jibladze, Aug 29 2018
a(n) = Sum_{k=1..n} 1/phi(n / gcd(n, k)). - Daniel Suteu, Nov 05 2018
a(k*n) = a(n)*(f(k,n)+2)/(f(k,n)+1), where f(k,n) is the exponent of the highest power of k dividing n and k is prime. - Gary Detlefs, Feb 08 2019
a(n) = 2*log(p(n))/log(n), n > 1, where p(n)= the product of the factors of n = A007955(n). - Gary Detlefs, Feb 15 2019
a(n) = (1/n) * Sum_{k=1..n} sigma(gcd(n,k)), where sigma(n) = sum of divisors of n. - Orges Leka, May 09 2019
a(n) = A001227(n)*(A007814(n) + 1) = A001227(n)*A001511(n). - Ivan N. Ianakiev, Nov 14 2019
From Richard L. Ollerton, May 11 2021: (Start)
a(n) = A038040(n) / n = (1/n)*Sum_{d|n} phi(d)*sigma(n/d), where phi = A000010 and sigma = A000203.
a(n) = (1/n)*Sum_{k=1..n} phi(gcd(n,k))*sigma(n/gcd(n,k))/phi(n/gcd(n,k)). (End)
From Ridouane Oudra, Nov 12 2021: (Start)
a(n) = Sum_{j=1..n} Sum_{k=1..j} (1/j)*cos(2*k*n*Pi/j);
a(n) = Sum_{j=1..n} Sum_{k=1..j} (1/j)*e^(2*k*n*Pi*i/j), where i^2=-1. (End)

Incorrect formula deleted by Ridouane Oudra, Oct 28 2021

A001222 Number of prime divisors of n counted with multiplicity (also called big omega of n, bigomega(n) or Omega(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 4, 1, 2, 3, 6, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 5, 4, 2, 1, 4, 2, 2, 2, 4, 1, 4, 2, 3, 2, 2, 2, 6, 1, 3, 3, 4, 1, 3, 1, 4, 3, 2, 1, 5, 1, 3, 2

Offset: 1

N. J. A. Sloane

Maximal number of terms in any factorization of n.
Number of prime powers (not including 1) that divide n.
Sum of exponents in prime-power factorization of n. - Daniel Forgues, Mar 29 2009
Sum_{d|n} 2^(-A001221(d) - a(n/d)) = Sum_{d|n} 2^(-a(d) - A001221(n/d)) = 1 (see Dressler and van de Lune link). - Michel Marcus, Dec 18 2012
Row sums in A067255. - Reinhard Zumkeller, Jun 11 2013
Conjecture: Let f(n) = (x+y)^a(n), and g(n) = x^a(n), and h(n) = (x+y)^A046660(n) * y^A001221(n) with x, y complex numbers and 0^0 = 1. Then f(n) = Sum_{d|n} g(d)*h(n/d). This is proved for x = 1-y (see Dressler and van de Lune link). - Werner Schulte, Feb 10 2018
Let r, s be some fixed integers. Then we have:
  (1) The sequence b(n) = Dirichlet convolution of r^bigomega(n) and s^bigomega(n) is multiplicative with b(p^e) = (r^(e+1)-s^(e+1))/(r-s) for prime p and e >= 0. The case r = s leads to b(p^e) = (e+1)*r^e.
  (2) The sequence c(n) = Dirichlet convolution of r^bigomega(n) and mu(n)*s^bigomega(n) is multiplicative with c(p^e) = (r-s)*r^(e-1) and c(1) = 1 for prime p and e > 0 where mu(n) = A008683(n). - Werner Schulte, Feb 20 2019
a(n) is also the length of the composition series for every solvable group of order n. - Miles Englezou, Apr 25 2024

			16=2^4, so a(16)=4; 18=2*3^2, so a(18)=3.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 119, #12, omega(n).
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, pp. 48-57.
M. Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Monograph 12, Math. Assoc. Amer., 1959, see p. 64.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 92.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from N. J. A. Sloane)
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. 844.
Benoit Cloitre, A tauberian approach to RH, arXiv:1107.0812 [math.NT], 2011.
Robert E. Dressler and Jan van de Lune, Some remarks concerning the number theoretic functions omega and Omega, Proc. Amer. Math. Soc. 41 (1973), 403-406.
G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number, Quart. J. Math. 48 (1917), 76-92. Also Collected papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI (2000): 262-275.
Douglas E. Iannucci and Urban Larsson, Game values of arithmetic functions, arXiv:2101.07608 [math.NT], 2021. Section 1.1.1. pp. 4-5.
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4, 1.10.
Eric Weisstein's World of Mathematics, Prime Factor
Eric Weisstein's World of Mathematics, Roundness
Wolfram Research, First 50 numbers factored
Index entries for "core" sequences
Index entries for sequences computed from exponents in factorization of n
Index entries for sequences computed from indices in prime factorization

Cf. A001221 (omega, primes counted without multiplicity), A008836 (Liouville's lambda, equal to (-1)^a(n)), A046660, A144494, A074946, A134334.
Bisections give A091304 and A073093. A086436 is essentially the same sequence. Cf. A022559 (partial sums), A066829 (parity), A092248 (parity of omega).
Sequences listing n such that a(n) = r: A000040 (r = 1), A001358 (r = 2), A014612 (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13),  A069275 (r = 14),  A069276 (r = 15),  A069277 (r = 16),  A069278 (r = 17),  A069279 (r = 18),  A069280 (r = 19),  A069281 (r = 20). - Jason Kimberley, Oct 02 2011
Cf. A079149 (primes adj. to integers with at most 2 prime factors, a(n)<=2).
Cf. A000120, A020639, A091202, A091222, A156552, A267116, A268387.
Cf. A027748 (without repetition).
Cf. A000010.

Concatenation([0],List([2..150],n->Length(Factors(n)))); # Muniru A Asiru, Feb 21 2019

import Math.NumberTheory.Primes.Factorisation (factorise)
a001222 = sum . snd . unzip . factorise
-- Reinhard Zumkeller, Nov 28 2015

using Nemo
function NumberOfPrimeFactors(n; distinct=true)
    distinct && return length(factor(ZZ(n)))
    sum(e for (p, e) in factor(ZZ(n)); init=0)
end
println([NumberOfPrimeFactors(n, distinct=false) for n in 1:60])  # Peter Luschny, Jan 02 2024

[n eq 1 select 0 else &+[p[2]: p in Factorization(n)]: n in [1..120]]; // Bruno Berselli, Nov 27 2013

with(numtheory): seq(bigomega(n), n=1..111);

Array[ Plus @@ Last /@ FactorInteger[ # ] &, 105]
PrimeOmega[Range[120]] (* Harvey P. Dale, Apr 25 2011 *)

PARI
```
vector(100,n,bigomega(n))
```

from sympy import primeomega
def a(n): return primeomega(n)
print([a(n) for n in range(1, 112)]) # Michael S. Branicky, Apr 30 2022

[sloane.A001222(n) for n in (1..120)] # Giuseppe Coppoletta, Jan 19 2015

[gp.bigomega(n) for n in range(1,131)] # G. C. Greubel, Jul 13 2024

(define (A001222 n) (let loop ((n n) (z 0)) (if (= 1 n) z (loop (/ n (A020639 n)) (+ 1 z)))))
;; Requires also A020639 for which an equally naive implementation can be found under that entry. - Antti Karttunen, Apr 12 2017

n = Product_(p_j^k_j) -> a(n) = Sum_(k_j).
Dirichlet g.f.: ppzeta(s)*zeta(s). Here ppzeta(s) = Sum_{p prime} Sum_{k>=1} 1/(p^k)^s. Note that ppzeta(s) = Sum_{p prime} 1/(p^s-1) and ppzeta(s) = Sum_{k>=1} primezeta(k*s). - Franklin T. Adams-Watters, Sep 11 2005
Totally additive with a(p) = 1.
a(n) = if n=1 then 0 else a(n/A020639(n)) + 1. - Reinhard Zumkeller, Feb 25 2008
a(n) = Sum_{k=1..A001221(n)} A124010(n,k). - Reinhard Zumkeller, Aug 27 2011
a(n) = A022559(n) - A022559(n-1).
G.f.: Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k)). - Ilya Gutkovskiy, Jan 25 2017
a(n) = A091222(A091202(n)) = A000120(A156552(n)). - Antti Karttunen, circa 2004 and Mar 06 2017
a(n) >= A267116(n) >= A268387(n). - Antti Karttunen, Apr 12 2017
Sum_{k=1..n} 2^(-A001221(gcd(n,k)) - a(n/gcd(n,k)))/phi(n/gcd(n,k)) = Sum_{k=1..n} 2^(-a(gcd(n,k)) - A001221(n/gcd(n,k)))/phi(n/gcd(n,k)) = 1, where phi = A000010. - Richard L. Ollerton, May 13 2021
a(n) = a(A046523(n)) = A007814(A108951(n)) = A061395(A122111(n)) = A056239(A181819(n)) = A048675(A293442(n)). - Antti Karttunen, Apr 30 2022

More terms from David W. Wilson

A001221 Number of distinct primes dividing n (also called omega(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 3, 2, 1, 2, 1, 3, 2

Offset: 1

N. J. A. Sloane

From Peter C. Heinig (algorithms(AT)gmx.de), Mar 08 2008: (Start)
This is also the number of maximal ideals of the ring (Z/nZ,+,*). Since every finite integral domain must be a field, every prime ideal of Z/nZ is a maximal ideal and since in general each maximal ideal is prime, there are just as many prime ideals as maximal ones in Z/nZ, so the sequence gives the number of prime ideals of Z/nZ as well.
The reason why this number is given by the sequence is that the ideals of Z/nZ are precisely the subgroups of (Z/nZ,+). Hence for an ideal to be maximal it has form a maximal subgroup of (Z/nZ,+) and this is equivalent to having prime index in (Z/nZ) and this is equivalent to being generated by a single prime divisor of n.
Finally, all the groups arising in this way have different orders, hence are different, so the number of maximal ideals equals the number of distinct primes dividing n. (End)
Equals double inverse Mobius transform of A143519, where A051731 = the inverse Mobius transform. - Gary W. Adamson, Aug 22 2008
a(n) is the number of unitary prime power divisors of n (not including 1). - Jaroslav Krizek, May 04 2009 [corrected by Ilya Gutkovskiy, Oct 09 2019]
Sum_{d|n} 2^(-A001221(d) - A001222(n/d)) = Sum_{d|n} 2^(-A001222(d) - A001221(n/d)) = 1 (see Dressler and van de Lune link). - Michel Marcus, Dec 18 2012
Up to 2*3*5*7*11*13*17*19*23*29  - 1 = 6469693230 - 1, also the decimal expansion of the constant 0.01111211... = Sum_{k>=0} 1/(10 ^ A000040(k) - 1) (see A073668). - Eric Desbiaux, Jan 20 2014
The average order of a(n): Sum_{k=1..n} a(k) ~ Sum_{k=1..n} log log k. - Daniel Forgues, Aug 13-16 2015
From Peter Luschny, Jul 13 2023: (Start)
We can use A001221 and A001222 to classify the positive integers as follows.
   A001222(n) = A001221(n) = 0 singles out {1}.
Restricting to n > 1:
   A001222(n)^A001221(n) = 1: A000040, prime numbers.
   A001221(n)^A001222(n) = 1: A246655, prime powers.
   A001222(n)^A001221(n) > 1: A002808, the composite numbers.
   A001221(n)^A001222(n) > 1: A024619, complement of A246655.
   n^(A001222(n) - A001221(n)) = 1: A144338, products of distinct primes. (End)
Inverse Möbius transform of the characteristic function of primes (A010051). - Wesley Ivan Hurt, Jun 22 2024
Dirichlet convolution of A010051(n) and 1. - Wesley Ivan Hurt, Jul 15 2025

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, pp. 48-57.
J. Peters, A. Lodge and E. J. Ternouth, E. Gifford, Factor Table (n<100000) (British Association Mathematical Tables Vol.V), Burlington House/Cambridge University Press London 1935.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 from NJAS)
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].
Henry Bottomley, Prime factors calculator
J. Brennen, Prime Factoring Applet
J. Britton, Prime Factorization Machine
A. Dendane, Prime Factors Calculator
Robert E. Dressler and Jan van de Lune, Some remarks concerning the number theoretic functions omega and Omega, Proc. Amer. Math. Soc. 41 (1973), 403-406
J. Flament, Decomposition d'un nombre en facteurs premiers
G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number, Quart. J. Math. 48 (1917), 76-92. Also Collected papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI (2000): 262-275.
A. Hodges, Java Applet for Factorisation
M. Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Monograph 12, Math. Assoc. Amer., 1959, see p. 64.
S. O. S. Math, Prime factorization of the first 1000 integers
K. Matthews, Factorization and calculating phi(n),omega(n),d(n),sigma(n) and mu(n)
J. Moyer, "Prime Factors of Integers" server for numbers up to 10^36
Primefan, The First 2500 Integers,Factored
Primefan, Factorer
F. Richman, Factoring into Primes
Eric Weisstein's World of Mathematics, Distinct Prime Factors
Eric Weisstein's World of Mathematics, Moebius Transform
Eric Weisstein's World of Mathematics, Prime Factor
Eric Weisstein's World of Mathematics, Prime zeta function primezeta(s).
Wikipedia, Prime factor
Wikipedia, Table of prime factors
G. Xiao, WIMS server, Factoris
Index entries for "core" sequences
Index entries for sequences computed from exponents in factorization of n

Cf. A001222 (primes counted with multiplicity), A046660, A285577, A346617. Partial sums give A013939.
Cf. also A125666, A069010, A087624, A091202, A091221, A143519, A144494, A158210, A156542, A156552, A000010, A008683.
Sum of the k-th powers of the primes dividing n for k=0..10: this sequence (k=0), A008472 (k=1), A005063 (k=2), A005064 (k=3), A005065 (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), A351196 (k=8), A351197 (k=9), A351198 (k=10).
Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k=0..10: this sequence (k=0), A069359 (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

import Math.NumberTheory.Primes.Factorisation (factorise)
a001221 = length . snd . unzip . factorise
-- Reinhard Zumkeller, Nov 28 2015

using Nemo
function NumberOfPrimeFactors(n; distinct=true)
    distinct && return length(factor(ZZ(n)))
    sum(e for (p, e) in factor(ZZ(n)); init=0)
end
println([NumberOfPrimeFactors(n) for n in 1:60]) # Peter Luschny, Jan 02 2024

[#PrimeDivisors(n): n in [1..120]]; // Bruno Berselli, Oct 15 2021

A001221 := proc(n) local t1, i; if n = 1 then return 0 else t1 := 0; for i to n do if n mod ithprime(i) = 0 then t1 := t1 + 1 end if end do end if; t1 end proc;
A001221 := proc(n) nops(numtheory[factorset](n)) end proc: # Emeric Deutsch
omega := n -> NumberTheory:-NumberOfPrimeFactors(n, 'distinct'): # Peter Luschny, Jun 15 2025

Array[ Length[ FactorInteger[ # ] ]&, 100 ]
PrimeNu[Range[120]]  (* Harvey P. Dale, Apr 26 2011 *)

func(nops(numlib::primedivisors(n)), n):

numlib::omega(n)$ n=1..110 // Zerinvary Lajos, May 13 2008

PARI
```
a(n)=omega(n)
```

from sympy.ntheory import primefactors
print([len(primefactors(n)) for n in range(1, 1001)])  # Indranil Ghosh, Mar 19 2017

def A001221(n): return sum(1 for p in divisors(n) if is_prime(p))
[A001221(n) for n in (1..80)] # Peter Luschny, Feb 01 2012

[sloane.A001221(n) for n in (1..111)] # Giuseppe Coppoletta, Jan 19 2015

[gp.omega(n) for n in range(1,101)] # G. C. Greubel, Jul 13 2024

G.f.: Sum_{k>=1} x^prime(k)/(1-x^prime(k)). - Benoit Cloitre, Apr 21 2003; corrected by Franklin T. Adams-Watters, Sep 01 2009
Dirichlet generating function: zeta(s)*primezeta(s). - Franklin T. Adams-Watters, Sep 11 2005
Additive with a(p^e) = 1.
a(1) = 0, a(p) = 1, a(pq) = 2, a(pq...z) = k, a(p^k) = 1, where p, q, ..., z are k distinct primes and k natural numbers. - Jaroslav Krizek, May 04 2009
a(n) = log_2(Sum_{d|n} mu(d)^2). - Enrique Pérez Herrero, Jul 09 2012
a(A002110(n)) = n, i.e., a(prime(n)#) = n. - Jean-Marc Rebert, Jul 23 2015
a(n) = A091221(A091202(n)) = A069010(A156552(n)). - Antti Karttunen, circa 2004 & Mar 06 2017
L.g.f.: -log(Product_{k>=1} (1 - x^prime(k))^(1/prime(k))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018
a(n) = log_2(Sum_{k=1..n} mu(gcd(n,k))^2/phi(n/gcd(n,k))) = log_2(Sum_{k=1..n} mu(n/gcd(n,k))^2/phi(n/gcd(n,k))), where phi = A000010 and mu = A008683. - Richard L. Ollerton, May 13 2021
Sum_{k=1..n} 2^(-a(gcd(n,k)) - A001222(n/gcd(n,k)))/phi(n/gcd(n,k)) = Sum_{k=1..n} 2^(-A001222(gcd(n,k)) - a(n/gcd(n,k)))/phi(n/gcd(n,k)) = 1, where phi = A000010. - Richard L. Ollerton, May 13 2021
a(n) = A005089(n) + A005091(n) + A059841(n) = A005088(n) +A005090(n) +A079978(n). - R. J. Mathar, Jul 22 2021
From Wesley Ivan Hurt, Jun 22 2024: (Start)
a(n) = Sum_{p|n, p prime} 1.
a(n) = Sum_{d|n} c(d), where c = A010051. (End)

A008683 Möbius (or Moebius) function mu(n). mu(1) = 1; mu(n) = (-1)^k if n is the product of k different primes; otherwise mu(n) = 0.

Original entry on oeis.org

1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1, 0, -1, 0, 1, 1, -1, 0, 0, 1, 0, 0, -1, -1, -1, 0, 1, 1, 1, 0, -1, 1, 1, 0, -1, -1, -1, 0, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 1, 0, 1, 1, -1, 0, -1, 1, 0, 0, 1, -1, -1, 0, 1, -1, -1, 0, -1, 1, 0, 0, 1, -1

Offset: 1

N. J. A. Sloane

Moebius inversion: f(n) = Sum_{d|n} g(d) for all n <=> g(n) = Sum_{d|n} mu(d)*f(n/d) for all n.
a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3 * 3 and 375 = 3 * 5^3 both have prime signature (3, 1).
A008683 = A140579^(-1) * A140664. - Gary W. Adamson, May 20 2008
Coons & Borwein prove that Sum_{n>=1} mu(n) z^n is transcendental. - Jonathan Vos Post, Jun 11 2008; edited by Charles R Greathouse IV, Sep 06 2017
Equals row sums of triangle A144735 (the square of triangle A054533). - Gary W. Adamson, Sep 20 2008
Conjecture: a(n) is the determinant of Redheffer matrix A143104 where T(n, n) = 0. Verified for the first 50 terms. - Mats Granvik, Jul 25 2008
From Mats Granvik, Dec 06 2008: (Start)
The Editorial Office of the Journal of Number Theory kindly provided (via B. Conrey) the following proof of the conjecture: Let A be A143104 and B be A143104 where T(n, n) = 0.
"Suppose you expand det(B_n) along the bottom row. There is only a 1 in the first position and so the answer is (-1)^n times det(C_{n-1}) say, where C_{n-1} is the (n-1) by (n-1) matrix obtained from B_n by deleting the first column and the last row. Now the determinant of the Redheffer matrix is det(A_n) = M(n) where M(n) is the sum of mu(m) for 1 <= m <= n. Expanding det(A_n) along the bottom row, we see that det(A_n) = (-1)^n * det(C_{n-1}) + M(n-1). So we have det(B_n) = (-1)^n * det(C_{n-1}) = det(A_n) - M(n-1) = M(n) - M(n-1) = mu(n)." (End)
Conjecture: Consider the table A051731 and treat 1 as a divisor. Move the value in the lower right corner vertically to a divisor position in the transpose of the table and you will find that the determinant is the Moebius function. The number of permutation matrices that contribute to the Moebius function appears to be A074206. - Mats Granvik, Dec 08 2008
Convolved with A152902 = A000027, the natural numbers. - Gary W. Adamson, Dec 14 2008
[Pickover, p. 226]: "The probability that a number falls in the -1 mailbox turns out to be 3/Pi^2 - the same probability as for falling in the +1 mailbox". - Gary W. Adamson, Aug 13 2009
Let A = A176890 and B = A * A * ... * A, then the leftmost column in matrix B converges to the Moebius function. - Mats Granvik, Gary W. Adamson, Apr 28 2010 and May 28 2020
Equals row sums of triangle A176918. - Gary W. Adamson, Apr 29 2010
Calculate matrix powers: A175992^0 - A175992^1 + A175992^2 - A175992^3 + A175992^4 - ... Then the Mobius function is found in the first column. Compare this to the binomial series for (1+x)^-1 = 1 - x + x^2 - x^3 + x^4 - ... . - Mats Granvik, Gary W. Adamson, Dec 06 2010
From Richard L. Ollerton, May 08 2021: (Start)
Formulas for the numerous OEIS entries involving the Möbius transform (Dirichlet convolution of a(n) and some sequence h(n)) can be derived using the following (n >= 1):
Sum_{d|n} mu(d)*h(n/d) = Sum_{k=1..n} h(gcd(n,k))*mu(n/gcd(n,k))/phi(n/gcd(n,k)) = Sum_{k=1..n} h(n/gcd(n,k))*mu(gcd(n,k))/phi(n/gcd(n,k)), where phi = A000010.
Use of gcd(n,k)*lcm(n,k) = n*k provides further variations. (End)
Formulas for products corresponding to the sums above are also available for sequences f(n) > 0: Product_{d|n} f(n/d)^mu(d) = Product_{k=1..n} f(gcd(n,k))^(mu(n/gcd(n,k))/phi(n/gcd(n,k))) = Product_{k=1..n} f(n/gcd(n,k))^(mu(gcd(n,k))/phi(n/gcd(n,k))). - Richard L. Ollerton, Nov 08 2021

			G.f. = x - x^2 - x^3 - x^5 + x^6 - x^7 + x^10 - x^11 - x^13 + x^14 + x^15 + ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 24.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 161, #16.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, pp. 64-65.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 262 and 287.
Clifford A. Pickover, "The Math Book, from Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics", Sterling Publishing, 2009, p. 226. - Gary W. Adamson, Aug 13 2009
G. Pólya and G. Szegő, Problems and Theorems in Analysis Volume II. Springer_Verlag 1976.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 98-99.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from N. J. A. Sloane)
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 826.
All Angles, What is the Moebius function?, video, 2024.
Joerg Arndt, Matters Computational (The Fxtbook), pp. 705-707.
Yu Hin (Gary) Au, Decompositions of Unit Hypercubes and the Reversion of a Generalized Möbius Series, arXiv:2205.03680 [math.CO], 2022.
Olivier Bordellès, Some Explicit Estimates for the Mobius Function , J. Int. Seq. 18 (2015) 15.11.1
G. J. Chaitin, Thoughts on the Riemann hypothesis arXiv:math/0306042 [math.HO], 2003.
Michael Coons and Peter Borwein, Transcendence of Power Series for Some Number Theoretic Functions, arXiv:0806.1563 [math.NT], 2008.
Marc Deléglise and Joël Rivat, Computing the summation of the Mobius function, Experiment. Math. 5:4 (1996), pp. 291-295.
Tom Edgar, Posets and Möbius Inversion, slides, (2008).
Mats Granvik, Inverse of a triangular matrix using determinants, Inverse of a triangular matrix using matrix multiplication, Inverse of a triangular matrix as a binomial series, The ordinary generating function for the Mobius function.
Keith Matthews, Factorizing n and calculating phi(n),omega(n),d(n),sigma(n) and mu(n).
A. F. Möbius, Über eine besondere Art von Umkehrung der Reihen. Journal für die reine und angewandte Mathematik 9 (1832), 105-123.
Ed Pegg Jr., The Mobius function (and squarefree numbers).
Anders Björner and Richard P. Stanley, A combinatorial miscellany.
Paul Tarau, Emulating Primality with Multiset Representations of Natural Numbers, in Theoretical Aspects Of Computing, ICTAC 2011, Lecture Notes in Computer Science, 2011, Volume 6916/2011, 218-238.
Paul Tarau, Towards a generic view of primality through multiset decompositions of natural numbers, Theoretical Computer Science, Volume 537, Jun 05 2014, pp. 105-124.
Gerard Villemin's Almanac of Numbers, Nombres de Moebius et de Mertens.
Eric Weisstein's World of Mathematics, Moebius Function.
Eric Weisstein's World of Mathematics, Redheffer Matrix.
Wikipedia, Moebius function.
Index entries for "core" sequences
Index entries for sequences computed from exponents in factorization of n

Variants of a(n) are A178536, A181434, A181435.
Cf. A000010, A001221, A008966, A007423, A080847, A002321 (partial sums), A069158, A055615, A129360, A140579, A140664, A140254, A143104, A152902, A206706, A063524, A007427, A007428, A124010, A073776, A074206, A132971, A156552.
Cf. A059956 (Dgf at s=2), A088453 (Dgf at s=3), A215267 (Dgf at s=4), A343308 (Dgf at s=5).

Axiom
```
[moebiusMu(n) for n in 1..100]
```

import Math.NumberTheory.Primes.Factorisation (factorise)
a008683 = mu . snd . unzip . factorise where
mu [] = 1; mu (1:es) = - mu es; mu (_:es) = 0
-- Reinhard Zumkeller, Dec 13 2015, Oct 09 2013

a008683 1 = 1
a008683 n = - sum [a008683 d | d <- [1..(n-1)], n `mod` d == 0]
-- Harry Richman, Jun 13 2025

Magma
```
[ MoebiusMu(n) : n in [1..100]];
```

with(numtheory): A008683 := n->mobius(n);
with(numtheory): [ seq(mobius(n), n=1..100) ];
# Note that older versions of Maple define mobius(0) to be -1.
# This is unwise! Moebius(0) is better left undefined.
with(numtheory):
mu:= proc(n::posint) option remember; `if`(n=1, 1,
       -add(mu(d), d=divisors(n) minus {n}))
     end:
seq(mu(n), n=1..100);  # Alois P. Heinz, Aug 13 2008

Array[ MoebiusMu, 100]
(* Second program: *)
m = 100; A[_] = 0;
Do[A[x_] = x - Sum[A[x^k], {k, 2, m}] + O[x]^m // Normal, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Oct 20 2019, after Ilya Gutkovskiy *)

A008683(n):=moebius(n)$ makelist(A008683(n),n,1,30); /* Martin Ettl, Oct 24 2012 */

PARI
```
a=n->if(n<1,0,moebius(n));
```

{a(n) = if( n<1, 0, direuler( p=2, n, 1 - X)[n])};

list(n)=my(v=vector(n,i,1)); forprime(p=2, sqrtint(n), forstep(i=p, n, p, v[i]*=-1); forstep(i=p^2, n, p^2, v[i]=0)); forprime(p=sqrtint(n)+1, n, forstep(i=p, n, p, v[i]*=-1)); v \\ Charles R Greathouse IV, Apr 27 2012

from sympy import mobius
print([mobius(i) for i in range(1, 101)])  # Indranil Ghosh, Mar 18 2017

@cached_function
def mu(n):
    if n < 2: return n
    return -sum(mu(d) for d in divisors(n)[:-1])
# Changing the sign of the sum gives the number of ordered factorizations of n A074206.
print([mu(n) for n in (1..96)])  # Peter Luschny, Dec 26 2016

Sum_{d|n} mu(d) = 1 if n = 1 else 0.
Dirichlet generating function: Sum_{n >= 1} mu(n)/n^s = 1/zeta(s). Also Sum_{n >= 1} mu(n)*x^n/(1-x^n) = x.
In particular, Sum_{n > 0} mu(n)/n = 0. - Franklin T. Adams-Watters, Jun 20 2014
phi(n) = Sum_{d|n} mu(d)*n/d.
a(n) = A091219(A091202(n)).
Multiplicative with a(p^e) = -1 if e = 1; 0 if e > 1. - David W. Wilson, Aug 01 2001
abs(a(n)) = Sum_{d|n} 2^A001221(d)*a(n/d). - Benoit Cloitre, Apr 05 2002
Sum_{d|n} (-1)^(n/d)*mobius(d) = 0 for n > 2. - Emeric Deutsch, Jan 28 2005
a(n) = (-1)^omega(n) * 0^(bigomega(n) - omega(n)) for n > 0, where bigomega(n) and omega(n) are the numbers of prime factors of n with and without repetition (A001222, A001221, A046660). - Reinhard Zumkeller, Apr 05 2003
Dirichlet generating function for the absolute value: zeta(s)/zeta(2s). - Franklin T. Adams-Watters, Sep 11 2005
mu(n) = A129360(n) * (1, -1, 0, 0, 0, ...). - Gary W. Adamson, Apr 17 2007
mu(n) = -Sum_{d < n, d|n} mu(d) if n > 1 and mu(1) = 1. - Alois P. Heinz, Aug 13 2008
a(n) = A174725(n) - A174726(n). - Mats Granvik, Mar 28 2010
a(n) = first column in the matrix inverse of a triangular table with the definition: T(1, 1) = 1, n > 1: T(n, 1) is any number or sequence, k = 2: T(n, 2) = T(n, k-1) - T(n-1, k), k > 2 and n >= k: T(n,k) = (Sum_{i = 1..k-1} T(n-i, k-1)) - (Sum_{i = 1..k-1} T(n-i, k)). - Mats Granvik, Jun 12 2010
Product_{n >= 1} (1-x^n)^(-a(n)/n) = exp(x) (product form of the exponential function). - Joerg Arndt, May 13 2011
a(n) = Sum_{k=1..n, gcd(k,n)=1} exp(2*Pi*i*k/n), the sum over the primitive n-th roots of unity. See the Apostol reference, p. 48, Exercise 14 (b). - Wolfdieter Lang, Jun 13 2011
mu(n) = Sum_{k=1..n} A191898(n,k)*exp(-i*2*Pi*k/n)/n. (conjecture). - Mats Granvik, Nov 20 2011
Sum_{k=1..n} a(k)*floor(n/k) = 1 for n >= 1. - Peter Luschny, Feb 10 2012
a(n) = floor(omega(n)/bigomega(n))*(-1)^omega(n) = floor(A001221(n)/A001222(n))*(-1)^A001221(n). - Enrique Pérez Herrero, Apr 27 2012
Multiplicative with a(p^e) = binomial(1, e) * (-1)^e. - Enrique Pérez Herrero, Jan 19 2013
G.f. A(x) satisfies: x^2/A(x) = Sum_{n>=1} A( x^(2*n)/A(x)^n ). - Paul D. Hanna, Apr 19 2016
a(n) = -A008966(n)*A008836(n)/(-1)^A005361(n) = -floor(rad(n)/n)Lambda(n)/(-1)^tau(n/rad(n)). - Anthony Browne, May 17 2016
a(n) = Kronecker delta of A001221(n) and A001222(n) (which is A008966) multiplied by A008836(n). - Eric Desbiaux, Mar 15 2017
a(n) = A132971(A156552(n)). - Antti Karttunen, May 30 2017
Conjecture: a(n) = Sum_{k>=0} (-1)^(k-1)*binomial(A001222(n)-1, k)*binomial(A001221(n)-1+k, k), for n > 1. Verified for the first 100000 terms. - Mats Granvik, Sep 08 2018
From Peter Bala, Mar 15 2019: (Start)
Sum_{n >= 1} mu(n)*x^n/(1 + x^n) = x - 2*x^2. See, for example, Pólya and Szegő, Part V111, Chap. 1, No. 71.
Sum_{n >= 1} (-1)^(n+1)*mu(n)*x^n/(1 - x^n) = x + 2*(x^2 + x^4 + x^8 + x^16 + ...).
Sum_{n >= 1} (-1)^(n+1)*mu(n)*x^n/(1 + x^n) = x - 2*(x^4 + x^8 + x^16 + x^32 + ...).
Sum_{n >= 1} |mu(n)|*x^n/(1 - x^n) = Sum_{n >= 1} (2^w(n))*x^n, where w(n) is the number of different prime factors of n (Hardy and Wright, Chapter XVI, Theorem 264).
Sum_{n odd} |mu(n)|*x^n/(1 + x^(2*n)) = Sum_{n in S_1} (2^w_1(n))*x^n, where S_1 = {1, 5, 13, 17, 25, 29, ...} is the multiplicative semigroup of positive integers generated by 1 and the primes p = 1 (mod 4), and w_1(n) is the number of different prime factors p = 1 (mod 4) of n.
Sum_{n odd} (-1)^((n-1)/2)*mu(n)*x^n/(1 - x^(2*n)) = Sum_{n in S_3} (2^w_3(n))*x^n, where S_3 = {1, 3, 7, 9, 11, 19, 21, ...} is the multiplicative semigroup of positive integers generated by 1 and the primes p = 3 (mod 4), and where w_3(n) is the number of different prime factors p = 3 (mod 4) of n. (End)
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} A(x^k). - Ilya Gutkovskiy, May 11 2019
a(n) = sign(A023900(n)) * [A007947(n) = n] where [] is the Iverson bracket. - I. V. Serov, May 15 2019
a(n) = Sum_{k = 1..n} gcd(k, n)*a(gcd(k, n)) = Sum_{d divides n} a(d)*d*phi(n/d). - Peter Bala, Jan 16 2024

A003961 Completely multiplicative with a(prime(k)) = prime(k+1).

Original entry on oeis.org

1, 3, 5, 9, 7, 15, 11, 27, 25, 21, 13, 45, 17, 33, 35, 81, 19, 75, 23, 63, 55, 39, 29, 135, 49, 51, 125, 99, 31, 105, 37, 243, 65, 57, 77, 225, 41, 69, 85, 189, 43, 165, 47, 117, 175, 87, 53, 405, 121, 147, 95, 153, 59, 375, 91, 297, 115, 93, 61, 315, 67, 111, 275, 729, 119

Offset: 1

Marc LeBrun

Meyers (see Guy reference) conjectures that for all r >= 1, the least odd number not in the set {a(i): i < prime(r)} is prime(r+1). - N. J. A. Sloane, Jan 08 2021
Meyers' conjecture would be refuted if and only if for some r there were such a large gap between prime(r) and prime(r+1) that there existed a composite c for which prime(r) < c < a(c) < prime(r+1), in which case (by Bertrand's postulate) c would necessarily be a term of A246281. - Antti Karttunen, Mar 29 2021
a(n) is odd for all n and for each odd m there exists a k with a(k) = m (see A064216). a(n) > n for n > 1: bijection between the odd and all numbers. - Reinhard Zumkeller, Sep 26 2001
a(n) and n have the same number of distinct primes with (A001222) and without multiplicity (A001221). - Michel Marcus, Jun 13 2014
From Antti Karttunen, Nov 01 2019: (Start)
More generally, a(n) has the same prime signature as n, A046523(a(n)) = A046523(n). Also A246277(a(n)) = A246277(n) and A287170(a(n)) = A287170(n).
Many permutations and other sequences that employ prime factorization of n to encode either polynomials, partitions (via Heinz numbers) or multisets in general can be easily defined by using this sequence as one of their constituent functions. See the last line in the Crossrefs section for examples.
(End)

			a(12) = a(2^2 * 3) = a(prime(1)^2 * prime(2)) = prime(2)^2 * prime(3) = 3^2 * 5 = 45.
a(A002110(n)) = A002110(n + 1) / 2.

Richard K. Guy, editor, Problems From Western Number Theory Conferences, Labor Day, 1983, Problem 367 (Proposed by Leroy F. Meyers, The Ohio State U.).

Indranil Ghosh, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Index entries for sequences computed from indices in prime factorization.
Index entries for sequences related to Heinz numbers.

See A045965 for another version.
Row 1 of table A242378 (which gives the "k-th powers" of this sequence), row 3 of A297845 and of A306697. See also arrays A066117, A246278, A255483, A308503, A329050.
Cf. A064989 (a left inverse), A064216, A000040, A002110, A000265, A027746, A046523, A048673 (= (a(n)+1)/2), A108228 (= (a(n)-1)/2), A191002 (= a(n)*n), A252748 (= a(n)-2n), A286385 (= a(n)-sigma(n)), A283980 (= a(n)*A006519(n)), A341529 (= a(n)*sigma(n)), A326042, A049084, A001221, A001222, A122111, A225546, A260443, A245606, A244319, A246269 (= A065338(a(n))), A322361 (= gcd(n, a(n))), A305293.
Cf. A191555, A252738.
Cf. A249734, A249735 (bisections).
Cf. A246261 (a(n) is of the form 4k+1), A246263 (of the form 4k+3), A246271, A246272, A246259, A246281 (n such that a(n) < 2n), A246282 (n such that a(n) > 2n), A252742.
Cf. A275717 (a(n) > a(n-1)), A275718 (a(n) < a(n-1)).
Cf. A003972 (Möbius transform), A003973 (Inverse Möbius transform), A318321.
Cf. A300841, A305421, A322991, A250469, A269379 for analogous shift-operators in other factorization and quasi-factorization systems.
Cf. also following permutations and other sequences that can be defined with the help of this sequence: A005940, A163511, A122111, A260443, A206296, A265408, A265750, A275733, A275735, A297845, A091202 & A091203, A250245 & A250246, A302023 & A302024, A302025 & A302026.
A version for partition numbers is A003964, strict A357853.
A permutation of A005408.
Applying the same transformation again gives A357852.
Other multiplicative sequences: A064988, A357977, A357978, A357980, A357983.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000720, A076610, A296150.

a003961 1 = 1
a003961 n = product $ map (a000040 . (+ 1) . a049084) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012, Oct 09 2011
(MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)
(require 'factor)
(define (A003961 n) (apply * (map A000040 (map 1+ (map A049084 (factor n))))))
;; Antti Karttunen, May 20 2014

a:= n-> mul(nextprime(i[1])^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 13 2017

a[p_?PrimeQ] := a[p] = Prime[ PrimePi[p] + 1]; a[1] = 1; a[n_] := a[n] = Times @@ (a[#1]^#2& @@@ FactorInteger[n]); Table[a[n], {n, 1, 65}] (* Jean-François Alcover, Dec 01 2011, updated Sep 20 2019 *)
Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[n == 1], {n, 65}] (* Michael De Vlieger, Mar 24 2017 *)

a(n)=local(f); if(n<1,0,f=factor(n); prod(k=1,matsize(f)[1],nextprime(1+f[k,1])^f[k,2]))

a(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Michel Marcus, May 17 2014

use ntheory ":all";  sub a003961 { vecprod(map { next_prime($) } factor(shift)); }  # _Dana Jacobsen, Mar 06 2016

from sympy import factorint, prime, primepi, prod
def a(n):
    f=factorint(n)
    return 1 if n==1 else prod(prime(primepi(i) + 1)**f[i] for i in f)
[a(n) for n in range(1, 11)] # Indranil Ghosh, May 13 2017

If n = Product p(k)^e(k) then a(n) = Product p(k+1)^e(k).
Multiplicative with a(p^e) = A000040(A000720(p)+1)^e. - David W. Wilson, Aug 01 2001
a(n) = Product_{k=1..A001221(n)} A000040(A049084(A027748(n,k))+1)^A124010(n,k). - Reinhard Zumkeller, Oct 09 2011 [Corrected by Peter Munn, Nov 11 2019]
A064989(a(n)) = n and a(A064989(n)) = A000265(n). - Antti Karttunen, May 20 2014 & Nov 01 2019
A001221(a(n)) = A001221(n) and A001222(a(n)) = A001222(n). - Michel Marcus, Jun 13 2014
From Peter Munn, Oct 31 2019: (Start)
a(n) = A225546((A225546(n))^2).
a(A225546(n)) = A225546(n^2).
(End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-nextprime(p))) = 2.06399637... . - Amiram Eldar, Nov 18 2022

A049084 a(n) = pi(n) if n is prime, otherwise 0.

Original entry on oeis.org

0, 1, 2, 0, 3, 0, 4, 0, 0, 0, 5, 0, 6, 0, 0, 0, 7, 0, 8, 0, 0, 0, 9, 0, 0, 0, 0, 0, 10, 0, 11, 0, 0, 0, 0, 0, 12, 0, 0, 0, 13, 0, 14, 0, 0, 0, 15, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 17, 0, 18, 0, 0, 0, 0, 0, 19, 0, 0, 0, 20, 0, 21, 0, 0, 0, 0, 0, 22, 0, 0, 0, 23, 0, 0, 0, 0, 0, 24, 0, 0, 0

Offset: 1

N. J. A. Sloane

pi(n) is the prime counting function, A000720.

Equals row sums of triangle A143541. - Gary W. Adamson, Aug 23 2008

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

a(n) = A091227(A091202(n)).

Cf. A143541.

import Data.List (unfoldr)
a049084 n = a049084_list !! (fromInteger n - 1)
a049084_list = unfoldr x (1, 1, a000040_list) where
   x (i, z, ps'@(p:ps)) | i == p = Just (z, (i + 1, z + 1, ps))
                        | i /= p = Just (0, (i + 1, z, ps'))
-- Reinhard Zumkeller, Apr 17 2012, Mar 31 2012, Sep 15 2011

A049084 := proc(n)
    local i;
    if isprime(n) then
        for i from 1 do
            if ithprime(i) = n then
                return i;
            end if;
        end do;
    else
        return 0 ;
    fi;
end proc:
seq(A049084(n),n=1..120) ;

Table[PrimePi[n] * Boole[PrimeQ[n]], {n, 92}] (* Jean-François Alcover, Nov 07 2011, after R. J. Mathar *)
Table[If[PrimeQ[n],PrimePi[n],0],{n,100}] (* Harvey P. Dale, Jan 09 2022 *)

a(n)=if(isprime(n),primepi(n),0) \\ Charles R Greathouse IV, Jan 08 2013

a(n) = pi(n)*(pi(n) - pi(n-1)), pi = A000720. - Reinhard Zumkeller, Nov 30 2003
a(n) = A000720(n*A010051(n)). - Labos Elemer, Jan 09 2004
a(n) = A000720(n)*A010051(n). - R. J. Mathar, Mar 01 2011

Name clarified by Alonso del Arte, Feb 07 2020 at the suggestion of David A. Corneth

A048720 Multiplication table {0..i} X {0..j} of binary polynomials (polynomials over GF(2)) interpreted as binary vectors, then written in base 10; or, binary multiplication without carries.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 4, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 5, 8, 5, 0, 0, 6, 10, 12, 12, 10, 6, 0, 0, 7, 12, 15, 16, 15, 12, 7, 0, 0, 8, 14, 10, 20, 20, 10, 14, 8, 0, 0, 9, 16, 9, 24, 17, 24, 9, 16, 9, 0, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 0, 11, 20, 27, 32, 27, 20, 27, 32, 27, 20, 11, 0

Offset: 0

Antti Karttunen, Apr 26 1999

Essentially same as A091257 but computed starting from offset 0 instead of 1.
Each polynomial in GF(2)[X] is encoded as the number whose binary representation is given by the coefficients of the polynomial, e.g., 13 = 2^3 + 2^2 + 2^0 = 1101_2 encodes 1*X^3 + 1*X^2 + 0*X^1 + 1*X^0 = X^3 + X^2 + X^0. - Antti Karttunen and Peter Munn, Jan 22 2021
To listen to this sequence, I find instrument 99 (crystal) works well with the other parameters defaulted. - Peter Munn, Nov 01 2022

			Top left corner of array:
  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ...
  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
  0  2  4  6  8 10 12 14 16 18 20 22 24 26 28 30 ...
  0  3  6  5 12 15 10  9 24 27 30 29 20 23 18 17 ...
  ...
From _Antti Karttunen_ and _Peter Munn_, Jan 23 2021: (Start)
Multiplying 10 (= 1010_2) and 11 (= 1011_2), in binary results in:
     1011
  *  1010
  -------
   c1011
  1011
  -------
  1101110  (110 in decimal),
and we see that there is a carry-bit (marked c) affecting the result.
In carryless binary multiplication, the second part of the process (in which the intermediate results are summed) looks like this:
    1011
  1011
  -------
  1001110  (78 in decimal).
(End)

Alois P. Heinz, Antidiagonals n = 0..200, flattened
Antti Karttunen, Scheme-program for computing this sequence.
N. J. A. Sloane, Transforms: Maple implementation of binary eXclusive OR (XORnos).
Index entries for sequences related to carryless arithmetic
Index entries for sequences related to polynomials in ring GF(2)[X]

Cf. A051776 (Nim-product), A091257 (subtable).
Carryless multiplication in other bases: A325820 (3), A059692 (10).
Ordinary {0..i} * {0..j} multiplication table: A004247 and its differences from this: A061858 (which lists further sequences related to presence/absence of carry in binary multiplication).
Carryless product of the prime factors of n: A234741.
Binary irreducible polynomials ("X-primes"): A014580, factorization table: A256170, table of "X-powers": A048723, powers of 3: A001317, rearranged subtable with distinct terms (comparable to A054582): A277820.
See A014580 for further sequences related to the difference between factorization into GF(2)[X] irreducibles and ordinary prime factorization of the integer encoding.
Row/column 3: A048724 (even bisection of A003188), 5: A048725, 6: A048726, 7: A048727; main diagonal: A000695.
Associated additive operation: A003987.
Equivalent sequences, as compared with standard integer multiplication: A048631 (factorials), A091242 (composites), A091255 (gcd), A091256 (lcm), A280500 (division).
See A091202 (and its variants) and A278233 for maps from/to ordinary multiplication.
See A115871, A115872 and A277320 for tables related to cross-domain congruences.

trinv := n -> floor((1+sqrt(1+8*n))/2); # Gives integral inverses of the triangular numbers
# Binary multiplication of nn and mm, but without carries (use XOR instead of ADD):
Xmult := proc(nn,mm) local n,m,s; n := nn; m := mm; s := 0; while (n > 0) do if(1 = (n mod 2)) then s := XORnos(s,m); fi; n := floor(n/2); # Shift n right one bit. m := m*2; # Shift m left one bit. od; RETURN(s); end;

trinv[n_] := Floor[(1 + Sqrt[1 + 8*n])/2];
Xmult[nn_, mm_] := Module[{n = nn, m = mm, s = 0}, While[n > 0, If[1 == Mod[n, 2], s = BitXor[s, m]]; n = Floor[n/2]; m = m*2]; Return[s]];
a[n_] := Xmult[(trinv[n] - 1)*((1/2)*trinv[n] + 1) - n, n - (trinv[n]*(trinv[n] - 1))/2];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 16 2015, updated Mar 06 2016 after Maple *)

up_to = 104;
A048720sq(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A048720list(up_to) = { my(v = vector(1+up_to), i=0); for(a=0, oo, for(col=0, a, i++; if(i > up_to, return(v)); v[i] = A048720sq(col, a-col))); (v); };
v048720 = A048720list(up_to);
A048720(n) = v048720[1+n]; \\ Antti Karttunen, Feb 15 2021

a(n) = Xmult( (((trinv(n)-1)*(((1/2)*trinv(n))+1))-n), (n-((trinv(n)*(trinv(n)-1))/2)) );
T(2b, c)=T(c, 2b)=T(b, 2c)=2T(b, c); T(2b+1, c)=T(c, 2b+1)=2T(b, c) XOR c - Henry Bottomley, Mar 16 2001
For n >= 0, A003188(2n) = T(n, 3); A003188(2n+1) = T(n, 3) XOR 1, where XOR is the bitwise exclusive-or operator, A003987. - Peter Munn, Feb 11 2021

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A106444 Exponent-recursed cross-domain bijection from N to GF(2)[X]. Variant of A091202 and A106442.

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A234747 Self-inverse and multiplicative permutation of natural numbers, A091202-conjugate of Blue code: a(n) = A091203(A193231(A091202(n))).

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A000005 d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.

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A001222 Number of prime divisors of n counted with multiplicity (also called big omega of n, bigomega(n) or Omega(n)).

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A001221 Number of distinct primes dividing n (also called omega(n)).

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