A035116 -id:A035116 - cp's OEIS frontend

A349483 Length of cycle reached when iterating the mapping x-> n*A035116(x) on 1.

1, 2, 2, 2, 4, 2, 5, 2, 2, 7, 2, 1, 2, 5, 6, 1, 2, 2, 2, 3, 2, 2, 2, 1, 1, 2, 4, 1, 2, 1, 2, 3, 1, 2, 1, 1, 2, 2, 1, 3, 2, 4, 2, 1, 3, 2, 2, 4, 3, 6, 1, 1, 2, 2, 3, 3, 1, 2, 2, 4, 2, 2, 1, 3, 3, 3, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 3, 2, 8, 1, 2, 2, 3, 3, 2, 1, 3, 2, 3, 1, 1, 1, 2, 3, 1, 2, 4, 1, 2

Offset: 1

Tejo Vrush, Nov 19 2021

nonn

The terms 1-25 all appear below 10^8; the last of these are a(12545280) = 21, a(12684672) = 24, and a(96940800) = 25. - Charles R Greathouse IV, Nov 23 2021

			For n = 2, 1 --> 2 --> 8 --> 32 --> 72 --> 288 --> 648 --> 800 --> 648. The cycle reached has just two terms: 648 and 800. Therefore, a(2) = 2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A035116.

Similar sequences: A349410.

a[n_] := Module[{s = NestWhileList[n*DivisorSigma[0, #]^2 &, 1, UnsameQ, All]}, Differences[Position[s, s[[-1]]]][[1, 1]]]; Array[a, 100] (* Amiram Eldar, Nov 19 2021 *)

brent(f,x)=my(pow=1,lam=1,tortoise=x,hare=f(x)); while(tortoise!=hare, if(pow==lam, tortoise=hare; pow<<=1; lam=0); hare=f(hare); lam++); lam
a(n)=brent(k->n*numdiv(k)^2,1) \\ Charles R Greathouse IV, Nov 19 2021

A048691 a(n) = d(n^2), where d(k) = A000005(k) is the number of divisors of k.

Original entry on oeis.org

1, 3, 3, 5, 3, 9, 3, 7, 5, 9, 3, 15, 3, 9, 9, 9, 3, 15, 3, 15, 9, 9, 3, 21, 5, 9, 7, 15, 3, 27, 3, 11, 9, 9, 9, 25, 3, 9, 9, 21, 3, 27, 3, 15, 15, 9, 3, 27, 5, 15, 9, 15, 3, 21, 9, 21, 9, 9, 3, 45, 3, 9, 15, 13, 9, 27, 3, 15, 9, 27, 3, 35, 3, 9, 15, 15, 9, 27, 3, 27

Offset: 1

David Johnson-Davies

Inverse Moebius transform of A034444: Sum_{d|n} 2^omega(d), where omega(n) = A001221(n) is the number of distinct primes dividing n.
Number of elements in the set {(x,y): x|n, y|n, gcd(x,y)=1}.
Number of elements in the set {(x,y): lcm(x,y)=n}.
Also gives total number of positive integral solutions (x,y), order being taken into account, to the optical or parallel resistor equation 1/x + 1/y = 1/n. Indeed, writing the latter as X*Y=N, with X=x-n, Y=y-n, N=n^2, the one-to-one correspondence between solutions (X, Y) and (x, y) is obvious, so that clearly, the solution pairs (x, y) are tau(N)=tau(n^2) in number. - Lekraj Beedassy, May 31 2002
Number of ordered pairs of positive integers (a,c) such that n^2 - ac = 0. Therefore number of quadratic equations of the form ax^2 + 2nx + c = 0 where a,n,c are positive integers and each equation has two equal (rational) roots, -n/a. (If a and c are positive integers, but, instead, the coefficient of x is odd, it is impossible for the equation to have equal roots.) - Rick L. Shepherd, Jun 19 2005
Problem A1 on the 21st Putnam competition in 1960 (see John Scholes link) asked for the number of pairs of positive integers (x,y) such that xy/(x+y) = n: the answer is a(n); for n = 4, the a(4) = 5 solutions (x,y) are (5,20), (6,12), (8,8), (12,6), (20,5). - Bernard Schott, Feb 12 2023
Numbers k such that a(k)/d(k) is an integer are in A217584 and the corresponding quotients are in A339055. - Bernard Schott, Feb 15 2023

A. M. Gleason et al., The William Lowell Putnam Mathematical Competitions, Problems & Solutions:1938-1960 Soln. to Prob. 1 1960, p. 516, MAA, 1980.
Ross Honsberger, More Mathematical Morsels, Morsel 43, pp. 232-3, DMA No. 10 MAA, 1991.
Loren C. Larson, Problem-Solving Through Problems, Prob. 3.3.7, p. 102, Springer 1983.
Alfred S. Posamentier and Charles T. Salkind, Challenging Problems in Algebra, Prob. 9-9 pp. 143 Dover NY, 1988.
D. O. Shklarsky et al., The USSR Olympiad Problem Book, Soln. to Prob. 123, pp. 28, 217-8, Dover NY.
Wacław Sierpiński, Elementary Theory of Numbers, pp. 71-2, Elsevier, North Holland, 1988.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 91.
Charles W. Trigg, Mathematical Quickies, Question 194, pp. 53, 168, Dover, 1985.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000
Ovidiu Bagdasar, On Some Functions Involving the lcm and gcd of Integer Tuples.
Umberto Cerruti, Percorsi tra i numeri (in Italian), pages 3-4.
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seq., Vol. 6 (2003), Article 03.1.6.
John Scholes, Problem A1 of 21st Putnam Competition 1960, 2002. [Wayback Machine link]
Wacław Sierpiński, Elementary Theory of Numbers, Warszawa 1964.
Index to sequences related to Olympiads and other Mathematical competitions.

Cf. A000005, A034444, Mobius transf. of A035116, A048785, A061391, A018805, A002088, A015614, A018892, A063647, A182139.
Partial sums give A061503.
For similar LCM sequences, see A070919, A070920, A070921.
Cf. A005408, A124010.
For the earliest occurrence of 2n-1 see A016017.
Cf. A001620, A082633, A217584, A339055.

a048691 = product . map (a005408 . fromIntegral) . a124010_row
-- Reinhard Zumkeller, Jul 12 2012

A048691[n_]:=DivisorSigma[0,n^2] (* Enrique Pérez Herrero, May 30 2010 *)
DivisorSigma[0,Range[80]^2] (* Harvey P. Dale, Apr 08 2015 *)

numlib::tau (n^2)$ n=1..90 // Zerinvary Lajos, May 13 2008

A048691(n)=prod(i=1,#n=factor(n)[,2],n[i]*2+1) /* or, of course, a(n)=numdiv(n^2) */ \\ M. F. Hasler, Dec 30 2007

for(n=1, 100, print1(direuler(p=2, n, (1 - X^2)/(1 - X)^3)[n], ", ")) \\ Vaclav Kotesovec, Aug 21 2021

[sigma(n**2, 0) for n in range(1, 81)] # Stefano Spezia, Jul 14 2025

a(n) = A000005(A000290(n)).
tau(n^2) = Sum_{d|n} mu(n/d)*tau(d)^2, where mu(n) = A008683(n), cf. A061391.
Multiplicative with a(p^e) = 2e+1. - Vladeta Jovovic, Jul 23 2001
Also a(n) = Sum_{d|n} (tau(d)*moebius(n/d)^2), Dirichlet convolution of A000005 and A008966. - Benoit Cloitre, Sep 08 2002
a(n) = A055205(n) + A000005(n). - Reinhard Zumkeller, Dec 08 2009
Dirichlet g.f.: (zeta(s))^3/zeta(2s). - R. J. Mathar, Feb 11 2011
a(n) = Sum_{d|n} 2^omega(d). Inverse Mobius transform of A034444. - Enrique Pérez Herrero, Apr 14 2012
G.f.: Sum_{k>=1} 2^omega(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 10 2018
Sum_{k=1..n} a(k) ~ n*(6/Pi^2)*(log(n)^2/2 + log(n)*(3*gamma - 1) + 1 - 3*gamma + 3*gamma^2 - 3*gamma_1 + (2 - 6*gamma - 2*log(n))*zeta'(2)/zeta(2) + (2*zeta'(2)/zeta(2))^2 - 2*zeta''(2)/zeta(2)), where gamma is Euler's constant (A001620) and gamma_1 is the first Stieltjes constant (A082633). - Amiram Eldar, Jan 26 2023

Additional comments from Vladeta Jovovic, Apr 29 2001

A062319 Number of divisors of n^n, or of A000312(n).

Original entry on oeis.org

1, 1, 3, 4, 9, 6, 49, 8, 25, 19, 121, 12, 325, 14, 225, 256, 65, 18, 703, 20, 861, 484, 529, 24, 1825, 51, 729, 82, 1653, 30, 29791, 32, 161, 1156, 1225, 1296, 5329, 38, 1521, 1600, 4961, 42, 79507, 44, 4005, 4186, 2209, 48, 9457, 99, 5151, 2704, 5565, 54

Offset: 0

Jason Earls, Jul 05 2001

From Gus Wiseman, May 02 2021: (Start)
Conjecture: The number of divisors of n^n equals the number of pairwise coprime ordered n-tuples of divisors of n. Confirmed up to n = 30. For example, the a(1) = 1 through a(5) = 6 tuples are:
  (1)  (1,1)  (1,1,1)  (1,1,1,1)  (1,1,1,1,1)
       (1,2)  (1,1,3)  (1,1,1,2)  (1,1,1,1,5)
       (2,1)  (1,3,1)  (1,1,1,4)  (1,1,1,5,1)
              (3,1,1)  (1,1,2,1)  (1,1,5,1,1)
                       (1,1,4,1)  (1,5,1,1,1)
                       (1,2,1,1)  (5,1,1,1,1)
                       (1,4,1,1)
                       (2,1,1,1)
                       (4,1,1,1)
The unordered case (pairwise coprime n-multisets of divisors of n) is counted by A343654.
(End)

			From _Gus Wiseman_, May 02 2021: (Start)
The a(1) = 1 through a(5) = 6 divisors:
  1  1  1   1    1
     2  3   2    5
     4  9   4    25
        27  8    125
            16   625
            32   3125
            64
            128
            256
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Harry J. Smith)

Cf. A046798, A077592, A035116. - Enrique Pérez Herrero, Nov 09 2010
Number of divisors of A000312(n).
Taking Omega instead of sigma gives A066959.
Positions of squares are A173339.
Diagonal n = k of the array A343656.
A000005 counts divisors.
A059481 counts k-multisets of elements of {1..n}.
A334997 counts length-k strict chains of divisors of n.
A343658 counts k-multisets of divisors.
Pairwise coprimality:
- A018892 counts coprime pairs of divisors.
- A084422 counts pairwise coprime subsets of {1..n}.
- A100565 counts pairwise coprime triples of divisors.
- A225520 counts pairwise coprime sets of divisors.
- A343652 counts maximal pairwise coprime sets of divisors.
- A343653 counts pairwise coprime non-singleton sets of divisors > 1.
- A343654 counts pairwise coprime sets of divisors > 1.
Cf. A000169, A000272, A002064, A002109, A009998, A048691, A143773, A146291, A176029, A327527, A343657.

[NumberOfDivisors(n^n): n in  [0..60]]; // Vincenzo Librandi, Nov 09 2014

A062319[n_IntegerQ]:=DivisorSigma[0,n^n]; (* Enrique Pérez Herrero, Nov 09 2010 *)
Join[{1},DivisorSigma[0,#^#]&/@Range[60]] (* Harvey P. Dale, Jun 06 2024 *)

je=[]; for(n=0,200,je=concat(je,numdiv(n^n))); je

{ for (n=0, 1000, write("b062319.txt", n, " ", numdiv(n^n)); ) } \\ Harry J. Smith, Aug 04 2009

a(n)=local(fm);fm=factor(n);prod(k=1,matsize(fm)[1],fm[k,2]*n+1) \\ Franklin T. Adams-Watters, May 03 2011

a(n) = if(n==0, 1, sumdiv(n, d, n^omega(d))); \\ Seiichi Manyama, May 12 2021

from math import prod
from sympy import factorint
def A062319(n): return prod(n*d+1 for d in factorint(n).values()) # Chai Wah Wu, Jun 03 2021

a(n) = A000005(A000312(n)). - Enrique Pérez Herrero, Nov 09 2010
a(2^n) = A002064(n). - Gus Wiseman, May 02 2021
a(prime(n)) = prime(n) + 1. - Gus Wiseman, May 02 2021
a(n) = Product_{i=1..s} (1 + n * m_i) where (m_1,...,m_s) is the sequence of prime multiplicities (prime signature) of n. - Gus Wiseman, May 02 2021
a(n) = Sum_{d|n} n^omega(d) for n > 0. - Seiichi Manyama May 12 2021

A184389 a(n) = Sum_{k=1..tau(n)} k, where tau is the number of divisors of n (A000005).

Original entry on oeis.org

1, 3, 3, 6, 3, 10, 3, 10, 6, 10, 3, 21, 3, 10, 10, 15, 3, 21, 3, 21, 10, 10, 3, 36, 6, 10, 10, 21, 3, 36, 3, 21, 10, 10, 10, 45, 3, 10, 10, 36, 3, 36, 3, 21, 21, 10, 3, 55, 6, 21, 10, 21, 3, 36, 10, 36, 10, 10, 3, 78, 3, 10, 21, 28, 10, 36, 3, 21, 10, 36, 3, 78

Offset: 1

Jaroslav Krizek, Jan 12 2011

nonn

Length of row n in triangle A187207. - Omar E. Pol, Aug 07 2011

Number of pairs of even divisors of 2n, (d1,d2), such that d1<=d2. - Wesley Ivan Hurt, Aug 24 2020

			For n = 4; tau(4) = 3; a(4) = 1+2+3 = 6.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000

Cf. A000005 (tau), A000217 (triangular numbers).
Cf. A184387, A184388, A184390, A184391, A130674.
Cf. A035116, A066446.
Cf. A187207, A135539, A337362, A129308.

a184389 = a000217 . a000005'  -- Reinhard Zumkeller, Sep 08 2015

A184389 := proc(n)
    A000217(numtheory[tau](n)) ;
end proc: # R. J. Mathar, Oct 04 2014

((#+1)#)/2&/@DivisorSigma[0,Range[80]] (* Harvey P. Dale, Feb 27 2013 *)

a(n) = my(nd=numdiv(n)); nd*(nd+1)/2; \\ Michel Marcus, Jun 25 2016

a(n) = A000217(A000005(n)) = (1/2)*A000005(n)*(A000005(n)+1).
a(n) = A066446(n) + A000005(n) = A035116(n) - A066446(n). - Reinhard Zumkeller, Sep 08 2015
Dirichlet g.f.: zeta(s)^2*(zeta(s)^2 + zeta(2*s))/(2*zeta(2*s)). - Ilya Gutkovskiy, Jun 25 2016
a(n) = Sum_{d1|(2*n), d2|(2*n), d1 and d2 even, d1<=d2} 1. - Wesley Ivan Hurt, Aug 24 2020
a(n) = Sum_{d|n} A018892(d). - Daniel Suteu, Jan 08 2021
a(n) = Sum_{d|n} A135539(n,d). - Ridouane Oudra, May 29 2025
a(n) = A337362(n) + A129308(n). - Ridouane Oudra, May 30 2025

A061502 a(n) = Sum_{k<=n} tau(k)^2, where tau = number of divisors function A000005.

Original entry on oeis.org

1, 5, 9, 18, 22, 38, 42, 58, 67, 83, 87, 123, 127, 143, 159, 184, 188, 224, 228, 264, 280, 296, 300, 364, 373, 389, 405, 441, 445, 509, 513, 549, 565, 581, 597, 678, 682, 698, 714, 778, 782, 846, 850, 886, 922, 938, 942, 1042, 1051, 1087

Offset: 1

N. J. A. Sloane, Jun 14 2001

nonn

R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; Chapter II, Problem 56.

N. J. A. Sloane, Table of n, a(n) for n = 1..1024
Adrian Dudek, On the Success of Mishandling Euclid's Lemma, arXiv:1602.03555 [math.HO], 2016. See B(n) p. 2.
Chaohua Jia and Ayyadurai Sankaranarayanan, The mean square of the divisor function, Acta Arithmetica 164 (2014), 181-208.
Michaela Cully-Hugill and Timothy Trudgian, Two explicit divisor sums, arXiv:1911.07369 [math.NT], Nov 19 2019
Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)
Florian Luca and László Tóth, The r-th moment of the divisor function: an elementary approach, Journal of Integer Sequences 20 (2017), Article 17.7.4, 8 pp.
Adolf Piltz, Über das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst, 1881.
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, Formula (3).
D. Suryanarayana and R. Rama Chandra Rao, On an Asymptotic Formula of Ramanujan, Mathematica Scandinavica, 32, 258-264, 1973.
B. M. Wilson, Proofs of some formulas enunciated by Ramanujan, Proc. London Math. Soc. (2) 21 (1922) 235-255.

Cf. A000005, A035116, A061503, A318755.
Cf. A092742 (A), A245074 (B), A319090 (C), A319091 (D).
Cf. A057434, A072379, A074789.

[&+[NumberOfDivisors(k^2)*Floor(n/k): k in [1..n]]: n in [1..60]]; // Vincenzo Librandi, Sep 10 2016

Table[Sum[DivisorSigma[0, k^2]*Floor[n/k], {k, 1, n}], {n, 1, 50}] (* Vaclav Kotesovec, Aug 30 2018 *)
Accumulate[Table[DivisorSigma[0, n]^2, {n, 1, 50}]] (* Vaclav Kotesovec, Sep 10 2018 *)

for (n=1, 1024, write("b061502.txt", n, " ", sum(k=1, n, numdiv(k)^2)) ) \\ Harry J. Smith, Jul 23 2009

vector(60, n, sum(k=1, n, numdiv(k)^2)) \\ Michel Marcus, Jul 23 2015

first(n)=my(v=vector(n),s); forfactored(k=1,n, v[k[1]] = s += numdiv(k)^2); v; \\ Charles R Greathouse IV, Nov 28 2018

a(n) = Sum_{k=1..n} tau(k^2)*floor(n/k).
Asymptotic to A*n*log(n)^3 + B*n*log(n)^2 + C*n*log(n) + D*n + O(n^(1/2+eps)) where A = 1/Pi^2 and B = (12*gamma-3)/Pi^2 - 36*zeta'(2)/Pi^4. [corrected by Vaclav Kotesovec, Aug 30 2018]
C = 36*gamma^2/Pi^2 - (288*z1/Pi^4 + 24/Pi^2)*gamma + (864*z1^2/Pi^6 + 72*z1/Pi^4 - 72/Pi^4*z2 + 6/Pi^2) - 24*g1/Pi^2 and D = 24*gamma^3/Pi^2 - (432*z1 /Pi^4+ 36/Pi^2)*gamma^2 + (3456*z1^2/Pi^6 + 288*(z1-z2)/Pi^4 + 24/Pi^2 - 72*g1/Pi^2)*gamma + g1*(288*z1/Pi^4 + 24/Pi^2)-10368*z1^3/Pi^8 - 864*z1^2/Pi^6 + 1728*z2*z1/Pi^6 + 72*(z2-z1)/Pi^4- 48*z3/Pi^4 + (12*g2-6)/Pi^2, where gamma is the Euler-Mascheroni constant A001620, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994, z3 = Zeta'''(2) = A201995 and g1, g2 are the Stieltjes constants, see A082633 and A086279. - Vaclav Kotesovec, Sep 10 2018
See Cully-Hugill & Trudgian, Theorem 2, for an explicit version of the asymptotic given above. - Charles R Greathouse IV, Nov 19 2019

Definition corrected by N. J. A. Sloane, May 25 2008

A066446 Number of unordered divisor pairs of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 6, 3, 6, 1, 15, 1, 6, 6, 10, 1, 15, 1, 15, 6, 6, 1, 28, 3, 6, 6, 15, 1, 28, 1, 15, 6, 6, 6, 36, 1, 6, 6, 28, 1, 28, 1, 15, 15, 6, 1, 45, 3, 15, 6, 15, 1, 28, 6, 28, 6, 6, 1, 66, 1, 6, 15, 21, 6, 28, 1, 15, 6, 28, 1, 66, 1, 6, 15, 15, 6, 28, 1, 45, 10, 6, 1, 66, 6, 6, 6, 28

Offset: 1

Robert G. Wilson v, Dec 28 2001

			The divisors of 6 are 1, 2, 3 & 6. In unordered pairs they are {1, 2}, {1, 3}, {1, 6}, {2, 3}, {2, 6}, & {3, 6}. Since there are six pairs, a(6) = 6. Also d(6) = 4. 4*3/2 = 6.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (terms 1..1000 from Harry J. Smith)

Cf. A000005, A000217, A129510.

Cf. A035116, A184389, A063647.

a066446 = a000217 . subtract 1 . a000005'
-- Reinhard Zumkeller, Sep 08 2015

with(numtheory): seq(tau(n)*(tau(n)-1)/2, n=1..60); # Ridouane Oudra, Apr 15 2023

Table[ Binomial[ DivisorSigma[0, n], 2], {n, 1, 100}]

{ for (n=1, 1000, a=binomial(numdiv(n), 2); write("b066446.txt", n, " ", a) ) } \\ Harry J. Smith, Feb 15 2010

a(p) = 1 iff p is a prime.
Combinations of d(n), the number of divisors of n (A000005), taken two at a time. If the canonical factorization of n into prime powers is Product p^e(p) then d(n) = Product (e(p) + 1). Therefore a(n) = C(d(n), 2) = d(n)*{ d(n)-1 }/2 which is a triangular number (A000217).
a(n) = A184389(n) - A000005(n) = A035116(n) - A184389(n). - Reinhard Zumkeller, Sep 08 2015
a(n) = A000217(A000005(n)-1). - Antti Karttunen, Sep 21 2018
a(n) = Sum_{k|n, i|n, i < k} 1. - Wesley Ivan Hurt, Aug 20 2020
a(n) = Sum_{d|n} A063647(d). - Ridouane Oudra, Apr 15 2023

A061391 a(n) = t(n,3) = Sum_{d|n} tau(d^3), where tau(n) = number of divisors of n, cf. A000005.

Original entry on oeis.org

1, 5, 5, 12, 5, 25, 5, 22, 12, 25, 5, 60, 5, 25, 25, 35, 5, 60, 5, 60, 25, 25, 5, 110, 12, 25, 22, 60, 5, 125, 5, 51, 25, 25, 25, 144, 5, 25, 25, 110, 5, 125, 5, 60, 60, 25, 5, 175, 12, 60, 25, 60, 5, 110, 25, 110, 25, 25, 5, 300, 5, 25, 60, 70, 25, 125, 5, 60, 25, 125, 5, 264

Offset: 1

Vladeta Jovovic, Apr 29 2001

Inverse Mobius transform of A048785. - R. J. Mathar, Feb 09 2011

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. t(n, 0) = A000005(n), t(n, 1) = A007425(n), t(n, 2) = A035116(n).

Cf. A048691.

f[p_, e_] := (3*e^2 + 5*e + 2)/2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

A061391 = n -> sumdiv(n, d, numdiv(d^3));
for(n=1, 10000, write("b061391.txt", n, " ", A061391(n)));
\\ Antti Karttunen, Jan 17 2017

for(n=1, 100, print1(direuler(p=2, n, (1 + 2*X)/(1 - X)^3)[n], ", ")) \\ Vaclav Kotesovec, May 15 2021

for(n=1, 100, print1(direuler(p=2, n, (1 - 3*X^2 + 2*X^3)/(1 - X)^5)[n], ", ")) \\ Vaclav Kotesovec, Aug 20 2021

t(n, k) = Sum_{d|n} tau(d^k) is multiplicative: if the canonical factorization of n = Product p^e(p) over primes then t(n, k) = Product t(p^e(p), k), t(p^e(p), k) = (1/2) *(k*e(p)+2)*(e(p)+1).
For k=2 we get an interesting identity: Sum_{d|n} tau(d^2)=(tau(n))^2, cf. A048691, A035116.
a(n) = Sum_{d|n} tau(n*d). - Benoit Cloitre, Nov 30 2002
G.f.: Sum_{n>=1} tau(n^3)*x^n/(1-x^n). - Joerg Arndt, Jan 01 2011
Dirichlet g.f.: zeta(s)^3 * Product_{primes p} (1 + 2/p^s). - Vaclav Kotesovec, May 15 2021
Dirichlet g.f.: zeta(s)^5 * Product_{primes p} (1 - 3/p^(2*s) + 2/p^(3*s)). - Vaclav Kotesovec, Aug 20 2021

A062367 Multiplicative with a(p^e) = (e+1)(e+2)(2*e+3)/6.

Original entry on oeis.org

1, 5, 5, 14, 5, 25, 5, 30, 14, 25, 5, 70, 5, 25, 25, 55, 5, 70, 5, 70, 25, 25, 5, 150, 14, 25, 30, 70, 5, 125, 5, 91, 25, 25, 25, 196, 5, 25, 25, 150, 5, 125, 5, 70, 70, 25, 5, 275, 14, 70, 25, 70, 5, 150, 25, 150, 25, 25, 5, 350, 5, 25, 70, 140, 25, 125, 5, 70, 25, 125, 5

Offset: 1

Vladeta Jovovic, Jul 07 2001

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000005, A000012, A000330, A029939, A035116, A048691, A060648, A062368.

A062367 := proc(n)
    add(numtheory[tau](d)^2,d=numtheory[divisors](n)) ;
end proc:
seq(A062367(n),n=1..40) ; # R. J. Mathar, May 15 2025

{1}~Join~Array[Times @@ Map[((# + 1) (# + 2) (2 # + 3))/6 &, FactorInteger[#][[All, -1]] ] &, 70, 2] (* or *)
Array[DivisorSum[#, DivisorSigma[0, #]^2 &] &, 71] (* Michael De Vlieger, Mar 05 2021 *)

a(n) = sumdiv(n, d, numdiv(d)^2) \\ Michel Marcus, Jun 17 2013

a(n) = Sum_{i|n, j|n} tau(gcd(i, j)) = Sum_{d|n} tau(d)^2.
a(n) = Sum_{i|n, j|n} tau(i)*tau(j)/tau(lcm(i, j)), where tau(n) = number of divisors of n, cf. A000005.
Dirichlet convolution of A035116 and A000012 (i.e., inverse Mobius transform of A035116). Dirichlet g.f.: zeta^5(s)/zeta(2s). - R. J. Mathar, Feb 03 2011
G.f.: Sum_{n>=1} A000005(n)^2*x^n/(1-x^n). - Mircea Merca, Feb 26 2014
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(tau(k)^2/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 23 2018
Dirichlet convolution of A007426 and A008966. Dirichlet convolution of A007425 and A034444. - R. J. Mathar, Jun 05 2020
Let b(n), n > 0, be Dirichlet inverse of a(n). Then b(n) is multiplicative with b(p^e) = (-1)^e*(Sum_{i=0..e} binomial(4,i)) for prime p and e >= 0, where binomial(n,k)=0 if n < k; abs(b(n)) is multiplicative and has the Dirichlet g.f.: (zeta(s))^5/(zeta(2*s))^4. - Werner Schulte, Feb 07 2021
a(n) = Sum_{d divides n} tau(d^2)*tau(n/d), Dirichlet convolution of A048691 and A000005. - Peter Bala, Jan 26 2024

A062369 Dirichlet convolution of n and tau^2(n).

Original entry on oeis.org

1, 6, 7, 21, 9, 42, 11, 58, 30, 54, 15, 147, 17, 66, 63, 141, 21, 180, 23, 189, 77, 90, 27, 406, 54, 102, 106, 231, 33, 378, 35, 318, 105, 126, 99, 630, 41, 138, 119, 522, 45, 462, 47, 315, 270, 162, 51, 987, 86, 324, 147, 357, 57, 636, 135, 638, 161, 198, 63, 1323

Offset: 1

Vladeta Jovovic, Jul 07 2001

Dirichlet convolution of A000027 and A035116.

Inverse Mobius transform of A060724. - R. J. Mathar, Oct 15 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A000203, A060648.

Cf. A000005, A060724, A064950, A062369, A062368, A062380.

[&+[d*#Divisors(Floor(n/d))^2:d in Divisors(n)]:n in [1..60]]; // Marius A. Burtea, Aug 25 2019

a[n_] := Sum[ DivisorSigma[1, i]*DivisorSigma[1, j] / DivisorSigma[1, LCM[i, j]], {i, Divisors[n]}, {j, Divisors[n]}]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 26 2013 *)

a(n) = sumdiv(n, d, d*numdiv(n/d)^2); \\ Michel Marcus, Nov 03 2018

a(n) = Sum_{i|n, j|n} sigma(i)*sigma(j)/sigma(lcm(i,j)), where sigma(n) = sum of divisors of n.
a(n) = Sum_{i|d, j|d} sigma(gcd(i, j));
a(n) = Sum_{d|n} d*tau(n/d)^2, where tau(n) = number of divisors of n.
Multiplicative with a(p^e) = (1-p^(3+e)-p^(2+e)+e^2+4*p^2+p^2*e^2+2*e-3*p+4*p^2*e-6*e*p-2*e^2*p)/(1-p)^3.
Dirichlet g.f.: (zeta(s))^4*zeta(s-1)/zeta(2*s). - R. J. Mathar, Feb 09 2011
G.f.: Sum_{k>=1} tau(k)^2*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Nov 02 2018
Sum_{k=1..n} a(k) ~ 5 * Pi^4 * n^2 / 144. - Vaclav Kotesovec, Jan 28 2019
a(n) = Sum_{d|n} tau(d^2)*sigma(n/d), where tau(n) = number of divisors of n, and sigma(n) = sum of divisors of n. - Ridouane Oudra, Aug 25 2019

A321348 a(n) = Sum_{d|n} tau(d^n), where tau() is the number of divisors (A000005).

Original entry on oeis.org

1, 4, 5, 15, 7, 64, 9, 52, 30, 144, 13, 546, 15, 256, 289, 165, 19, 1140, 21, 1386, 529, 576, 25, 3848, 78, 784, 166, 2610, 31, 32768, 33, 486, 1225, 1296, 1369, 12321, 39, 1600, 1681, 10248, 43, 85184, 45, 6210, 6486, 2304, 49, 24250, 150, 7956

Offset: 1

Ilya Gutkovskiy, Nov 06 2018

nonn

a(n) is prime iff n is in A001359, which makes the sequence a supersequence of A006512. - Ivan N. Ianakiev, Nov 07 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A000005, A007425, A035116, A061391.

Cf. A001359, A006512, A052147.

[&+[NumberOfDivisors(d^n): d in Divisors(n)]: n in [1..50]]; // Vincenzo Librandi, Nov 08 2018

with(numtheory): seq(coeff(series(add(tau(k^n)*x^k/(1-x^k),k=1..n),x,n+1), x, n), n = 1 .. 50); # Muniru A Asiru, Nov 25 2018

Table[Sum[DivisorSigma[0, d^n], {d, Divisors[n]}], {n, 50}]
a[n_] := Times @@ ((#[[2]] + 1) (n #[[2]] + 2)/2 & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 50}]

a(n) = sumdiv(n, d, numdiv(d^n)); \\ Michel Marcus, Nov 06 2018

from math import prod
from sympy import factorint
def A321348(n): return prod((e+1)*(n*e+2)>>1 for e in factorint(n).values()) # Chai Wah Wu, Dec 13 2022

a(n) = [x^n] Sum_{k>=1} tau(k^n)*x^k/(1 - x^k).
If n = Product (p_j^k_j) then a(n) = Product ((k_j + 1)*(n*k_j + 2)/2).
a(prime(n)) = prime(n) + 2 = A052147(n). - Michel Marcus, Nov 25 2018

cp's OEIS Frontend

A349483 Length of cycle reached when iterating the mapping x-> n*A035116(x) on 1.

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A048691 a(n) = d(n^2), where d(k) = A000005(k) is the number of divisors of k.

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A062319 Number of divisors of n^n, or of A000312(n).

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A184389 a(n) = Sum_{k=1..tau(n)} k, where tau is the number of divisors of n (A000005).

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A061502 a(n) = Sum_{k<=n} tau(k)^2, where tau = number of divisors function A000005.

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A066446 Number of unordered divisor pairs of n.

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A062367 Multiplicative with a(p^e) = (e+1)(e+2)(2*e+3)/6.