A000005 -id:A000005 - cp's OEIS frontend

A035033 Numbers k such that k <= d(k)^2, where d() = number of divisors (A000005).

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 72, 80, 84, 90, 96, 108, 120, 126, 132, 140, 144, 168, 180, 192, 210, 216, 240, 252, 288, 300, 336, 360, 420, 480, 504, 540, 720, 840, 1260

Offset: 1

Erich Friedman, Labos Elemer

Cf. A000005, A035034, A035035, A033950, A036763, A034884, A273323.

Select[Range[1300],#<=DivisorSigma[0,#]^2&] (* Harvey P. Dale, Mar 16 2012 *)

is(n)=numdiv(n)^2>=n \\ Charles R Greathouse IV, Aug 03 2012

A036763 Numbers k such that k*d(x) = x has no solution for x, where d (A000005) is the number of divisors; sequence gives impossible x/d(x) quotients in order of magnitude.

Original entry on oeis.org

18, 27, 30, 45, 63, 64, 72, 99, 105, 112, 117, 144, 153, 160, 162, 165, 171, 195, 207, 225, 243, 252, 255, 261, 279, 285, 288, 294, 320, 333, 336, 345, 352, 360, 369, 387, 396, 405, 416, 423, 435, 441, 465, 468, 477, 490, 504, 531, 544, 549, 555, 567, 576

Offset: 1

Labos Elemer

nonn

A special case of a bound on d(n) by Erdős and Suranyi (1960) was used to get a limit: a = x/d(x) > 0.5*sqrt(x) and below 4194304 a computer test shows these values did not occur as x = a*d(x). For larger x this is impossible since if d(x) < sqrt(x), then x/d(x) > sqrt(4194304) = 2048 > the given terms.
A051521(a(n)) = 0. - Reinhard Zumkeller, Dec 28 2011
This sequence contains all numbers of the form k = 9p, p prime (i.e., k = 18, 27, 45, 63, 99, ...). - Jianing Song, Nov 25 2018

			No natural number x exists for which x = 18*d(x), so 18 is a term.

P. Erdős and J. Suranyi, Selected Topics in Number Theory, Tankonyvkiado, Budapest, 1960 (in Hungarian).
P. Erdős and J. Suranyi, Selected Topics in Number Theory, Springer, New York, 2003 (in English).

Donovan Johnson, Table of n, a(n) for n = 1..5000

Cf. A000005, A033950, A036761, A036762, A036764, A051278, A051279, A051280, A051521.

a036763 n = a036763_list !! (n-1)
a036763_list = filter ((== 0) . a051521) [1..]
-- Reinhard Zumkeller, Dec 28 2011

with(numtheory): A036763 := proc(n) local k,p: for k from 1 to 4*n^2 do p:=n*k: if(p=n*tau(p))then return NULL: fi: od: return n: end: seq(A036763(n),n=1..100); # Nathaniel Johnston, May 04 2011

noSolQ[n_] := !AnyTrue[Range[4*n^2], # == DivisorSigma[0, n*#]& ];
Reap[Do[If[noSolQ[n], Print[n]; Sow[n]], {n, 600}]][[2, 1]] (* Jean-François Alcover, Jan 30 2018 *)

Definition corrected by N. J. A. Sloane, May 18 2022 at the suggestion of David James Sycamore.

A062249 a(n) = n + d(n), where d(n) = number of divisors of n, cf. A000005.

Original entry on oeis.org

2, 4, 5, 7, 7, 10, 9, 12, 12, 14, 13, 18, 15, 18, 19, 21, 19, 24, 21, 26, 25, 26, 25, 32, 28, 30, 31, 34, 31, 38, 33, 38, 37, 38, 39, 45, 39, 42, 43, 48, 43, 50, 45, 50, 51, 50, 49, 58, 52, 56, 55, 58, 55, 62, 59, 64, 61, 62, 61, 72, 63, 66, 69, 71, 69, 74, 69, 74, 73, 78, 73

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 01 2001

nonn

Number of cyclic subgroups of dihedral group with 2n elements.

a(n) is the n-th smallest number not a divisor of n. - J. Lowell, Apr 06 2008

Harry J. Smith, Table of n, a(n) for n=1..1000

Cf. A007503, A060710, A000005, A049820.

Cf. A064491 (iteration, start=1).

a062249 n = a000005 n + n  -- Reinhard Zumkeller, Mar 29 2014

with(numtheory):seq(n+tau(n), n=1..71) ; # Zerinvary Lajos, Jun 04 2008

Table[n + DivisorSigma[0, n], {n, 100}] (* Indranil Ghosh, Apr 12 2017 *)

a(n) = n + numdiv(n) \\ Harry J. Smith, Aug 03 2009

from sympy.ntheory import divisor_count
[n + divisor_count(n) for n in range(101)] # Indranil Ghosh, Apr 12 2017

a(n) = n + A000005(n). - Omar E. Pol, Dec 12 2008
From Ilya Gutkovskiy, Apr 12 2017: (Start)
G.f.: x/(1 - x)^2 + Sum_{k>=1} x^k/(1 - x^k).
Dirichlet g.f.: zeta(s)^2 + zeta(s-1). (End)

Formula and more terms from Vladeta Jovovic, Jul 03 2001

A034695 Tau_6 (the 6th Piltz divisor function), the number of ordered 6-factorizations of n; Dirichlet convolution of number-of-divisors function (A000005) with A007426.

Original entry on oeis.org

1, 6, 6, 21, 6, 36, 6, 56, 21, 36, 6, 126, 6, 36, 36, 126, 6, 126, 6, 126, 36, 36, 6, 336, 21, 36, 56, 126, 6, 216, 6, 252, 36, 36, 36, 441, 6, 36, 36, 336, 6, 216, 6, 126, 126, 36, 6, 756, 21, 126, 36, 126, 6, 336, 36, 336, 36, 36, 6, 756, 6, 36, 126, 462, 36, 216, 6, 126

Offset: 1

N. J. A. Sloane

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pages 29 and 38
Leveque, William J., Fundamentals of Number Theory. New York:Dover Publications, 1996, ISBN 9780486689067, p .167-Exercise 5.b.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Enrique Pérez Herrero)
E. Pérez Herrero, Piltz Divisor functions (1), Psychedelic Geometry Blogspot, Dec 21 2009.
E. Pérez Herrero, Piltz Divisor functions (2), Psychedelic Geometry Blogspot, Dec 24 2009.

Cf. A000005 (tau_2), A007425 (tau_3), A007426 (tau_4), A061200 (tau_5).
Cf. A061204.
Column k=6 of A077592.

tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[ tau[n, 6], {n, 68}] (* Robert G. Wilson v, Nov 02 2005 *)
tau[1, k_] := 1; tau[n_, k_] := Times @@ (Binomial[Last[#]+k-1, k-1]& /@ FactorInteger[n]); Table[tau[n, 6], {n, 1, 100}] (* Amiram Eldar, Sep 13 2020 *)

a(n) = my(f=factor(n)); for (i=1, #f~, f[i,1] = binomial(f[i,2] + 5, f[i,2]); f[i,2]=1); factorback(f); \\ Michel Marcus, Jun 09 2014

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^6)[n]), ", ")) \\ Vaclav Kotesovec, May 06 2025

from math import prod, comb
from sympy import factorint
def A034695(n): return prod(comb(5+e,5) for e in factorint(n).values()) # Chai Wah Wu, Dec 22 2024

Dirichlet g.f.: zeta^6(s).
Multiplicative with a(p^e) = binomial(e+5, e). - Mitch Harris, Jun 27 2005
The Piltz divisor functions hold for tau_j(*)tau_k = tau_{j+k}, where (*) means Dirichlet convolution.
G.f.: Sum_{k>=1} tau_5(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Oct 30 2018

More terms from Robert G. Wilson v, Nov 02 2005

A221530 Triangle read by rows: T(n,k) = A000005(k)*A000041(n-k).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 4, 2, 3, 5, 6, 4, 3, 2, 7, 10, 6, 6, 2, 4, 11, 14, 10, 9, 4, 4, 2, 15, 22, 14, 15, 6, 8, 2, 4, 22, 30, 22, 21, 10, 12, 4, 4, 3, 30, 44, 30, 33, 14, 20, 6, 8, 3, 4, 42, 60, 44, 45, 22, 28, 10, 12, 6, 4, 2, 56, 84, 60, 66, 30, 44, 14, 20, 9, 8, 2, 6

Offset: 1

Omar E. Pol, Jan 19 2013

T(n,k) is the number of partitions of n that contain k as a part multiplied by the number of divisors of k.
It appears that T(n,k) is also the total number of appearances of k in the last k sections of the set of partitions of n multiplied by the number of divisors of k.
T(n,k) is also the number of partitions of k into equal parts multiplied by the number of ones in the j-th section of the set of partitions of n, where j = (n - k + 1).
For another version see A245095. - Omar E. Pol, Jul 15 2014

			For n = 6:
  -------------------------
  k   A000005        T(6,k)
  1      1  *  7   =    7
  2      2  *  5   =   10
  3      2  *  3   =    6
  4      3  *  2   =    6
  5      2  *  1   =    2
  6      4  *  1   =    4
  .         A000041
  -------------------------
So row 6 is [7, 10, 6, 6, 4, 2]. Note that the sum of row 6 is 7+10+6+6+2+4 = 35 equals A006128(6).
.
Triangle begins:
  1;
  1,   2;
  2,   2,  2;
  3,   4,  2,  3;
  5,   6,  4,  3,  2;
  7,  10,  6,  6,  2,  4;
  11, 14, 10,  9,  4,  4,  2;
  15, 22, 14, 15,  6,  8,  2,  4;
  22, 30, 22, 21, 10, 12,  4,  4,  3;
  30, 44, 30, 33, 14, 20,  6,  8,  3,  4;
  42, 60, 44, 45, 22, 28, 10, 12,  6,  4,  2;
  56, 84, 60, 66, 30, 44, 14, 20,  9,  8,  2,  6;
  ...

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened)

Similar to A221529.
Columns 1-2: A000041, A139582. Leading diagonals 1-3: A000005, A000005, A062011. Row sums give A006128.
Cf. A027293, A135010, A138137, A182703, A245095, A245099.

A221530row[n_]:=DivisorSigma[0,Range[n]]PartitionsP[n-Range[n]];Array[A221530row,10] (* Paolo Xausa, Sep 04 2023 *)

row(n) = vector(n, i, numdiv(i)*numbpart(n-i)); \\ Michel Marcus, Jul 18 2014

T(n,k) = d(k)*p(n-k) = A000005(k)*A027293(n,k).

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

A372713 Number of divisors of 3n; a(n) = tau(3n) = A000005(3n).

Original entry on oeis.org

2, 4, 3, 6, 4, 6, 4, 8, 4, 8, 4, 9, 4, 8, 6, 10, 4, 8, 4, 12, 6, 8, 4, 12, 6, 8, 5, 12, 4, 12, 4, 12, 6, 8, 8, 12, 4, 8, 6, 16, 4, 12, 4, 12, 8, 8, 4, 15, 6, 12, 6, 12, 4, 10, 8, 16, 6, 8, 4, 18, 4, 8, 8, 14, 8, 12, 4, 12, 6, 16, 4, 16, 4, 8, 9, 12, 8, 12, 4, 20

Offset: 1

Vaclav Kotesovec, May 11 2024

nonn

In general, for p prime, Sum_{j=1..n} tau(j*p) ~ (2*p - 1) * n * (log(n) - 1 + 2*gamma)/p + n*log(p)/p, where gamma is the Euler-Mascheroni constant A001620.

If n is in A033428, then a(n) is odd and vice versa. - R. J. Mathar, Amiram Eldar, May 20 2024.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000005, A001620, A033428, A099777, A372714, A372715, A372786, A372789, A372792.

Cf. also A144613.

Table[DivisorSigma[0, 3*n], {n, 1, 150}]

a(n) = numdiv(3*n); \\ Michel Marcus, May 20 2024

Sum_{k=1..n} a(k) ~ n * (5*(log(n) + 2*gamma - 1) + log(3)) / 3, where gamma is the Euler-Mascheroni constant A001620.

A221531 Triangle read by rows: T(n,k) = A000005(n-k+1)*A000041(k-1), n>=1, k>=1.

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 3, 2, 4, 3, 2, 3, 4, 6, 5, 4, 2, 6, 6, 10, 7, 2, 4, 4, 9, 10, 14, 11, 4, 2, 8, 6, 15, 14, 22, 15, 3, 4, 4, 12, 10, 21, 22, 30, 22, 4, 3, 8, 6, 20, 14, 33, 30, 44, 30, 2, 4, 6, 12, 10, 28, 22, 45, 44, 60, 42, 6, 2, 8, 9, 20, 14, 44, 30, 66, 60, 84, 56

Offset: 1

Omar E. Pol, Jan 19 2013

			For n = 6:
-------------------------
k   A000041        T(6,k)
1      1  *  4   =    4
2      1  *  2   =    2
3      2  *  3   =    6
4      3  *  2   =    6
5      5  *  2   =   10
6      7  *  1   =    7
.         A000005
-------------------------
So row 6 is [4, 2, 6, 6, 10, 7]. Note that the sum of row 6 is 4+2+6+6+10+7 = 35 equals A006128(6).
.
Triangle begins:
1;
2,  1;
2,  2,  2;
3,  2,  4,  3;
2,  3,  4,  6, 5;
4,  2,  6,  6, 10, 7;
2,  4,  4,  9, 10, 14, 11;
4,  2,  8,  6, 15, 14, 22, 15;
3,  4,  4, 12, 10, 21, 22, 30, 22;
4,  3,  8,  6, 20, 14, 33, 30, 44, 30;
2,  4,  6, 12, 10, 28, 22, 45, 44, 60, 42;
6,  2,  8,  9, 20, 14, 44, 30, 66, 60, 84, 56;
...

Mirror of A221530. Columns 1-3: A000005, A000005, A062011. Leading diagonals 1-2: A000041, A139582. Row sums give A006128.

Cf. A140207, A182703.

T(n,k) = d(n-k+1)*p(k-1), n>=1, k>=1.

A015996 (tau(n^4) + 3)/4, where tau = A000005.

Original entry on oeis.org

1, 2, 2, 3, 2, 7, 2, 4, 3, 7, 2, 12, 2, 7, 7, 5, 2, 12, 2, 12, 7, 7, 2, 17, 3, 7, 4, 12, 2, 32, 2, 6, 7, 7, 7, 21, 2, 7, 7, 17, 2, 32, 2, 12, 12, 7, 2, 22, 3, 12, 7, 12, 2, 17, 7, 17, 7, 7, 2, 57, 2, 7, 12, 7, 7, 32, 2, 12, 7, 32, 2, 30, 2, 7, 12, 12, 7, 32, 2, 22, 5, 7, 2, 57, 7, 7

Offset: 1

Robert G. Wilson v

nonn

If n is prime, a(n) = 2 since a(p) = (tau(p^4)+3)/4 = (5+3)/4 = 2. - Wesley Ivan Hurt, Nov 16 2013

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

Cf. A000005, A018892, A015995, A015999, A016001, A016002, A016003, A016005, A016006, A016007, A016008, A016009, A016012, A016020.

A015996 := proc(n)
        (numtheory[tau](n^4)+3)/4 ;
end proc; # R. J. Mathar, May 09 2013

Table[(DivisorSigma[0, n^4] + 3)/4, {n, 100}] (* Wesley Ivan Hurt, Nov 16 2013 *)

A015996(n) = (numdiv(n^4)+3)/4;
for(n=1, 10000, write("b015996.txt", n, " ", A015996(n)));
\\ Antti Karttunen, Jan 17 2017

a(n) = (A000005(n^4) + 3)/4.

Definition corrected by Vladeta Jovovic, Sep 03 2005

A063446 Integers m such that d(m+1) = 2*d(m), where d(m) = A000005(m).

Original entry on oeis.org

1, 5, 7, 13, 37, 39, 49, 55, 61, 65, 69, 73, 77, 87, 129, 134, 157, 183, 185, 193, 194, 221, 237, 247, 249, 254, 265, 275, 277, 295, 309, 313, 321, 327, 343, 363, 365, 397, 398, 417, 421, 437, 454, 455, 457, 458, 469, 471, 473, 475, 482, 493, 497, 505, 517

Offset: 1

Labos Elemer, Jul 24 2001

nonn

			For m = 77, 2*d(77) = 2*4 = 8 = d(78).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000005, A063449, A063450.

Filtered([1..520],n->Tau(n+1)=2*Tau(n)); # Muniru A Asiru, Aug 20 2018

[n: n in [1..600] | NumberOfDivisors(n+1) eq (2* NumberOfDivisors(n))]; // Vincenzo Librandi, Aug 04 2018

Select[Range@ 520, DivisorSigma[0, # + 1] == 2 DivisorSigma[0, #] &] (* Michael De Vlieger, Feb 19 2017 *)

{ n=0; for (m=1, 10^9, if (numdiv(m + 1) == 2*numdiv(m), write("b063446.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 21 2009

cp's OEIS Frontend

A035033 Numbers k such that k <= d(k)^2, where d() = number of divisors (A000005).

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A036763 Numbers k such that k*d(x) = x has no solution for x, where d (A000005) is the number of divisors; sequence gives impossible x/d(x) quotients in order of magnitude.

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A062249 a(n) = n + d(n), where d(n) = number of divisors of n, cf. A000005.

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A034695 Tau_6 (the 6th Piltz divisor function), the number of ordered 6-factorizations of n; Dirichlet convolution of number-of-divisors function (A000005) with A007426.

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A221530 Triangle read by rows: T(n,k) = A000005(k)*A000041(n-k).

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

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A372713 Number of divisors of 3n; a(n) = tau(3*n) = A000005(3*n).

A372713 Number of divisors of 3n; a(n) = tau(3n) = A000005(3n).