A023645 -id:A023645 - cp's OEIS frontend

A338653 Number of divisors of n which are greater than 9.

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

Offset: 1

Ilya Gutkovskiy, Apr 22 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences related to divisors of numbers.

Column k=10 of A135539.

Cf. A000005, A001620, A023645, A032741, A321014, A338648, A338649, A338650, A338651, A338652.

Table[DivisorSum[n, 1 &, # > 9 &], {n, 1, 110}]
nmax = 110; CoefficientList[Series[Sum[x^(10 k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Drop[#, 1] &
nmax = 110; CoefficientList[Series[-Log[Product[(1 - x^k)^(1/k), {k, 10, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Drop[#, 1] &
Table[Count[Divisors[n],?(#>9&)],{n,120}] (* _Harvey P. Dale, Jan 09 2025 *)

a(n) = sumdiv(n, d, d>9); \\ Michel Marcus, Apr 22 2021

my(N=100, x='x+O('x^N)); concat([0, 0, 0, 0, 0, 0, 0, 0, 0], Vec(sum(k=10, N, x^k/(1-x^k)))) \\ Seiichi Manyama, Jan 07 2023

G.f.: Sum_{k>=1} x^(10*k) / (1 - x^k).
L.g.f.: -log( Product_{k>=10} (1 - x^k)^(1/k) ).
G.f.: Sum_{k>=10} x^k/(1 - x^k). - Seiichi Manyama, Jan 07 2023
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 9649/2520), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 08 2024

a(1)-a(9) prepended by David A. Corneth, Jun 13 2022

A049992 a(n) is the number of arithmetic progressions of 3 or more positive integers, nondecreasing with sum n.

Original entry on oeis.org

0, 0, 1, 1, 1, 3, 1, 2, 4, 3, 1, 7, 1, 3, 8, 4, 1, 10, 1, 6, 10, 4, 1, 14, 4, 4, 12, 7, 1, 19, 1, 6, 14, 5, 7, 22, 1, 5, 16, 12, 1, 24, 1, 8, 25, 6, 1, 27, 4, 12, 21, 9, 1, 29, 9, 12, 23, 7, 1, 40, 1, 7, 30, 11, 10, 35, 1, 10, 27, 21, 1, 42, 1, 8, 39, 11, 7, 40, 1, 22, 35, 9, 1, 49, 12, 9, 34

Offset: 1

Clark Kimberling

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12580
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions, Rostok. Math. Kolloq. 64 (2009), 11-16.
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions with an odd common difference, Integers 9(1) (2009), 77-81.
Jon Maiga, Computer-generated formulas for A049992, Sequence Machine.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
Augustine O. Munagi and Temba Shonhiwa, On the partitions of a number into arithmetic progressions, Journal of Integer Sequences 11 (2008), Article 08.5.4.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
Wikipedia, Arithmetic progression.

Cf. A007862, A014405, A023645, A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A049991, A049992, A049994, A175676, A240026, A240027, A307824, A320466.

A049992(n) = (A014405(n)+A023645(n)); \\ (Uses the programs given in A014405 and A023645) - Antti Karttunen, Feb 20 2023

G.f.: Sum_{k>=3} x^k/(1-x^(k*(k-1)/2))/(1-x^k). [Leroy Quet from A049988] - Petros Hadjicostas, Sep 29 2019

a(n) = A014405(n) + A023645(n) = A049994(n) + A175676(n). [Two of the formulas listed by Sequence Machine, both obviously true] - Antti Karttunen, Feb 20 2023

More terms from Petros Hadjicostas, Sep 29 2019

A076984 Number of Fibonacci numbers that are divisors of the n-th Fibonacci number.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 2, 3, 3, 3, 2, 5, 2, 3, 4, 4, 2, 5, 2, 5, 4, 3, 2, 7, 3, 3, 4, 5, 2, 7, 2, 5, 4, 3, 4, 8, 2, 3, 4, 7, 2, 7, 2, 5, 6, 3, 2, 9, 3, 5, 4, 5, 2, 7, 4, 7, 4, 3, 2, 11, 2, 3, 6, 6, 4, 7, 2, 5, 4, 7, 2, 11, 2, 3, 6, 5, 4, 7, 2, 9, 5, 3, 2, 11, 4, 3, 4, 7, 2, 11, 4, 5, 4, 3, 4, 11, 2, 5, 6, 8, 2, 7, 2

Offset: 1

Amarnath Murthy, Oct 25 2002

nonn

a(A001605(n)) = 2; a(A105802(n)) = n.

It is well known that if k is a divisor of n then F(k) divides F(n). Hence if n has d divisors, one expects that a(n)=d. However because F(1)=F(2)=1, there is one fewer Fibonacci divisor when n is even. So for even n, a(n)=d-1. - T. D. Noe, Jan 18 2006

			n=12, A000045(12)=144: 5 of the 15 divisors of 144 are also Fibonacci numbers, a(12) = #{1, 2, 3, 8, 144} = 5.

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fibonacci Number

Cf. A000005, A063375, A000045, A105800.
Cf. A076985.
Cf. A068499.

with(combinat, fibonacci):a[1] := 1:for i from 2 to 229 do s := 0:for j from 2 to i do if((fibonacci(i) mod fibonacci(j))=0) then s := s+1:fi:od:a[i] := s:od:seq(a[l],l=2..229);

Table[s=DivisorSigma[0, n]; If[OddQ[n], s, s-1], {n, 100}] (Noe)

{a(n)=if(n<1, 0, numdiv(n)+n%2-1)} /* Michael Somos, Sep 03 2006 */

{a(n)=if(n<1, 0, sumdiv(n,d, d!=2))} /* Michael Somos, Sep 03 2006 */

a(n) = A023645(n) + 1. - T. D. Noe, Jan 18 2006
a(n) = tau(n) - [n is even] = A000005(n) - A059841(n). Proof: gcd(Fib(m), Fib(n)) = Fib(gcd(m, n)) and Fib(2) = 1. - Olivier Wittenberg, following a conjecture of Ralf Stephan, Sep 28 2004
The number of divisors of n excluding 2.
a(2n) = A066660(n). a(2n-1) = A099774(n). - Michael Somos, Sep 03 2006
a(3*2^(Prime(n-1)-1)) = 2n + 1 for n > 3. a(3*2^A068499[n]) = 2n + 1, where A068499(n) = {1,2,3,4,6,10,12,16,18,...}. - Alexander Adamchuk, Sep 15 2006

Corrected and extended by Sascha Kurz, Jan 26 2003

Edited by N. J. A. Sloane, Sep 14 2006. Some of the comments and formulas may need to be adjusted to reflect the new offset.

A072528 Table T(n,k) read by rows, giving number of occurrences of the remainder k when n is divided by i=1,2,3,...,n.

Original entry on oeis.org

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

Offset: 1

Amarnath Murthy, Aug 01 2002

The n-th row adds to n.

			The table begins
1
2
2 1
3 1
2 2 1
4 1 1
2 3 1 1
4 1 2 1

Cf. A023645 for T(n, 2) and A072527 for T(n, 3).

Cf. A230374, A230399.

Let a(m) be the m-th term in the sequence. Then m=f(n)+k where f(1)=1 and f(n+1)=f(n)+floor((n+1)/2). n is the number being divided by the various i's and k is the remainder under consideration. f(n) has the generating function F(x)= (x(1+2x^2-2x^3))/((1-x)^2(1+x^2)) - Bruce Corrigan (scentman(AT)myfamily.com), Oct 22 2002

G.f. for k-th column: Sum_{m>0} x^((k+1)*m+k)/(1-x^m). - Vladeta Jovovic, Dec 16 2002

Edited by Bruce Corrigan (scentman(AT)myfamily.com), Oct 22 2002

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 25 2003

A086369 Number of factors over Q in the factorization of T_n(x) - 1 where T_n(x) is the Chebyshev polynomial of the first kind.

Original entry on oeis.org

1, 2, 3, 4, 3, 6, 3, 6, 5, 6, 3, 10, 3, 6, 7, 8, 3, 10, 3, 10, 7, 6, 3, 14, 5, 6, 7, 10, 3, 14, 3, 10, 7, 6, 7, 16, 3, 6, 7, 14, 3, 14, 3, 10, 11, 6, 3, 18, 5, 10, 7, 10, 3, 14, 7, 14, 7, 6, 3, 22, 3, 6, 11, 12, 7, 14, 3, 10, 7, 14, 3, 22, 3, 6, 11, 10, 7, 14, 3, 18

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 08 2003

nonn

If p is an odd prime then a(p) = 3.

a(n) is also the cardinality of the set T containing the divisors d of n and those m > 0 satisfying m + d = n (see the R. J. Mathar formula). Another way of defining a(n) is: if S is the set of nondivisors of n such that r and s belong to S if r + s = n, then a(n) = n - |S|. This second 'co-construction' (since n = |T| + |S|) of a(n) via S is more natural than the direct construction via T, as it avoids two ambiguities in the direct approach. Let f be an involutive function f(x) = y mapping distinct nonzero elements x, y of a set to a pair (x,y) in a set of pairs if x + y = n. Considering T, for m and d in T such that m <> n or d <> n, and m <> d, we have f(m) = d; however, n itself is a member of T yet there exists no valid function f'(n) = 0 since 0 is not a member of T; furthermore, if n is even then there is a unique d in T for which we have to define another function f''(d) = d, valid only for d. Whereas considering S, f(r) = s for every r and s in S and therefore f is a surjective map between S and the set of pairs; then, as stated, n - |S| = |T| = a(n). - Miles Englezou, Jun 22 2025

Antti Karttunen, Table of n, a(n) for n = 1..2049
Yusuf Z. Gürtaş, Chebyshev polynomials and the minimal polynomial of cos(2pi/n), Am. Math. Monthly 124 (1) (2017) 73-78, Theorem 1.

Cf. A000005, A001620, A023645, A086374.

a[n_] := 2 * DivisorSigma[0, n] + Mod[n, 2] - 2; Array[a, 100] (* Amiram Eldar, Jun 30 2025 *)

a(n)={vecsum(factor(polchebyshev(n, 1, x)-1)[, 2])} \\ Andrew Howroyd, Jul 10 2018

a(n) = if(n%2==1, 1+2*sumdiv(n, d, dMiles Englezou, Jun 22 2025

a(n) = 2 * numdiv(n) + n % 2 - 2; \\ Amiram Eldar, Jun 30 2025

a(n) = 1+2*A023645(n) for n odd, = 2+2*A023645(n) for n even. [Gürtaş] - R. J. Mathar, Mar 03 2023
a(p^m) = 2*m+1 for prime p > 2 and m >= 1. - Miles Englezou, Jun 22 2025
From Amiram Eldar, Jun 30 2025: (Start)
a(n) = 2*tau(n) + (n mod 2) - 2, where tau(n) = A000005(n).
Sum_{k=1..n} a(k) ~ 2*n * (log(n) + 2*gamma - 7/4), where gamma is Euler's constant (A001620). (End)

a(14) corrected and a(21)-a(80) added by Andrew Howroyd, Jul 10 2018

A296955 Sum of the smaller parts of the partitions of n into two distinct parts such that the smaller part divides the larger.

Original entry on oeis.org

0, 0, 1, 1, 1, 3, 1, 3, 4, 3, 1, 10, 1, 3, 9, 7, 1, 12, 1, 12, 11, 3, 1, 24, 6, 3, 13, 14, 1, 27, 1, 15, 15, 3, 13, 37, 1, 3, 17, 30, 1, 33, 1, 18, 33, 3, 1, 52, 8, 18, 21, 20, 1, 39, 17, 36, 23, 3, 1, 78, 1, 3, 41, 31, 19, 45, 1, 24, 27, 39, 1, 87, 1, 3, 49, 26, 19, 51, 1, 66, 40, 3, 1, 98, 23, 3, 33, 48, 1, 99, 21, 30, 35, 3, 25, 108, 1

Offset: 1

Wesley Ivan Hurt, Dec 22 2017

The number of partitions of n into 3 parts whose "middle" part divides n. - Wesley Ivan Hurt, Oct 21 2021

			a(12) = 10; the partitions of 12 into two distinct parts are (11,1), (10,2), (9,3), (8,4) and (7,5). 1 divides 11, 2 divides 10, 3 divides 9 and 4 divides 8, so the sum of the smaller parts gives 1 + 2 + 3 + 4 = 10.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
Index entries for sequences related to partitions

Cf. A297024, A001065, A023645, A000203, A080512.

with(numtheory):
a := n -> add( d, d = divisors(n) minus {floor((n+1)/2), n} ):
seq(a(n), n = 1..100); # Peter Bala, Jan 13 2021

Table[Sum[i (Floor[n/i] - Floor[(n - 1)/i]), {i, Floor[(n - 1)/2]}], {n, 100}]
f[n_] := Plus @@ Select[Divisors@n, 2 # < n &]; Array[f, 75] (* Robert G. Wilson v, Dec 23 2017 *)

A296955(n) = sumdiv(n,d,(d<(n/2))*d); \\ Antti Karttunen, Sep 25 2018

a(n) = Sum_{i=1..floor((n-1)/2)} i * (floor(n/i) - floor((n-1)/i)).
a(n) = the sum of the divisors < n/2. - Robert G. Wilson v, Dec 23 2017
a(n) = 1 iff n is an odd prime or n=4. - Robert G. Wilson v, Dec 23 2017
G.f.: Sum_{k>=1} k * x^(3*k) / (1 - x^k). - Ilya Gutkovskiy, May 30 2020
G.f.: Sum_{k >= 3} x^k/(1 - x^k)^2. Cf. A023645. - Peter Bala, Jan 13 2021
Faster converging g.f.: Sum_{n >= 1} q^(n*(n+2))*( n*q^(3*n+4) - (n + 1)*q^(2*n+2) - (n - 1)*q^(n+2) + n )/( (1 - q^n )*(1 - q^(n+2))^2 ). (In equation 1 in Arndt, after combining the two n = 0 summands to get t/(1 - t), apply the operator t*d/dt and then set t = q^2 and x = 1. Cf. A001065.) - Peter Bala, Jan 22 2021
a(n) = A000203(n) - A080512(n). - Ridouane Oudra, Aug 15 2024

More terms from Antti Karttunen, Sep 25 2018

A366968 a(n) = Sum_{k=3..n} floor(n/k).

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 6, 8, 10, 12, 13, 17, 18, 20, 23, 26, 27, 31, 32, 36, 39, 41, 42, 48, 50, 52, 55, 59, 60, 66, 67, 71, 74, 76, 79, 86, 87, 89, 92, 98, 99, 105, 106, 110, 115, 117, 118, 126, 128, 132, 135, 139, 140, 146, 149, 155, 158, 160, 161, 171, 172, 174, 179, 184

Offset: 1

Seiichi Manyama, Oct 30 2023

Column k=3 of A134867.
Partial sums of A023645.
Cf. A006218, A366969, A366970, A366971.

PARI
```
a(n) = sum(k=3, n, n\k);
```

from math import isqrt
def A366968(n): return -(s:=isqrt(n))**2+(sum(n//k for k in range(3,s+1))<<1)+n+(n>>1) if n>3 else int(n>2) # Chai Wah Wu, Oct 30 2023

G.f.: 1/(1-x) * Sum_{k>=1} x^(3*k)/(1-x^k) = 1/(1-x) * Sum_{k>=3} x^k/(1-x^k).

a(n) = A006218(n)-n-floor(n/2). - Chai Wah Wu, Oct 30 2023

A095374 One less than the number of divisors of 2*n + 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 3, 1, 1, 3, 3, 1, 3, 1, 1, 5, 1, 2, 3, 1, 3, 3, 1, 1, 5, 3, 1, 3, 1, 1, 5, 3, 1, 4, 1, 3, 3, 1, 3, 3, 3, 1, 5, 1, 1, 7, 1, 1, 3, 1, 3, 5, 3, 2, 3, 3, 1, 3, 1, 3, 7, 1, 1, 3, 3, 3, 5, 1, 1, 5, 3, 1, 3, 3, 1, 7, 1, 2, 5, 1, 5

Offset: 1

Labos Elemer, Jun 07 2004

Number of special divisors of A095372(n) with A095372(k) form.

			A095372(22) is divisible by {91, 9091, 90909091, 90909090909091, A095372(22)}, thus a(22)=5.
G.f.= x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + 3*x^7 + x^8 + x^9 + 3*x^10 + x^11 + 2*x^12 + ...

Michael De Vlieger, Table of n, a(n) for n = 1..5000
Gerzson Kéri, The factorization of compressed Chebyshev polynomials and other polynomials related to multiple-angle formulas, Annales Univ. Sci. Budapest (Hungary, 2022) Sect. Comp., Vol. 53, 93-108.

Cf. A000005, A001620, A095372, A095373, A023645, A099774.

g[x_]:=1+90*(100^x-1)/99 t=Table[1+90*(100^n-1)/99, {n, 1, 35}]; Do[Print[{w, is=Intersection[Divisors[g[w]], t], Length[is]}], {w, 1, 35}]
Table[DivisorSigma[0,2n+1],{n,90}]-1 (* Harvey P. Dale, Oct 31 2015 *)
Table[Sum[1 - Ceiling@ # + Floor@ # &[(n + i)/(n - i + 1)], {i, n}], {n, 87}] (* Michael De Vlieger, Feb 27 2017 *)

{a(n) = if( n<0, 0, numdiv(2*n + 1) - 1)} /* Michael Somos, Aug 30 2012 */

a(n) = A023645(2*n + 1) = A000005(2*n + 1) - 1.
a(n) = Sum_{i=1..n} 1-ceiling((n+i)/(n-i+1))+floor((n+i)/(n-i+1)). - Wesley Ivan Hurt, Feb 26 2017
O.g.f.: Sum_{n >= 1} x^n/(1 - x^(2*n+1)) = x/(1 - x) + Sum_{n >= 1} x^(2*n*(n+1))*(1 + x^(2*n+1))/(1 - x^(2*n+1)). - Peter Bala, Mar 04 2019
a(n) = A099774(n+1) - 1. - Bernard Schott, Mar 04 2019
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 3 + 3*log(2)) / 2, where gamma is Euler's constant (A001620). - Amiram Eldar, Mar 15 2025

A367588 Number of integer partitions of n with exactly two distinct parts, both appearing with the same multiplicity.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 3, 4, 5, 6, 5, 9, 6, 9, 10, 11, 8, 15, 9, 16, 14, 15, 11, 23, 14, 18, 18, 23, 14, 30, 15, 26, 22, 24, 22, 38, 18, 27, 26, 38, 20, 42, 21, 37, 36, 33, 23, 53, 27, 42, 34, 44, 26, 54, 34, 53, 38, 42, 29, 74, 30, 45, 49, 57, 40, 66, 33, 58, 46

Offset: 0

Gus Wiseman, Dec 01 2023

nonn

The Heinz numbers of these partitions are given by A268390.

			The a(3) = 1 through a(12) = 9 partitions (A = 10, B = 11):
  (21)  (31)  (32)  (42)    (43)  (53)    (54)      (64)    (65)  (75)
              (41)  (51)    (52)  (62)    (63)      (73)    (74)  (84)
                    (2211)  (61)  (71)    (72)      (82)    (83)  (93)
                                  (3311)  (81)      (91)    (92)  (A2)
                                          (222111)  (3322)  (A1)  (B1)
                                                    (4411)        (4422)
                                                                  (5511)
                                                                  (333111)
                                                                  (22221111)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

For any multiplicities we have A002133, ranks A007774.
For any number of distinct parts we have A047966, ranks A072774.
For distinct multiplicities we have A182473, ranks A367589.
These partitions have ranks A268390.
A000041 counts integer partitions, strict A000009.
A072233 counts partitions by number of parts.
A116608 counts partitions by number of distinct parts.
Cf. A023645, A091602, A116861, A181819, A243978, A367582.

Table[Sum[Floor[(d-1)/2],{d,Divisors[n]}],{n,30}]

G.f.: Sum_{i, j>0} x^(j*(2*i+1))/(1-x^j). - John Tyler Rascoe, Feb 04 2024

A072527 Number of values of k such that n divided by k leaves a remainder 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 4, 2, 2, 1, 5, 2, 2, 2, 4, 1, 5, 1, 4, 2, 2, 3, 6, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 7, 2, 4, 2, 4, 1, 5, 3, 6, 2, 2, 1, 9, 1, 2, 4, 5, 3, 5, 1, 4, 2, 6, 1, 9, 1, 2, 4, 4, 3, 5, 1, 8, 3, 2, 1, 9, 3, 2, 2, 6, 1, 9, 3, 4, 2, 2, 3, 9, 1, 4, 4, 7, 1, 5

Offset: 1

Amarnath Murthy, Aug 01 2002

For n > 3, the number of divisors of (n - 3) that are greater than 3; equivalently, those that are less than (n - 3)/3. - Peter Munn, May 18 2017

			a(15) = 3 as 15 divided by exactly three numbers 4, 6 and 12 leaves a remainder 3.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A001620, A023645, A072528.

A072527[n_] := If[n>6, DivisorSum[n-3, 1&, #>3&], 0];
Array[A072527, 150] (* Paolo Xausa, Jan 18 2024 *)

a(n) = sum(k=1, n-1, (n % k) == 3); \\ Michel Marcus, May 25 2017

a(n)=if(n>6, numdiv(n-3) - if(n%6==3, 3, if(n%6==2 || n%6==4, 1, 2)), 0) \\ Charles R Greathouse IV, May 27 2017

a(n) = tau(n-3)-1 if n is congruent to {2, 4} mod 6, tau(n-3)-2 if n is congruent to {0, 1, 5} mod 6, tau(n-3)-3 if n is congruent to 3 mod 6; n<>3. - Vladeta Jovovic, Aug 06 2002
G.f.: Sum_{k>0} x^(4*k+3)/(1-x^k). - Vladeta Jovovic, Dec 15 2002
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 17/6), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 18 2024

More terms from Matthew Conroy, Sep 09 2002

Incorrect comment deleted by Peter Munn, May 25 2017

cp's OEIS Frontend

A338653 Number of divisors of n which are greater than 9.

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A049992 a(n) is the number of arithmetic progressions of 3 or more positive integers, nondecreasing with sum n.

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A076984 Number of Fibonacci numbers that are divisors of the n-th Fibonacci number.

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A072528 Table T(n,k) read by rows, giving number of occurrences of the remainder k when n is divided by i=1,2,3,...,n.

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A086369 Number of factors over Q in the factorization of T_n(x) - 1 where T_n(x) is the Chebyshev polynomial of the first kind.

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Mathematica

PARI

PARI

PARI

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A296955 Sum of the smaller parts of the partitions of n into two distinct parts such that the smaller part divides the larger.

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Maple

Mathematica

PARI

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A366968 a(n) = Sum_{k=3..n} floor(n/k).

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