A023645 -id:A023645 - cp's OEIS frontend

A097986 Number of strict integer partitions of n with a part dividing all the other parts.

1, 1, 2, 2, 2, 4, 3, 5, 5, 7, 6, 12, 9, 13, 15, 20, 18, 28, 26, 37, 39, 47, 49, 71, 68, 85, 94, 117, 120, 159, 160, 201, 216, 257, 277, 348, 357, 430, 470, 562, 592, 720, 758, 901, 981, 1134, 1220, 1457, 1542, 1798, 1952, 2250, 2419, 2819, 3023, 3482, 3773, 4291

Offset: 1

Vladeta Jovovic, Oct 23 2004

If n > 0, we can assume such a part is the smallest. - Gus Wiseman, Apr 23 2021

Also the number of uniform (constant multiplicity) partitions of n containing 1, ranked by A367586. The strict case is A096765. The version without 1 is A329436. - Gus Wiseman, Dec 01 2023

			From _Gus Wiseman_, Dec 01 2023: (Start)
The a(1) = 1 through a(8) = 5 strict partitions with a part dividing all the other parts:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)
            (2,1)  (3,1)  (4,1)  (4,2)    (6,1)    (6,2)
                                 (5,1)    (4,2,1)  (7,1)
                                 (3,2,1)           (4,3,1)
                                                   (5,2,1)
The a(1) = 1 through a(8) = 5 uniform partitions containing 1:
  (1)  (11)  (21)   (31)    (41)     (51)      (61)       (71)
             (111)  (1111)  (11111)  (321)     (421)      (431)
                                     (2211)    (1111111)  (521)
                                     (111111)             (3311)
                                                          (11111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 500 terms from John Tyler Rascoe)

The non-strict version is A083710.
The case with no 1's is A098965.
The Heinz numbers of these partitions are A339563.
The strict complement is counted by A341450.
The version for "divisible by" instead of "dividing" is A343347.
The case where there is also a part divisible by all the others is A343378.
The case where there is no part divisible by all the others is A343381.
A000005 counts divisors.
A000009 counts strict partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
Cf. A083711, A098743, A130689, A200745, A264401, A338470, A343377, A343379, A343380.
Cf. A023645, A025147, A047966, A072774, A096765, A329436, A367586.

Take[ CoefficientList[ Expand[ Sum[x^k*Product[1 + x^(k*i), {i, 2, 62}], {k, 62}]], x], {2, 60}] (* Robert G. Wilson v, Nov 01 2004 *)
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Or@@Table[And@@IntegerQ/@(#/x), {x,#}]&]], {n,0,30}] (* Gus Wiseman, Apr 23 2021 *)

A_x(N) = {my(x='x+O('x^N)); Vec(sum(k=1,N,x^k*prod(i=2,N-k, (1+x^(k*i)))))}
A_x(50) \\ John Tyler Rascoe, Nov 19 2024

a(n) = Sum_{d|n} A025147(d-1).
G.f.: Sum_{k>=1} (x^k*Product_{i>=2} (1+x^(k*i))).
a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Jul 06 2025

More terms from Robert G. Wilson v, Nov 01 2004

Name shortened by Gus Wiseman, Apr 23 2021

A135539 Triangle read by rows: T(n,k) = number of divisors of n that are >= k.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Oct 30 2007

Row sums give A000203.

Left border is A000005.

			First few rows of the triangle:
  1;
  2, 1;
  2, 1, 1;
  3, 2, 1, 1;
  2, 1, 1, 1, 1;
  4, 3, 2, 1, 1, 1;
  2, 1, 1, 1, 1, 1, 1;
  4, 3, 2, 2, 1, 1, 1, 1;
  3, 2, 2, 1, 1, 1, 1, 1, 1;
  4, 3, 2, 2, 2, 1, 1, 1, 1, 1;
  2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  6, 5, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1;
  ...

Seiichi Manyama, Rows n = 1..140, flattened

Column k=1..10 give A000005, A032741, A023645, A321014, A338648, A338649, A338650, A338651, A338652, A338653.
Cf. A051731, A000203.
Cf. A001008, A001620, A002805.

with(numtheory);
f1:=proc(n) local d,s1,t1,t2,i;
d:=tau(n);
s1:=sort(divisors(n));
t1:=Array(1..n,0);
for i from 1 to d do t1[n-s1[i]+1]:=1; od:
t2:=PSUM(convert(t1,list));
[seq(t2[n+1-i],i=1..n)];
end proc;
for n from 1 to 15 do lprint(f1(n)); od: # N. J. A. Sloane, Nov 09 2018

T[n_, k_] := DivisorSum[n, Boole[# >= k]&];
Table[T[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 15 2023 *)

row(n) = my(d=divisors(n)); vector(n, k, #select(x->(x>=k), d)); \\ Michel Marcus, Jul 23 2022

Triangle read by rows, partial sums of A051731 starting from the right. A051731 as a lower triangular matrix times an all 1's lower triangular matrix.
From Seiichi Manyama, Jan 07 2023: (Start)
G.f. of column k: Sum_{j>=1} x^(k*j)/(1 - x^j).
G.f. of column k: Sum_{j>=k} x^j/(1 - x^j). (End)
Sum_{j=1..n} T(j, k) ~ n * (log(n) + 2*gamma - 1 - H(k-1)), where gamma is Euler's constant (A001620), and H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Jan 08 2024

Clearer definition from N. J. A. Sloane, Nov 09 2018

A014405 Number of arithmetic progressions of 3 or more positive integers, strictly increasing with sum n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 3, 0, 1, 5, 1, 0, 6, 0, 2, 7, 2, 0, 8, 2, 2, 9, 3, 0, 13, 0, 2, 11, 3, 4, 15, 0, 3, 13, 6, 0, 18, 0, 4, 20, 4, 0, 19, 2, 8, 18, 5, 0, 23, 6, 6, 20, 5, 0, 30, 0, 5, 25, 6, 7, 29, 0, 6, 24, 15, 0, 32, 0, 6, 34, 7, 4, 34, 0, 14, 31, 7, 0, 39, 9, 7, 31, 9, 0, 49, 5, 9, 33, 8, 10, 42, 0, 12

Offset: 1

Clark Kimberling

nonn

			E.g., 15 = 1+2+3+4+5 = 1+5+9 = 2+5+8 = 3+5+7 = 4+5+6.

Antti Karttunen, Table of n, a(n) for n = 1..12580 (first 1000 terms from Fausto A. C. Cariboni)
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.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
Augustine O. Munagi, Combinatorics of integer partitions in arithmetic progression, Integers 10(1) (2010), 73-82.
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.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000217, A007862, A014406, A014407, A023645, A047966, A049982, A049983, A049986, A049987, A049992, A129654, A240026, A240027, A307824, A320466, A325325, A325328.

a(n)= t=0; st=0; forstep(s=(n-3)\3,1,-1, st++; for(c=1,st, m=3; w=m*(s+c); while(wRick L. Shepherd, Aug 30 2006

G.f.: Sum_{k >= 3} x^t(k)/(x^t(k) - x^t(k-1) - x^k + 1) = Sum_{k >= 3} x^t(k)/((1 - x^k) * (1 - x^t(k-1))), where t(k) = k*(k+1)/2 = A000217(k) is the k-th triangular number [Graeme McRae]. - Petros Hadjicostas, Sep 29 2019

a(n) = A049992(n) - A023645(n). - Antti Karttunen, Feb 20 2023

A049986 a(n) is the number of arithmetic progressions of 4 or more positive integers, strictly increasing with sum n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 2, 1, 2, 0, 1, 2, 2, 1, 3, 0, 4, 0, 2, 1, 3, 4, 4, 0, 3, 1, 6, 0, 5, 0, 4, 6, 4, 0, 4, 2, 8, 2, 5, 0, 6, 6, 6, 2, 5, 0, 11, 0, 5, 5, 6, 7, 8, 0, 6, 2, 15, 0, 9, 0, 6, 10, 7, 4, 9, 0, 14, 5, 7, 0, 12, 9, 7, 3

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 A049986, 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, Combinatorics of integer partitions in arithmetic progression, Integers 10(1) (2010), 73-82.
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.

Cf. A014405, A014406, A049980, A049981, A049982, A049983, A049987 (partial sums), A049988, A049989, A049990, A049991, A049994, A127938, A321014.

\\ Needs also code from A014405 and A023645:
A049994(n) = (A014405(n) + A023645(n) - if(n%3, 0, n/3));
A321014(n) = (numdiv(n) - 3 + !!(n%2) + !!(n%3)); \\ From A321014.
A049986(n) = (A049994(n)-A321014(n)); \\ Antti Karttunen, Feb 20 2023

G.f.: Sum_{k >= 4} x^t(k)/(x^t(k) - x^t(k-1) - x^k + 1) = Sum_{k >= 4} x^t(k)/((1 - x^k)*(1 - x^t(k-1))), where t(k) = k*(k+1)/2 = A000217(k) is the k-th triangular number [Graeme McRae]. - Petros Hadjicostas, Sep 29 2019

a(n) = A049994(n) - A321014(n). [Listed by Sequence Machine and obviously true] - Antti Karttunen, Feb 20 2023

A319367 Triangle read by rows: T(n,k) is the number of simple vertex transitive graphs with n nodes and valency k, (0 <= k < n).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 2, 3, 3, 2, 1, 1, 1, 0, 2, 0, 3, 0, 2, 0, 1, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 1, 1, 4, 7, 11, 13, 13, 11, 7, 4, 1, 1, 1, 0, 1, 0, 3, 0, 4, 0, 3, 0, 1, 0, 1

Offset: 1

Andrew Howroyd, Sep 17 2018

			Triangle begins, n >= 1, 0 <= k < n:
  1;
  1, 1;
  1, 0, 1;
  1, 1, 1, 1;
  1, 0, 1, 0,  1;
  1, 1, 2, 2,  1,  1;
  1, 0, 1, 0,  1,  0,  1;
  1, 1, 2, 3,  3,  2,  1,  1;
  1, 0, 2, 0,  3,  0,  2,  0,  1;
  1, 1, 2, 3,  4,  4,  3,  2,  1,  1;
  1, 0, 1, 0,  2,  0,  2,  0,  1,  0,  1;
  1, 1, 4, 7, 11, 13, 13, 11,  7,  4,  1,  1;
  1, 0, 1, 0,  3,  0,  4,  0,  3,  0,  1,  0,  1;
  1, 1, 2, 3,  6,  6,  9,  9,  6,  6,  3,  2,  1,  1;
  1, 0, 3, 0,  8,  0, 12,  0, 12,  0,  8,  0,  3,  0, 1;
  1, 1, 3, 7, 16, 27, 40, 48, 48, 40, 27, 16,  7,  3, 1, 1;
  1, 0, 1, 0,  4,  0,  7,  0, 10,  0,  7,  0,  4,  0, 1, 0, 1;
  1, 1, 4, 7, 16, 24, 38, 45, 54, 54, 45, 38, 24, 16, 7, 4, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..496
B. D. McKay and G. F. Royle, The transitive graphs with at most 26 vertices, Ars Combin. 30 (1990), 161-176. (Annotated scanned copy)
Gordon Royle, Transitive Graphs.
Eric Weisstein's World of Mathematics, Vertex-Transitive Graph.

Columns k=2..12 (even n only for odd k) are A023645, A023646, A023647, A023640, A023641, A023642, A023643, A023644, A023637, A023638, A023639.
Row sums are A006799.
Cf. A319368, A319372.

A338648 Number of divisors of n which are greater than 4.

Original entry on oeis.org

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

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=5 of A135539.

Cf. A000005, A001620, A023645, A032741, A083040, A321014, A338649, A338650, A338651, A338652, A338653.

Table[DivisorSum[n, 1 &, # > 4 &], {n, 1, 110}]
nmax = 110; CoefficientList[Series[Sum[x^(5 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, 5, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Drop[#, 1] &

a(n) = sumdiv(n, d, d>4); \\ Michel Marcus, Apr 22 2021; corrected Jun 13 2022

my(N=100, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=5, N, x^k/(1-x^k)))) \\ Seiichi Manyama, Jan 07 2023

G.f.: Sum_{k>=1} x^(5*k) / (1 - x^k).
L.g.f.: -log( Product_{k>=5} (1 - x^k)^(1/k) ).
a(n) = A000005(n) - A083040(n).
G.f.: Sum_{k>=5} x^k/(1 - x^k). - Seiichi Manyama, Jan 07 2023
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 37/12), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 08 2024

a(1)-a(4) prepended by David A. Corneth, Jun 13 2022

A338649 Number of divisors of n which are greater than 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, 5, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 6, 2, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 7, 1, 2, 4, 4, 2, 5, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 3, 5, 1, 6, 3, 2, 1, 8, 2, 2, 2, 5, 1, 8, 3, 3, 2, 2, 2, 8, 1, 4, 4, 5, 1, 5, 1, 5, 5, 2, 1, 8, 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=6 of A135539.

Cf. A000005, A001620, A023645, A032741, A321014, A338648, A338650, A338651, A338652, A338653.

Table[DivisorSum[n, 1 &, # > 5 &], {n, 1, 110}]
nmax = 110; CoefficientList[Series[Sum[x^(6 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, 6, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Drop[#, 1] &

a(n) = sumdiv(n, d, d>5); \\ Michel Marcus, Apr 22 2021

my(N=100, x='x+O('x^N)); concat([0, 0, 0, 0, 0], Vec(sum(k=6, N, x^k/(1-x^k)))) \\ Seiichi Manyama, Jan 07 2023

G.f.: Sum_{k>=1} x^(6*k) / (1 - x^k).
L.g.f.: -log( Product_{k>=6} (1 - x^k)^(1/k) ).
G.f.: Sum_{k>=6} x^k/(1 - x^k). - Seiichi Manyama, Jan 07 2023
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 197/60), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 08 2024

a(1)-a(5) prepended by David A. Corneth, Jun 13 2022

A338650 Number of divisors of n which are greater than 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 4, 2, 5, 2, 2, 1, 6, 1, 2, 4, 4, 2, 4, 1, 3, 2, 5, 1, 7, 1, 2, 3, 3, 3, 4, 1, 6, 3, 2, 1, 7, 2, 2, 2, 5, 1, 7, 3, 3, 2, 2, 2, 7, 1, 4, 4, 5, 1, 4, 1, 5, 5, 2, 1, 7, 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=7 of A135539.

Cf. A000005, A001620, A023645, A032741, A321014, A338648, A338649, A338651, A338652, A338653.

Table[DivisorSum[n, 1 &, # > 6 &], {n, 1, 110}]
nmax = 110; CoefficientList[Series[Sum[x^(7 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, 7, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Drop[#, 1] &

a(n) = sumdiv(n, d, d>6); \\ Michel Marcus, Apr 22 2021

my(N=100, x='x+O('x^N)); concat([0, 0, 0, 0, 0, 0], Vec(sum(k=7, N, x^k/(1-x^k)))) \\ Seiichi Manyama, Jan 07 2023

G.f.: Sum_{k>=1} x^(7*k) / (1 - x^k).
L.g.f.: -log( Product_{k>=7} (1 - x^k)^(1/k) ).
G.f.: Sum_{k>=7} x^k/(1 - x^k). - Seiichi Manyama, Jan 07 2023
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 69/20), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 08 2024

a(1)-a(6) prepended by David A. Corneth, Jun 13 2022

A338651 Number of divisors of n which are greater than 7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 6, 1, 2, 3, 4, 2, 4, 1, 3, 2, 4, 1, 7, 1, 2, 3, 3, 2, 4, 1, 6, 3, 2, 1, 6, 2, 2, 2, 5, 1, 7, 2, 3, 2, 2, 2, 7, 1, 3, 4, 5, 1, 4, 1, 5, 4, 2, 1, 7, 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=8 of A135539.

Cf. A000005, A001620, A023645, A032741, A321014, A338648, A338649, A338650, A338652, A338653.

Table[DivisorSum[n, 1 &, # > 7 &], {n, 1, 110}]
nmax = 110; CoefficientList[Series[Sum[x^(8 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, 8, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Drop[#, 1] &

a(n) = sumdiv(n, d, d>7); \\ Michel Marcus, Apr 22 2021

my(N=100, x='x+O('x^N)); concat([0, 0, 0, 0, 0, 0, 0], Vec(sum(k=8, N, x^k/(1-x^k)))) \\ Seiichi Manyama, Jan 07 2023

G.f.: Sum_{k>=1} x^(8*k) / (1 - x^k).
L.g.f.: -log( Product_{k>=8} (1 - x^k)^(1/k) ).
G.f.: Sum_{k>=8} x^k/(1 - x^k). - Seiichi Manyama, Jan 07 2023
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 503/140), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 08 2024

a(1)-a(7) prepended by David A. Corneth, Jun 13 2022

A338652 Number of divisors of n which are greater than 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 3, 1, 3, 1, 3, 3, 2, 1, 4, 1, 3, 2, 3, 1, 4, 2, 3, 2, 2, 1, 6, 1, 2, 3, 3, 2, 4, 1, 3, 2, 4, 1, 6, 1, 2, 3, 3, 2, 4, 1, 5, 3, 2, 1, 6, 2, 2, 2, 4, 1, 7, 2, 3, 2, 2, 2, 6, 1, 3, 4, 5, 1, 4, 1, 4, 4, 2, 1, 7, 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=9 of A135539.

Cf. A000005, A001620, A023645, A032741, A321014, A338648, A338649, A338650, A338651, A338653.

Table[DivisorSum[n, 1 &, # > 8 &], {n, 1, 110}]
nmax = 110; CoefficientList[Series[Sum[x^(9 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, 9, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Drop[#, 1] &

a(n) = sumdiv(n, d, d>8); \\ Michel Marcus, Apr 22 2021

my(N=100, x='x+O('x^N)); concat([0, 0, 0, 0, 0, 0, 0, 0], Vec(sum(k=9, N, x^k/(1-x^k)))) \\ Seiichi Manyama, Jan 07 2023

G.f.: Sum_{k>=1} x^(9*k) / (1 - x^k).
L.g.f.: -log( Product_{k>=9} (1 - x^k)^(1/k) ).
G.f.: Sum_{k>=9} x^k/(1 - x^k). - Seiichi Manyama, Jan 07 2023
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 1041/280), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 08 2024

a(1)-a(8) prepended by David A. Corneth, Jun 13 2022

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