A115584 -id:A115584 - cp's OEIS frontend

A276078 Numbers n in whose prime factorization no exponent of any prime(k) exceeds k.

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 98, 99, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 117, 118, 119, 121

Offset: 1

Antti Karttunen, Aug 18 2016

nonn

Numbers not divisible by p^(1+A000720(p)) for any prime p, where A000720(p) gives the index of prime p: 1 for 2, 2 for 3, 3 for 5, and so on.
Also Heinz numbers of integer partitions where the multiplicity of i does not exceed i for any i (A052335). Differs from A048103 in lacking {625, 1250, 1875, 3750, 4375, 5625, 6875, 8125, 8750, ...}. - Gus Wiseman, Mar 09 2019
Asymptotic density is Product_{i>=1} 1-prime(i)^(-1-i) = 0.72102334... - Amiram Eldar, Oct 20 2020

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

Positions of zeros in A276077.
Complement: A276079.
Sequence A276076 sorted into ascending order.
Cf. A000040, A000720.
Subsequence of A048103 from which it differs for the first time at n=451, where a(451) = 626, while A048103(451) = 625, a value missing from here.
Cf. A052335, A056239, A062457, A087153, A109298, A112798, A115584, A117144, A118914, A124010, A197987, A276078.
Cf. A324524, A324525, A324571, A324587, A324588.

Select[Range@ 121, Or[# == 1, AllTrue[FactorInteger[#], PrimePi[#1] >= #2 & @@ # &]] &] (* Michael De Vlieger, Jun 24 2017 *)

isok(n) = my(f=factor(n)); for (k=1, #f~, if (f[k, 2] > primepi(f[k, 1]), return(0))); return (1); \\ Michel Marcus, Jun 24 2017

is(n) = {my(t=1);forprime(p = 2, , t++; pp = p^t; if(n%pp==0, return(0)); if(pp > n, return(1)))} \\ David A. Corneth, Jun 24 2017

upto(n) = {my(v = vector(n,i,1), t=1, res=List()); forprime(p=2, , t++; pp = p^t; if(pp>n, break); for(i=1, n\pp, v[pp*i] = 0)); for(i=1, n, if(v[i]==1, listput(res, i))); res} \\ David A. Corneth, Jun 24 2017

from sympy import factorint, primepi
def ok(n):
    f = factorint(n)
    return all(f[i] <= primepi(i) for i in f)
print([n for n in range(1, 151) if ok(n)]) # Indranil Ghosh, Jun 24 2017

A114640 Number of partitions of n such that the set of parts and the set of multiplicities of parts are equal.

Original entry on oeis.org

1, 1, 0, 0, 2, 1, 1, 0, 1, 1, 3, 2, 3, 3, 5, 0, 4, 5, 2, 3, 8, 6, 5, 10, 9, 9, 16, 14, 12, 16, 17, 10, 17, 15, 16, 19, 35, 17, 34, 37, 40, 31, 54, 36, 60, 61, 58, 63, 88, 58, 88, 87, 91, 84, 115, 93, 116, 108, 115, 130, 190, 143, 165, 214, 219, 200, 255, 240

Offset: 0

Vladeta Jovovic, Feb 18 2006

nonn

The Heinz numbers of these partitions are given by A109297. - Gus Wiseman, Apr 02 2019

			From _Gus Wiseman_, Apr 02 2019: (Start)
The initial terms count the following integer partitions:
   0: ()
   1: (1)
   4: (22)
   4: (211)
   5: (221)
   6: (3111)
   8: (41111)
   9: (333)
  10: (511111)
  10: (3331)
  10: (322111)
  11: (332111)
  11: (322211)
  12: (6111111)
  12: (4221111)
  12: (33222)
  13: (33322)
  13: (333211)
  13: (332221)
  14: (71111111)
  14: (52211111)
  14: (4421111)
  14: (4222211)
  14: (333221)
(End)

Cf. A052335, A087153, A109297, A114639, A115584, A117144, A276429, A324572, A325132, A336031.

Table[Length[Select[IntegerPartitions[n],Union[#]==Union[Length/@Split[#]]&]],{n,0,30}] (* Gus Wiseman, Apr 02 2019 *)

More terms from Alois P. Heinz, Aug 09 2016

A087153 Number of partitions of n into nonsquares.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 3, 5, 5, 8, 9, 13, 15, 20, 24, 30, 37, 47, 55, 71, 83, 103, 123, 151, 178, 218, 257, 310, 366, 440, 515, 617, 722, 857, 1003, 1184, 1380, 1625, 1889, 2214, 2570, 3000, 3472, 4042, 4669, 5414, 6244, 7221, 8303, 9583, 10998, 12655, 14502

Offset: 0

Reinhard Zumkeller, Aug 21 2003

nonn

Also, number of partitions of n where there are fewer than k parts equal to k for all k. - Jon Perry and Vladeta Jovovic, Aug 04 2004. E.g. a(8)=5 because we have 8=6+2=5+3=4+4=3+3+2.
Convolution of A276516 and A000041. - Vaclav Kotesovec, Dec 30 2016
From Gus Wiseman, Apr 02 2019: (Start)
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The Heinz numbers of the integer partitions described in Perry and Jovovic's comment are given by A325128, while the Heinz numbers of the integer partitions described in the name are given by A325129. In the former case, the first 10 terms count the following integer partitions:
  ()  (2)  (3)  (4)  (5)   (6)   (7)   (8)    (9)
                     (32)  (33)  (43)  (44)   (54)
                           (42)  (52)  (53)   (63)
                                       (62)   (72)
                                       (332)  (432)
while in the latter case they count the following:
  ()  (2)  (3)  (22)  (5)   (6)    (7)    (8)     (63)
                      (32)  (33)   (52)   (53)    (72)
                            (222)  (322)  (62)    (333)
                                          (332)   (522)
                                          (2222)  (3222)
(End)

			n=7: 2+5 = 2+2+3 = 7: a(7)=3;
n=8: 2+6 = 2+2+2+2 = 2+3+3 = 3+5 = 8: a(8)=5;
n=9: 2+7 = 2+2+5 = 2+2+2+3 = 3+3+3 = 3+6: a(9)=5.

G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004. See page 48.

T. D. Noe and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Daniel I. A. Cohen, PIE-sums: a combinatorial tool for partition theory. J. Combin. Theory Ser. A 31 (1981), no. 3, 223--236. MR0635367 (82m:10026). See Cor. 5. - _N. J. A. Sloane_, Mar 27 2012
James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.

Cf. A087154, A001156, A000009, A000037, A052335 (<=k parts of k).
Cf. A115584, A172151, A225044, A264393, A276516.
Cf. A033461, A114639, A117144, A276429, A324572, A324588, A325128, A325129.

a087153 = p a000037_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Apr 25 2013

g:=product((1-x^(i^2))/(1-x^i),i=1..70):gser:=series(g,x=0,60):seq(coeff(gser,x^n),n=1..53); # Emeric Deutsch, Feb 09 2006

nn=54; CoefficientList[ Series[ Product[ Sum[x^(i*j), {j, 0, i - 1}], {i, 1, nn}], {x, 0, nn}], x] (* Robert G. Wilson v, Aug 05 2004 *)
nmax = 100; CoefficientList[Series[Product[(1 - x^(k^2))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 29 2016 *)

first(n)=my(x='x+O('x^(n+1))); Vec(prod(m=1,sqrtint(n), (1-x^m^2)/(1-x^m))*prod(m=sqrtint(n)+1,n,1/(1-x^m))) \\ Charles R Greathouse IV, Aug 28 2016

G.f.: Product_{m>0} (1-x^(m^2))/(1-x^m). - Vladeta Jovovic, Aug 21 2003
a(n) = (1/n)*Sum_{k=1..n} (A000203(k)-A035316(k))*a(n-k), a(0)=1. - Vladeta Jovovic, Aug 21 2003
G.f.: Product_{i>=1} (Sum_{j=0..i-1} x^(i*j)). - Jon Perry, Jul 26 2004
a(n) ~ exp(Pi*sqrt(2*n/3) - 3^(1/4) * Zeta(3/2) * n^(1/4) / 2^(3/4) - 3*Zeta(3/2)^2/(32*Pi)) * sqrt(Pi) / (2^(3/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Dec 30 2016

Zeroth term added by Franklin T. Adams-Watters, Jan 25 2010

A114639 Number of partitions of n such that the set of parts and the set of multiplicities of parts are disjoint.

Original entry on oeis.org

1, 0, 2, 2, 2, 3, 5, 4, 7, 7, 13, 16, 19, 23, 33, 34, 44, 58, 63, 80, 101, 112, 139, 171, 196, 234, 288, 328, 394, 478, 545, 658, 777, 881, 1050, 1236, 1414, 1666, 1936, 2216, 2592, 3018, 3428, 3992, 4604, 5243, 6069, 6986, 7951, 9139, 10447, 11892, 13625

Offset: 0

Vladeta Jovovic, Feb 18 2006

nonn

The Heinz numbers of these partitions are given by A325131. - Gus Wiseman, Apr 02 2019

			From _Gus Wiseman_, Apr 02 2019: (Start)
The a(2) = 2 through a(9) = 7 partitions:
  (2)   (3)    (4)     (5)      (6)       (7)        (8)         (9)
  (11)  (111)  (1111)  (32)     (33)      (43)       (44)        (54)
                       (11111)  (42)      (52)       (53)        (63)
                                (222)     (1111111)  (62)        (72)
                                (111111)             (2222)      (432)
                                                     (3311)      (222111)
                                                     (11111111)  (111111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A052335, A087153, A114640, A115584, A117144, A276429, A324572, A325130, A325131, A336032.

b:= proc(n, i, p, m) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1, p, select(x-> x x<=n-i*j, p union {i}),
         select(x-> x b(n$2, {}$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 09 2016

b[n_, i_, p_, m_] := b[n, i, p, m] = If[n == 0, 1, If[i<1, 0, b[n, i-1, p, Select[m, #Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Intersection[#,Length/@Split[#]]=={}&]],{n,0,30}] (* Gus Wiseman, Apr 02 2019 *)

a(0)=1 prepended and more terms from Alois P. Heinz, Aug 09 2016

A117144 Partitions of n in which each part k occurs at least k times.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 8, 9, 10, 12, 15, 16, 19, 21, 25, 28, 32, 34, 41, 46, 51, 55, 64, 70, 79, 86, 97, 106, 119, 129, 146, 159, 175, 190, 214, 232, 256, 277, 306, 334, 367, 394, 434, 472, 515, 556, 607, 654, 714, 770, 836, 901, 978, 1048, 1140, 1226, 1322

Offset: 0

Emeric Deutsch, Mar 06 2006

nonn

The Heinz numbers of these integer partitions are given by A324525. - Gus Wiseman, Mar 09 2019

			a(9)=5 because we have [3,3,3], [2,2,2,2,1], [2,2,2,1,1,1], [2,2,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1].
From _Gus Wiseman_, Mar 09 2019: (Start)
The a(1) = 1 through a(9) = 5 integer partitions:
  1  11  111  22    221    222     2221     2222      333
              1111  11111  2211    22111    22211     22221
                           111111  1111111  221111    222111
                                            11111111  2211111
                                                      111111111
(End)

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

Cf. A001462, A003114, A006141, A033461, A039900, A047993, A052335, A064174, A090858, A114638, A115584, A276078, A280204.

Cf. A324518, A324520, A324525, A324572.

g:=product((1-x^k+x^(k^2))/(1-x^k),k=1..100): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=0..66);
# second Maple program:
b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +add(b(n-i*j, i-1), j=i..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, Dec 28 2016

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + Sum[b[n-i*j, i-1], {j, i, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Feb 03 2017, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],And@@Table[Count[#,i]>=i,{i,Union[#]}]&]],{n,0,30}] (* Gus Wiseman, Mar 09 2019 *)
nmax = 100; CoefficientList[Series[Product[(1-x^k+x^(k^2))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 28 2024 *)

G.f.: Product_{k>=1} (1-x^k+x^(k^2))/(1-x^k).

A276429 Number of partitions of n containing no part i of multiplicity i.

Original entry on oeis.org

1, 0, 2, 2, 3, 5, 8, 9, 16, 19, 29, 36, 53, 65, 92, 115, 154, 195, 257, 318, 419, 516, 663, 821, 1039, 1277, 1606, 1963, 2441, 2978, 3675, 4454, 5469, 6603, 8043, 9688, 11732, 14066, 16963, 20260, 24310, 28953, 34586, 41047, 48857, 57802, 68528, 80862, 95534, 112388, 132391

Offset: 0

Emeric Deutsch, Sep 19 2016

nonn

The Heinz numbers of these partitions are given by A325130. - Gus Wiseman, Apr 02 2019

			a(4) = 3 because we have [1,1,1,1], [1,1,2], and [4]; the partitions [1,3], [2,2] do not qualify.
From _Gus Wiseman_, Apr 02 2019: (Start)
The a(2) = 2 through a(7) = 9 partitions:
  (2)   (3)    (4)     (5)      (6)       (7)
  (11)  (111)  (211)   (32)     (33)      (43)
               (1111)  (311)    (42)      (52)
                       (2111)   (222)     (511)
                       (11111)  (411)     (3211)
                                (3111)    (4111)
                                (21111)   (31111)
                                (111111)  (211111)
                                          (1111111)
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..12782 (terms 0..5000 from Alois P. Heinz)

Cf. A052335, A087153, A114639, A115584, A117144, A276427, A324572, A325130, A325131, A336269.

g := product(1/(1-x^i)-x^(i^2), i = 1 .. 100): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 0 .. 50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(i=j, 0, b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 19 2016

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[If[i == j, x, 1]*b[n - i*j, i - 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n][[1]], {n, 0, 60}] (* Jean-François Alcover, Nov 28 2016 after Alois P. Heinz's Maple code for A276427 *)
Table[Length[Select[IntegerPartitions[n],And@@Table[Count[#,i]!=i,{i,Union[#]}]&]],{n,0,30}] (* Gus Wiseman, Apr 02 2019 *)

a(n) = A276427(n,0).

G.f.: g(x) = Product_{i>=1} (1/(1-x^i) - x^{i^2}).

A054744 p-full numbers: numbers such that if any prime p divides it, then so does p^p.

Original entry on oeis.org

1, 4, 8, 16, 27, 32, 64, 81, 108, 128, 216, 243, 256, 324, 432, 512, 648, 729, 864, 972, 1024, 1296, 1728, 1944, 2048, 2187, 2592, 2916, 3125, 3456, 3888, 4096, 5184, 5832, 6561, 6912, 7776, 8192, 8748, 10368, 11664, 12500, 13824, 15552, 15625, 16384

Offset: 1

James Sellers, Apr 22 2000

A027748(a(n),k) <= A124010(a(n),k), 1<=k<=A001221(a(n)). [Reinhard Zumkeller, Apr 28 2012]

Heinz numbers of integer partitions where the multiplicity of each part k is at least prime(k). These partitions are counted by A325132. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Apr 02 2019

			8 is an element because 8 = 2^3 and 2<=3, while 25 is not an element because 25 = 5^2 and 5>2.
From _Gus Wiseman_, Apr 02 2019: (Start)
The sequence of terms together with their prime indices begins:
    1: {}
    4: {1,1}
    8: {1,1,1}
   16: {1,1,1,1}
   27: {2,2,2}
   32: {1,1,1,1,1}
   64: {1,1,1,1,1,1}
   81: {2,2,2,2}
  108: {1,1,2,2,2}
  128: {1,1,1,1,1,1,1}
  216: {1,1,1,2,2,2}
  243: {2,2,2,2,2}
  256: {1,1,1,1,1,1,1,1}
  324: {1,1,2,2,2,2}
  432: {1,1,1,1,2,2,2}
  512: {1,1,1,1,1,1,1,1,1}
  648: {1,1,1,2,2,2,2}
  729: {2,2,2,2,2,2}
  864: {1,1,1,1,1,2,2,2}
  972: {1,1,2,2,2,2,2}
(End)

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A048103, A001694, A036966.
Cf. A100716.
Cf. A056239, A109298, A112798, A115584, A117144, A124010, A276078, A324525, A324571, A325127, A325128, A325130, A325131, A325132.

a054744 n = a054744_list !! (n-1)
a054744_list = filter (\x -> and $
   zipWith (<=) (a027748_row x) (map toInteger $ a124010_row x)) [1..]
-- Reinhard Zumkeller, Apr 28 2012

Select[Range[1000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k>=p]&] (* Gus Wiseman, Apr 02 2019 *)

If n = Product p_i^e_i then p_i<=e_i for all i.

Sum_{n>=1} 1/a(n) = Product_{p prime} 1 + 1/(p^(p-1)*(p-1)) = 1.58396891058853238595.... - Amiram Eldar, Oct 24 2020

A325128 Numbers in whose prime factorization the exponent of prime(k) is less than k for all prime indices k.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 127, 129, 131, 133, 137, 139, 141

Offset: 1

Gus Wiseman, Apr 01 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions where each part k appears fewer than k times. Such partitions are counted by A087153.
The asymptotic density of this sequence is Product_{k>=1} (1 - 1/prime(k)^k) = 0.44070243286030291209... - Amiram Eldar, Feb 02 2021

			The sequence of terms together with their prime indices begins:
   1: {}
   3: {2}
   5: {3}
   7: {4}
  11: {5}
  13: {6}
  15: {2,3}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  29: {10}
  31: {11}
  33: {2,5}
  35: {3,4}
  37: {12}
  39: {2,6}
  41: {13}
  43: {14}
  47: {15}
  49: {4,4}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A056239, A062457, A087153, A109298, A112798, A115584, A118914, A276078.

Cf. A324524, A324525, A324571, A325127, A325130.

Select[Range[100],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k

A325127 Numbers in whose prime factorization the exponent of prime(k) is greater than k for all prime indices k.

Original entry on oeis.org

1, 4, 8, 16, 27, 32, 64, 81, 108, 128, 216, 243, 256, 324, 432, 512, 625, 648, 729, 864, 972, 1024, 1296, 1728, 1944, 2048, 2187, 2500, 2592, 2916, 3125, 3456, 3888, 4096, 5000, 5184, 5832, 6561, 6912, 7776, 8192, 8748, 10000, 10368, 11664, 12500, 13824, 15552

Offset: 1

Gus Wiseman, Apr 01 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions where each part k appears more than k times. Such partitions are counted by A115584.

			The sequence of terms together with their prime indices begins:
    1: {}
    4: {1,1}
    8: {1,1,1}
   16: {1,1,1,1}
   27: {2,2,2}
   32: {1,1,1,1,1}
   64: {1,1,1,1,1,1}
   81: {2,2,2,2}
  108: {1,1,2,2,2}
  128: {1,1,1,1,1,1,1}
  216: {1,1,1,2,2,2}
  243: {2,2,2,2,2}
  256: {1,1,1,1,1,1,1,1}
  324: {1,1,2,2,2,2}
  432: {1,1,1,1,2,2,2}
  512: {1,1,1,1,1,1,1,1,1}
  625: {3,3,3,3}
  648: {1,1,1,2,2,2,2}
  729: {2,2,2,2,2,2}
  864: {1,1,1,1,1,2,2,2}
  972: {1,1,2,2,2,2,2}

Amiram Eldar, Table of n, a(n) for n = 1..2847 (terms up to 10^12)

Cf. A000720, A056239, A062457, A109298, A112798, A115584, A118914, A276078.

Cf. A324524, A324525, A324571, A325128, A325130.

Select[Range[1000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k>PrimePi[p]]&]
With[{k = 4}, m = Prime[k]^(k + 1); s = {}; Do[p = Prime[i]; AppendTo[s, Join[{1}, p^Range[i + 1, Floor[Log[p, m]]]]], {i, 1, k}]; Union @ Select[Times @@@ Tuples[s], # <= m &]] (* Amiram Eldar, Oct 24 2020 *)

Sum_{n>=1} 1/a(n) = Product_{k>=1} 1 + 1/(prime(k)^k * (prime(k)-1)) = 1.58661114052385082598.... - Amiram Eldar, Oct 24 2020

A319758 Expansion of Product_{k>=1} 1/(1 - Sum_{j=1..k} x^(j*k)).

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 15, 20, 34, 48, 76, 103, 165, 222, 335, 461, 683, 919, 1352, 1813, 2611, 3519, 4985, 6651, 9408, 12501, 17401, 23165, 32009, 42312, 58241, 76748, 104725, 138017, 187155, 245521, 332135, 434536, 584023, 763799, 1022507, 1332549, 1779534, 2314437, 3077540, 3999825

Offset: 0

Ilya Gutkovskiy, Sep 27 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A052335, A115584.

a:=series(mul(1/(1-add(x^(j*k),j=1..k)),k=1..100),x=0,46): seq(coeff(a,x,n),n=0..45); # Paolo P. Lava, Apr 02 2019

nmax = 45; CoefficientList[Series[Product[1/(1 - Sum[x^(j k), {j, 1, k}]), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 - x^k)/(1 - 2*x^k + x^(k*(k+1))).
From Vaclav Kotesovec, Sep 27 2018: (Start)
a(n) ~ c * phi^(n/2), where
c = 188.4773924093125890061786423020365148584841831715... if n is even
c = 187.5693962190327254176348797865060646998844995050... if n is odd
phi = A001622 = (1+sqrt(5))/2 is the golden ratio. (End)

cp's OEIS Frontend

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