A001156 -id:A001156 - cp's OEIS frontend

A324525 Numbers divisible by prime(k)^k for each prime index k.

1, 2, 4, 8, 9, 16, 18, 27, 32, 36, 54, 64, 72, 81, 108, 125, 128, 144, 162, 216, 243, 250, 256, 288, 324, 432, 486, 500, 512, 576, 625, 648, 729, 864, 972, 1000, 1024, 1125, 1152, 1250, 1296, 1458, 1728, 1944, 2000, 2048, 2187, 2250, 2304, 2401, 2500, 2592

Offset: 1

Gus Wiseman, Mar 08 2019

nonn

These are a kind of self-describing numbers (cf. A001462, A304679).
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 prime signature of a number is the multiset of multiplicities (or exponents) in its prime factorization.
Also Heinz numbers of integer partitions where the multiplicity of k is at least k (A117144). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of terms together with their prime indices begins as follows. For example, 36 = prime(1) * prime(1) * prime(2) * prime(2) is a term because the prime multiplicities are {2,2}, which are greater than or equal to the prime indices {1,2}.
    1: {}
    2: {1}
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   16: {1,1,1,1}
   18: {1,2,2}
   27: {2,2,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   54: {1,2,2,2}
   64: {1,1,1,1,1,1}
   72: {1,1,1,2,2}
   81: {2,2,2,2}
  108: {1,1,2,2,2}
  125: {3,3,3}
  128: {1,1,1,1,1,1,1}

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

Cf. A001156, A033461, A056239, A062457, A112798, A117144, A118914 (prime signature), A124010, A181819, A276078, A304679.
Cf. A109298, A324524, A324571, A324572, A324587, A324588.
Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.

q:= n-> andmap(i-> i[2]>=numtheory[pi](i[1]), ifactors(n)[2]):
select(q, [$1..3000])[];  # Alois P. Heinz, Mar 08 2019

Select[Range[1000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k>=PrimePi[p]]&]
seq[max_] := Module[{ps = {2}, p, s = {1}, s1, s2, emax}, While[ps[[-1]]^Length[ps] < max, AppendTo[ps, NextPrime[ps[[-1]]]]]; Do[p = ps[[k]]; emax = Floor[Log[p, max]]; s1 = Join[{1}, p^Range[k, emax]]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= max &]; s = Union[s, s2], {k, 1, Length[ps]}]; s]; seq[3000] (* Amiram Eldar, Nov 23 2020 *)

Closed under multiplication.

Sum_{n>=1} 1/a(n) = Product_{k>=1} 1 + 1/(prime(k)^(k-1) * (prime(k)-1)) = 2.35782843100111139159... - Amiram Eldar, Nov 23 2020

A324524 Numbers where every prime index divides its multiplicity in the prime factorization. Numbers divisible by a power of prime(k)^k for each prime index k.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 18, 32, 36, 64, 72, 81, 125, 128, 144, 162, 250, 256, 288, 324, 500, 512, 576, 648, 729, 1000, 1024, 1125, 1152, 1296, 1458, 2000, 2048, 2250, 2304, 2401, 2592, 2916, 4000, 4096, 4500, 4608, 4802, 5184, 5832, 6561, 8000, 8192, 9000, 9216

Offset: 1

Gus Wiseman, Mar 07 2019

nonn

These are a kind of self-describing numbers (cf. A001462, A304679).
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 prime signature of a number is the multiset of multiplicities (or exponents) in its prime factorization.
Also Heinz numbers of integer partitions in which every part divides its multiplicity (counted by A001156). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Also products of elements of A062457.

			The sequence of terms together with their prime indices begins as follows. For example, we have 18: {1,2,2} because 18 = prime(1) * prime(2) * prime(2).
    1: {}
    2: {1}
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   16: {1,1,1,1}
   18: {1,2,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   64: {1,1,1,1,1,1}
   72: {1,1,1,2,2}
   81: {2,2,2,2}
  125: {3,3,3}
  128: {1,1,1,1,1,1,1}
  144: {1,1,1,1,2,2}
  162: {1,2,2,2,2}
  250: {1,3,3,3}
  256: {1,1,1,1,1,1,1,1}

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

Cf. A001156, A033461, A056239, A062457, A066328, A072873, A112798, A118914 (prime signature), A124010, A181819, A276078, A304679.
Cf. A109298, A324525, A324570, A324571, A324572, A324587, A324588.
Range of values of A090884.
Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.

q:= n-> andmap(i-> irem(i[2], numtheory[pi](i[1]))=0, ifactors(n)[2]):
select(q, [$1..10000])[];  # Alois P. Heinz, Mar 08 2019

Select[Range[1000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>Divisible[k,PrimePi[p]]]&]
v = Join[{1}, Prime[(r = Range[10])]^r]; n = Length[v]; vmax = 10^4; s = {1}; Do[v1 = v[[k]]; rmax = Floor[Log[v1, vmax]]; s1 = v1^Range[0, rmax]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= vmax &]; s = Union[s, s2], {k, 2, n}]; Length[s] (* Amiram Eldar, Sep 30 2020 *)

Closed under multiplication.

Sum_{n>=1} 1/a(n) = Product_{k>=1} 1/(1-prime(k)^(-k)) = 2.26910478689594012492... - Amiram Eldar, Sep 30 2020

A324571 Numbers whose ordered prime signature is equal to the set of distinct prime indices in decreasing order.

Original entry on oeis.org

1, 2, 9, 12, 40, 112, 125, 352, 360, 675, 832, 1008, 2176, 2401, 3168, 3969, 4864, 7488, 11776, 14000, 19584, 29403, 29696, 43776, 44000, 63488, 75600, 104000, 105984, 123201, 151552, 161051, 214375, 237600, 267264, 272000, 335872, 496125, 561600, 571392, 608000

Offset: 1

Gus Wiseman, Mar 08 2019

nonn

These are a kind of self-describing numbers (cf. A001462, A304679). The increasing case is A109298.
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 ordered prime signature (A124010) is the sequence of multiplicities (or exponents) in a number's prime factorization, taken in order of the prime base.
Also Heinz numbers of the integer partitions counted by A324572. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Each finite set of positive integers determines a unique term with those prime indices. For example, corresponding to {1,2,4,5} is 1397088 = prime(1)^5 * prime(2)^4 * prime(4)^2 * prime(5)^1.

			The sequence of terms together with their prime indices begins as follows. For example, we have 40: {1,1,1,3} because 40 = prime(1) * prime(1) * prime(1) * prime(3).
      1: {}
      2: {1}
      9: {2,2}
     12: {1,1,2}
     40: {1,1,1,3}
    112: {1,1,1,1,4}
    125: {3,3,3}
    352: {1,1,1,1,1,5}
    360: {1,1,1,2,2,3}
    675: {2,2,2,3,3}
    832: {1,1,1,1,1,1,6}
   1008: {1,1,1,1,2,2,4}
   2176: {1,1,1,1,1,1,1,7}
   2401: {4,4,4,4}
   3168: {1,1,1,1,1,2,2,5}
   3969: {2,2,2,2,4,4}
   4864: {1,1,1,1,1,1,1,1,8}
   7488: {1,1,1,1,1,1,2,2,6}
  11776: {1,1,1,1,1,1,1,1,1,9}
  14000: {1,1,1,1,3,3,3,4}
  19584: {1,1,1,1,1,1,1,2,2,7}

Cf. A001156, A033461, A056239, A062457, A109298, A112798, A117144, A118914, A124010, A181819, A276078.
Cf. A324524, A324525, A324572.
Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360, A304679.

Select[Range[1000],Reverse[PrimePi/@First/@If[#==1,{},FactorInteger[#]]]==Last/@If[#==1,{},FactorInteger[#]]&]

A046042 Number of partitions of n into fourth powers.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein

nonn

In general, the number of partitions of n into perfect s-th powers (s>=1) is asymptotic to (2*Pi)^(-(s+1)/2) * sqrt(s/(s+1)) * k * n^(1/(s+1)-3/2) * exp((s+1)*k*n^(1/(s+1))), where k = (Gamma(1 + 1/s) * Zeta(1 + 1/s) / s)^(s/(s+1)) [Hardy & Ramanujan, 1917]. - Vaclav Kotesovec, Dec 29 2016

			a(33) = 3 because we have [16,16,1], [16,1,1,...,1] (17 1's) and [1,1,...,1] (33 1's).

H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.
Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974.
Eric Weisstein's World of Mathematics, Partition

Cf. A000583, A002377, A003105.
Cf. A001156, A003108, A046042.
Cf. A037444, A259792, A259793.

a046042 = p $ tail a000583_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, May 18 2015   ~

g:=-1+1/product(1-x^(j^4),j=1..10): gser:=series(g,x=0,105): seq(coeff(gser,x,n),n=1..102); # Emeric Deutsch, Apr 06 2006

g = -1 + 1/Product[1 - x^(j^4), {j, 1, 10}]; gser =
Series[g, {x, 0, 105}]; Table[Coefficient[gser, x, n], {n, 1, 102}] (* Jean-François Alcover, Oct 29 2012, after Emeric Deutsch *)

G.f.: -1+1/product(1-x^(j^4),j=1..infinity). - Emeric Deutsch, Apr 06 2006
a(n) ~ exp(5 * (Gamma(1/4)*Zeta(5/4))^(4/5) * n^(1/5) / 2^(16/5)) * (Gamma(1/4)*Zeta(5/4))^(4/5) / (2^(47/10) * sqrt(5) * Pi^(5/2) * n^(13/10)) [Hardy & Ramanujan, 1917]. - Vaclav Kotesovec, Dec 29 2016
G.f.: Sum_{i>=1} x^(i^4) / Product_{j=1..i} (1 - x^(j^4)). - Ilya Gutkovskiy, May 07 2017

A103265 Number of partitions of n in which both even and odd square parts occur in 2 forms c, c* and with multiplicity 1. There is no restriction on parts which are twice squares.

Original entry on oeis.org

1, 2, 2, 2, 4, 6, 6, 6, 8, 12, 14, 14, 16, 22, 26, 26, 30, 38, 44, 46, 52, 62, 70, 74, 80, 96, 110, 116, 124, 146, 166, 174, 186, 210, 238, 254, 272, 302, 338, 362, 384, 426, 470, 502, 532, 588, 646, 686, 726, 792, 872, 926, 980, 1062

Offset: 0

Noureddine Chair, Feb 27 2005

Convolution of A001156 and A033461. - Vaclav Kotesovec, Aug 18 2015

			E.g. a(8)=8 because 8 can be written as 8, 44*, 422, 4*22, 4211*, 4*211*, 2222, 22211*.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A001156, A015128, A033461, A280263, A279227, A306147.

series(product((1+x^(k^2))/(1-x^(k^2)),k=1..100),x=0,100);

nmax = 50; CoefficientList[Series[Product[(1+x^(k^2)) / (1-x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 18 2015 *)

G.f.: Product_{k>0}((1+x^k^2)/(1-x^k^2)).

a(n) ~ exp(3 * ((4-sqrt(2))*zeta(3/2))^(2/3) * Pi^(1/3) * n^(1/3) / 4) * ((4-sqrt(2))*zeta(3/2))^(2/3) / (2^(7/2) * sqrt(3) * Pi^(7/6) * n^(7/6)). - Vaclav Kotesovec, Dec 29 2016

A259793 Number of partitions of n^4 into fourth powers.

Original entry on oeis.org

1, 1, 2, 7, 36, 253, 1886, 14800, 118238, 955639, 7750456, 62777522, 506272363, 4056634991, 32252971687, 254209569990, 1985108901344, 15352968310930, 117579612410477, 891596419221856, 6694250497509934, 49768995849050468, 366423320400440927, 2671969175372760210

Offset: 0

N. J. A. Sloane, Jul 06 2015

nonn

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..63 (terms 0..45 from Alois P. Heinz)
H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989
G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.

A row of the array in A259799.
Cf. A001156, A003108, A046042.
Cf. A037444, A259792.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1) +`if`(i^4>n, 0, b(n-i^4, i)))
    end:
a:= n-> b(n^4, n):
seq(a(n), n=0..23);  # Alois P. Heinz, Jul 10 2015

$RecursionLimit = 10^4; b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1] + If[i^4>n, 0, b[n-i^4, i]]]; a[n_] := b[n^4, n];  Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Dec 06 2016 after Alois P. Heinz *)

a(n) = [x^(n^4)] Product_{j>=1} 1/(1-x^(j^4)). - Alois P. Heinz, Jul 10 2015
a(n) = A046042(n^4). - Vaclav Kotesovec, Aug 19 2015
a(n) ~ exp(5 * (Gamma(1/4)*Zeta(5/4))^(4/5) * n^(4/5) / 2^(16/5)) * (Gamma(1/4)*Zeta(5/4))^(4/5) / (2^(47/10) * sqrt(5) * Pi^(5/2) * n^(26/5)) [after Hardy & Ramanujan, 1917]. - Vaclav Kotesovec, Dec 29 2016

More terms from Alois P. Heinz, Jul 10 2015

A112344 Number of partitions of n into perfect powers with each part > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 4, 2, 1, 0, 4, 2, 1, 0, 6, 5, 2, 2, 6, 5, 2, 2, 10, 8, 5, 4, 13, 8, 5, 4, 17, 14, 8, 9, 20, 17, 8, 9, 26, 24, 15, 14, 34, 27, 19, 14, 40, 38, 27, 25, 48, 47, 31, 30, 58, 59, 44, 42, 75, 68, 55, 47, 91, 86, 70, 67, 110, 106, 81, 81, 130, 134, 104

Offset: 1

Reinhard Zumkeller, Sep 05 2005

nonn

			a(20) = #{16+4, 8+8+4, 8+4+4+4, 4+4+4+4+4} = 4.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Perfect Power
Eric Weisstein's World of Mathematics, Partition

Cf. A001597, A000041, A001156, A078134, A062051, A112345.

Cf. A078635 (allowing 1).

N:= 200: # to get a(1) to a(N)
Pows:= {seq(seq(k^p, p=2..floor(log[k](N))),k=2..floor(sqrt(N)))}:
g:= proc(n,q) option remember; if n = 0 then 1 else `+`(seq(procname(n-r,r), r=select(`<=`,Pows,min(q,n)))) fi end proc:
seq(g(n,n), n=1..N); # Robert Israel, Nov 04 2015

M = 200; (* to get a(1) to a(M) *)
Pows = Table[k^p, {k, 2, Floor[Sqrt[M]]}, {p, 2, Floor[Log[k, M]]}] // Flatten // Union;
g[n_, q_] := g[n, q] = If[n == 0, 1, Plus @@ Table[g[n - r, r], {r, Select[Pows, # <= Min[q, n]&]}]];
Table[g[n, n], {n, 1, M}] (* Jean-François Alcover, Feb 03 2018, translated from Robert Israel's Maple code *)

leastp(n) = {while(!ispower(n), n--; if (n==0, return (0))); n;}
a(n) = {pmax = leastp(n); if (! pmax, return (0)); nb = 0; forpart(p=n, nb += (#select(x->ispower(x), Vec(p)) == #p), [4, pmax]); nb;} \\ Michel Marcus, Nov 04 2015

Name clarified by Sean A. Irvine, Jan 12 2025

A167661 Number of partitions of n into odd squares.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 11, 11, 12, 12, 13, 13, 13, 13, 14, 15, 15, 16, 16, 17, 17, 17, 17, 18, 19, 19, 20, 20, 21, 21, 22, 23, 24, 25, 25, 26, 26, 28, 28, 29, 30, 31

Offset: 0

Reinhard Zumkeller, Nov 08 2009

nonn

A167662 and A167663 give record values and where they occur: A167662(n)=a(A167663(n)) and a(m) < A167662(n) for m < A167663(n).

			a(10)=#{9+1,1+1+1+1+1+1+1+1+1+1}=2;
a(20)=#{9+9+1+1,9+1+1+1+1+1+1+1+1+1+1+1,20x1}=3;
a(30)=#{25+1+1+1+1+1,9+9+9+1+1+1,9+9+12x1,9+21x1,30x1}=5.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from R. Zumkeller)
Index entries for sequences related to sums of squares.

Cf. A001156, A000009, A101412, A016754, A167700.

g := 1/mul(1-x^((2*i-1)^2), i = 1 .. 150): gser := series(g, x = 0, 105): seq(coeff(gser, x, n), n = 0 .. 100);

nmax = 100; CoefficientList[Series[Product[1/(1 - x^((2*k-1)^2)), {k, 1, Floor[Sqrt[nmax]/2] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 18 2017 *)

a(n) = f(n,1,8) with f(x,y,z) = if x

G.f.: G = 1/Product_{i>=1}(1-x^{(2i-1)^2}). - Emeric Deutsch , Jan 26 2016

a(n) ~ exp(3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 4) * Zeta(3/2)^(1/3) / (4 * sqrt(3) * Pi^(1/3) * n^(5/6)). - Vaclav Kotesovec, Sep 18 2017

A162891 Expansion of 1 / Product_{k>=1} (1-x^k-x^(2*k)).

Original entry on oeis.org

1, 1, 3, 5, 11, 18, 36, 59, 109, 181, 318, 525, 902, 1481, 2492, 4087, 6788, 11090, 18274, 29776, 48772, 79332, 129411, 210172, 341958, 554728, 900872, 1460298, 2368555, 3837147, 6218652, 10070389, 16311432, 26407350, 42757335, 69208746, 112032256, 181316714

Offset: 0

Franklin T. Adams-Watters, Jul 16 2009

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

Cf. A000045, A000041, A001156, A003105, A263401, A276527.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/(&*[(1-x^k-x^(2*k)): k in [1..100]]))); // G. C. Greubel, Oct 24 2018

F:= n-> combinat[fibonacci](n+1):
b:= proc(n, i) option remember; `if`(n=0 or i=1, F(n),
      add((t-> b(t, min(t, i-1)))(n-i*j)*F(j), j=0..n/i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..39);  # Alois P. Heinz, Aug 24 2019

nmax = 50; CoefficientList[Series[1/Product[1-x^k-x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 16 2016 *)

al(n)=Vec(1/prod(k=1,n,1-x^k-x^(2*k)+x*O(x^n)))

a(n) ~ p / (sqrt(5) * r^(n+1)), where r = (sqrt(5)-1)/2 and p = Product_{n>1} 1/(1 - r^n - r^(2*n)) = 4.64451592505133910330213147... . - Vaclav Kotesovec, Nov 16 2016

A279225 Expansion of Product_{k>=1} 1/(1 - x^(k^2))^2.

Original entry on oeis.org

1, 2, 3, 4, 7, 10, 13, 16, 22, 30, 38, 46, 58, 74, 90, 106, 129, 158, 190, 222, 264, 314, 370, 426, 495, 580, 674, 772, 886, 1024, 1174, 1332, 1512, 1724, 1961, 2210, 2494, 2818, 3180, 3558, 3984, 4468, 5003, 5572, 6202, 6918, 7698, 8530, 9440, 10466, 11589

Offset: 0

Vaclav Kotesovec, Dec 08 2016

nonn

Number of partitions of n into squares of 2 kinds. - Ilya Gutkovskiy, Jan 23 2018

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A001156, A103265, A279226, A279227.

nmax = 100; CoefficientList[Series[Product[1/(1 - x^(k^2))^2, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 2^(2/3)) * Zeta(3/2) / (8 * sqrt(3) * Pi^2 * n^(3/2)). - Vaclav Kotesovec, Dec 29 2016

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cp's OEIS Frontend

A324525 Numbers divisible by prime(k)^k for each prime index k.

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A324524 Numbers where every prime index divides its multiplicity in the prime factorization. Numbers divisible by a power of prime(k)^k for each prime index k.

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A324571 Numbers whose ordered prime signature is equal to the set of distinct prime indices in decreasing order.

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A046042 Number of partitions of n into fourth powers.

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Haskell

Maple

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A103265 Number of partitions of n in which both even and odd square parts occur in 2 forms c, c* and with multiplicity 1. There is no restriction on parts which are twice squares.

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Maple

Mathematica

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A259793 Number of partitions of n^4 into fourth powers.

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Maple

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Extensions

A112344 Number of partitions of n into perfect powers with each part > 1.

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Maple