A106244 -id:A106244 - cp's OEIS frontend

A050361 Number of factorizations into distinct prime powers greater than 1.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1

Offset: 1

Christian G. Bower, Oct 15 1999

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

The number of unordered factorizations of n into 1 and exponentially odd prime powers, i.e., p^e where p is a prime and e is odd (A246551). - Amiram Eldar, Jun 12 2025

			From _Gus Wiseman_, Jul 30 2022: (Start)
The A000688(216) = 9 factorizations of 216 into prime powers are:
  (2*2*2*3*3*3)
  (2*2*2*3*9)
  (2*2*2*27)
  (2*3*3*3*4)
  (2*3*4*9)
  (2*4*27)
  (3*3*3*8)
  (3*8*9)
  (8*27)
Of these, the a(216) = 4 strict cases are:
  (2*3*4*9)
  (2*4*27)
  (3*8*9)
  (8*27)
(End)

Antti Karttunen, Table of n, a(n) for n = 1..100000 (first 10000 terms from Reinhard Zumkeller)
Index entries for sequences computed from exponents in factorization of n.

Cf. A000009, A050360, A050362, A050363, A050364, A101296, A246551.
Cf. A124010.
This is the strict case of A000688.
Positions of 1's are A004709, complement A046099.
The case of primes (instead of prime-powers) is A008966, non-strict A000012.
The non-strict additive version allowing 1's A023893, ranked by A302492.
The non-strict additive version is A023894, ranked by A355743.
The additive version (partitions) is A054685, ranked by A356065.
The additive version allowing 1's is A106244, ranked by A302496.
A001222 counts prime-power divisors.
A005117 lists all squarefree numbers.
A034699 gives maximal prime-power divisor.
A246655 lists all prime-powers (A000961 includes 1), towers A164336.
A296131 counts twice-factorizations of type PQR, non-strict A295935.
Cf. A001970, A002110, A025487, A055887, A063834, A076610, A085970, A279786, A302590, A302601, A354911, A355742.

a050361 = product . map a000009 . a124010_row
-- Reinhard Zumkeller, Aug 28 2014

A050361 := proc(n)
    local a,f;
    if n = 1 then
        1;
    else
        a := 1 ;
        for f in ifactors(n)[2] do
            a := a*A000009(op(2,f)) ;
        end do:
    end if;
end proc: # R. J. Mathar, May 25 2017

Table[Times @@ PartitionsQ[Last /@ FactorInteger[n]], {n, 99}] (* Arkadiusz Wesolowski, Feb 27 2017 *)

A000009(n,k=(n-!(n%2))) = if(!n,1,my(s=0); while(k >= 1, if(k<=n, s += A000009(n-k,k)); k -= 2); (s));
A050361(n) = factorback(apply(A000009,factor(n)[,2])); \\ Antti Karttunen, Nov 17 2019

Dirichlet g.f.: Product_{n is a prime power >1}(1 + 1/n^s).
Multiplicative with a(p^e) = A000009(e).
a(A002110(k))=1.
a(n) = A050362(A101296(n)). - R. J. Mathar, May 26 2017
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.26020571070524171076..., where f(x) = (1-x) * Product_{k>=1} (1 + x^k). - Amiram Eldar, Oct 03 2023

A023893 Number of partitions of n into prime power parts (1 included); number of nonisomorphic Abelian subgroups of symmetric group S_n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 36, 48, 63, 82, 105, 134, 171, 215, 269, 335, 415, 511, 626, 764, 929, 1125, 1356, 1631, 1953, 2333, 2776, 3296, 3903, 4608, 5427, 6377, 7476, 8744, 10205, 11886, 13818, 16032, 18565, 21463, 24768, 28536

Offset: 0

Olivier Gérard

nonn

			From _Gus Wiseman_, Jul 28 2022: (Start)
The a(0) = 1 through a(6) = 10 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (33)
           (11)  (21)   (22)    (32)     (42)
                 (111)  (31)    (41)     (51)
                        (211)   (221)    (222)
                        (1111)  (311)    (321)
                                (2111)   (411)
                                (11111)  (2211)
                                         (3111)
                                         (21111)
                                         (111111)
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for sequences related to groups

Cf. A009490, A023894 (first differences), A062297 (number of Abelian subgroups).
Cf. A018819, A062051, A131995.
The multiplicative version (factorizations) is A000688.
Not allowing 1's gives A023894, strict A054685, ranked by A355743.
The version for just primes (not prime-powers) is A034891, strict A036497.
The strict version is A106244.
These partitions are ranked by A302492.
A000041 counts partitions, strict A000009.
A001222 counts prime-power divisors.
A072233 counts partitions by sum and length.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
Cf. A001970, A055887, A063834, A085970, A279784, A295935, A356065.

Table[Length[Select[IntegerPartitions[n],Count[Map[Length,FactorInteger[#]], 1] == Length[#] &]], {n, 0, 35}] (* Geoffrey Critzer, Oct 25 2015 *)
nmax = 50; Clear[P]; P[m_] := P[m] = Product[Product[1/(1-x^(p^k)), {k, 1, m}], {p, Prime[Range[PrimePi[nmax]]]}]/(1-x)+O[x]^nmax // CoefficientList[ #, x]&; P[1]; P[m=2]; While[P[m] != P[m-1], m++]; P[m] (* Jean-François Alcover, Aug 31 2016 *)

lista(m) = {x = t + t*O(t^m); gf = prod(k=1, m, if (isprimepower(k), 1/(1-x^k), 1))/(1-x); for (n=0, m, print1(polcoeff(gf, n, t), ", "));} \\ Michel Marcus, Mar 09 2013

from functools import lru_cache
from sympy import factorint
@lru_cache(maxsize=None)
def A023893(n):
    @lru_cache(maxsize=None)
    def c(n): return sum((p**(e+1)-p)//(p-1) for p,e in factorint(n).items())+1
    return (c(n)+sum(c(k)*A023893(n-k) for k in range(1,n)))//n if n else 1 # Chai Wah Wu, Jul 15 2024

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

A023894 Number of partitions of n into prime power parts (1 excluded).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 6, 7, 9, 12, 15, 19, 23, 29, 37, 44, 54, 66, 80, 96, 115, 138, 165, 196, 231, 275, 322, 380, 443, 520, 607, 705, 819, 950, 1099, 1268, 1461, 1681, 1932, 2214, 2533, 2898, 3305, 3768, 4285, 4872, 5530, 6267, 7094, 8022, 9060

Offset: 0

Olivier Gérard

nonn

			From _Gus Wiseman_, Jul 28 2022: (Start)
The a(0) = 1 through a(9) = 7 partitions:
  ()  .  (2)  (3)  (4)   (5)   (33)   (7)    (8)     (9)
                   (22)  (32)  (42)   (43)   (44)    (54)
                               (222)  (52)   (53)    (72)
                                      (322)  (332)   (333)
                                             (422)   (432)
                                             (2222)  (522)
                                                     (3222)
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
E. Grosswald, Partitions into prime powers

The multiplicative version (factorizations) is A000688, coprime A354911.
Allowing 1's gives A023893, strict A106244, ranked by A302492.
The strict version is A054685.
The version for just primes is ranked by A076610, squarefree A356065.
Twice-partitions of this type are counted by A279784, factorizations A295935.
These partitions are ranked by A355743.
A000041 counts partitions, strict A000009.
A001222 counts prime-power divisors.
A072233 counts partitions by sum and length.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
Cf. A001970, A055887, A063834, A085970.

Table[Length[Select[IntegerPartitions[n],And@@PrimePowerQ/@#&]],{n,0,30}] (* Gus Wiseman, Jul 28 2022 *)

is_primepower(n)= {ispower(n, , &n); isprime(n)}
lista(m) = {x = t + t*O(t^m); gf = prod(k=1, m, if (is_primepower(k), 1/(1-x^k), 1)); for (n=0, m, print1(polcoeff(gf, n, t), ", "));}
\\ Michel Marcus, Mar 09 2013

from functools import lru_cache
from sympy import factorint
@lru_cache(maxsize=None)
def A023894(n):
    @lru_cache(maxsize=None)
    def c(n): return sum((p**(e+1)-p)//(p-1) for p,e in factorint(n).items())
    return (c(n)+sum(c(k)*A023894(n-k) for k in range(1,n)))//n if n else 1 # Chai Wah Wu, Jul 15 2024

G.f.: Prod(p prime, Prod(k >= 1, 1/(1-x^(p^k))))

A355743 Numbers whose prime indices are all prime-powers.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25, 27, 31, 33, 35, 41, 45, 49, 51, 53, 55, 57, 59, 63, 67, 69, 75, 77, 81, 83, 85, 93, 95, 97, 99, 103, 105, 109, 115, 119, 121, 123, 125, 127, 131, 133, 135, 147, 153, 155, 157, 159, 161, 165, 171, 175, 177, 179, 187

Offset: 1

Gus Wiseman, Jul 24 2022

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.

Also MM-numbers of multiset partitions into constant multisets, where the multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The terms together with their prime indices begin:
   1: {}
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  15: {2,3}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  31: {11}
  33: {2,5}
  35: {3,4}
  41: {13}
  45: {2,2,3}

Robert Price, Table of n, a(n) for n = 1..1410

The multiplicative version is A000688, strict A050361, coprime A354911.
The case of only primes (not all prime-powers) is A076610, strict A302590.
Allowing prime index 1 gives A302492.
These are the products of elements of A302493.
Requiring n to be a prime-power gives A302601.
These are the positions of 1's in A355741.
The squarefree case is A356065.
The complement is A356066.
A001222 counts prime-power divisors.
A023894 counts ptns into prime-powers, strict A054685, with 1's A023893.
A034699 gives maximal prime-power divisor.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A355742 chooses a prime-power divisor of each prime index.
Cf. A085970, A106244, A279784, A295935, A355731, A356064.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And@@PrimePowerQ/@primeMS[#]&]

A062051 Number of partitions of n into powers of 3.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 5, 7, 7, 7, 9, 9, 9, 12, 12, 12, 15, 15, 15, 18, 18, 18, 23, 23, 23, 28, 28, 28, 33, 33, 33, 40, 40, 40, 47, 47, 47, 54, 54, 54, 63, 63, 63, 72, 72, 72, 81, 81, 81, 93, 93, 93, 105, 105, 105, 117, 117, 117, 132, 132, 132, 147, 147, 147, 162

Offset: 0

Amarnath Murthy, Jun 06 2001

nonn

Number of different partial sums of 1+[1,*3]+[1,*3]+..., where [1,*3] means we can either add 1 or multiply by 3. E.g., a(6)=3 because we have 6=1+1+1+1+1+1=(1+1)*3=1*3+1+1+1. - Jon Perry, Jan 01 2004
Also number of partitions of n into distinct 3-smooth parts. E.g., a(10) = #{9+1, 8+2, 6+4, 6+3+1, 4+3+2+1} = #{9+1, 3+3+3+1, 3+3+1+1+1+1, 3+1+1+1+1+1+1+1, 1+1+1+1+1+1+1+1+1+1} = 5. - Reinhard Zumkeller, Apr 07 2005
Starts to differ from A008650 at a(81). - R. J. Mathar, Jul 31 2010
If m=ceiling(log_3(2k)) and n=(3^m+1)/2-k for k in the range (3^(m-1)+1)/2+(3^(m-2))<=k<=(3^m-1)/2, this sequence gives the number of "feasible" partitions described in the sequence A254296. For instance, the terms starting at 121st term of A254296 backwards to 68th term of A254296 provide the first 54 terms of this sequence. - Md. Towhidul Islam, Mar 01 2015
From Gary W. Adamson, Sep 03 2016: (Start)
Let M =
  1, 0, 0, 0, 0, ...
  1, 0, 0, 0, 0, ...
  1, 0, 0, 0, 0, ...
  1, 1, 0, 0, 0, ...
  1, 1, 0, 0, 0, ...
  1, 1, 0, 0, 0, ...
  1, 1, 1, 0, 0, ...
  1, 1, 1, 0, 0, ...
  ..., where the leftmost column is all 1's, and all other columns are 1's shifted down thrice. Lim_{k=1..inf} M^k has a single nonzero column, which gives the sequence. (End)

			a(4) = 2 and the partitions are 3+1, 1+1+1+1;
a(9) = 5 and the partitions are 9; 3+3+3; 3+3+1+1+1; 3+1+1+1+1+1+1; 1+1+1+1+1+1+1+1+1.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Vassil Dimitrov, Laurent Imbert, and Andrew Zakaluzny, Multiplication by a Constant is Sublinear, 18th IEEE Symposium on Computer Arithmetic (2007). See Theorem 1.
Md Towhidul Islam and Md Shahidul Islam, Number of Partitions of an n-kilogram Stone into Minimum Number of Weights to Weigh All Integral Weights from 1 to n kg(s) on a Two-pan Balance, arXiv:1502.07730 [math.CO], 2015.
M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DM-CCG), 2001, 215-228.
M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DM-CCG), 2001, 215-228. [Cached copy, with permission]

A005704 with terms repeated 3 times.

Cf. A000123, A018819, A000009, A003586, A105420, A039966, A023893, A105420, A106244, A131995, A179051, A254296.

nn=70;a=Product[1/(1-x^(3^i)),{i,0,4}];CoefficientList[Series[a,{x,0,nn}],x] (* Geoffrey Critzer, Oct 30 2012 *)

{ n=15; v=vector(n); for (i=1,n,v[i]=vector(2^(i-1))); v[1][1]=1; for (i=2,n, k=length(v[i-1]); for (j=1,k, v[i][j]=v[i-1][j]+1; v[i][j+k]=v[i-1][j]*3)); c=vector(n); for (i=1,n, for (j=1,2^(i-1), if (v[i][j]<=n, c[v[i][j]]++))); c } \\ Jon Perry

from functools import lru_cache
@lru_cache(maxsize=None)
def A062051(n): return A062051(n-1)+(0 if n%3 else A062051(n//3)) if n>2 else 1 # Chai Wah Wu, Sep 21 2022

a(n) = A005704([n/3]).
G.f.: Product_{k>=0} 1/(1-x^(3^k)). - R. J. Mathar, Jul 31 2010
If m = ceiling(log_3(2k)), define n = (3^m + 1)/2 - k for k in the range (3^(m-1)+1)/2 + (3^(m-2)) <= k <= (3^m-1)/2. Then, a(n) = Sum_{s=ceiling((k-1)/3)..(3^(m-1)-1)/2} a(s). This gives the first 2(3^(m-1))/3 terms. - Md. Towhidul Islam, Mar 01 2015
G.f.: 1 + Sum_{i>=0} x^(3^i) / Product_{j=0..i} (1 - x^(3^j)). - Ilya Gutkovskiy, May 07 2017

More terms from Larry Reeves (larryr(AT)acm.org), Jun 11 2001

A054685 Number of partitions of n into distinct prime powers (1 not considered a power).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 3, 2, 4, 3, 5, 5, 6, 7, 7, 10, 9, 12, 12, 15, 15, 18, 19, 22, 24, 26, 30, 32, 36, 39, 43, 48, 51, 57, 61, 68, 73, 79, 87, 93, 103, 108, 121, 127, 140, 148, 162, 173, 187, 200, 215, 232, 247, 266, 283, 306, 324, 348, 371, 397, 423, 450, 480, 512, 543, 579, 614

Offset: 0

David W. Wilson, Apr 19 2000

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)

Cf. A051613.
Cf. A106244.
Cf. A000961.

import Data.MemoCombinators (memo2, integral)
a054685 n = a054685_list !! n
a054685_list = map (p' 2) [0..] where
   p' = memo2 integral integral p
   p _ 0 = 1
   p k m = if m < pp then 0 else p' (k + 1) (m - pp) + p' (k + 1) m
           where pp = a000961 k
-- Reinhard Zumkeller, Nov 23 2015

CoefficientList[Series[Product[Product[1 +x^(Prime[n]^k), {k, 1, 9}], {n, 1, 25}], {x, 0, 100}], x] (* G. C. Greubel, May 09 2019 *)

G.f.: Product_{p prime} Product_{k >= 1} (1 + x^(p^k)).

A051613 a(n) = partitions of n into powers of distinct primes (1 not considered a power).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 0, 3, 2, 3, 2, 4, 3, 4, 4, 4, 8, 4, 8, 6, 9, 8, 10, 10, 13, 12, 13, 16, 16, 19, 17, 21, 23, 23, 25, 29, 31, 31, 31, 37, 40, 42, 44, 48, 49, 54, 55, 64, 67, 68, 70, 77, 84, 90, 92, 99, 102, 108, 115, 127, 133, 135, 138, 150, 165, 171, 183, 186, 198, 201, 220

Offset: 0

Vladeta Jovovic

			a(16) = 8 because we can write 16 = 2^4 = 3+13 = 5+11 = 3^2+7 = 2+3+11 = 2+3^2+5 = 2^3+3+5 = 2^2+5+7.

T. D. Noe, Table of n, a(n) for n = 0..1000
J. Bamberg, G. Cairns and D. Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, 110 (March 2003), 202-209.

Cf. A023894, A009490, A054685, A008475, A106245, A000961, A025473, A106244.

import Data.MemoCombinators (memo3, integral)
a051613' = p 1 2 where
   p x _ 0 = 1
   p x k m | m < qq       = 0
           | mod x q == 0 = p x (k + 1) m
           | otherwise    = p (q * x) (k + 1) (m - qq) + p x (k + 1) m
           where q = a025473 k; qq = a000961 k
-- Reinhard Zumkeller, Nov 23 2015

b:= proc(n,i) option remember; local p;
      p:= `if`(i<1, 1, ithprime(i));
      `if`(n=0, 1, `if`(i<1, 0, b(n,i-1)+
      add(b(n-p^j, i-1), j=1..ilog[p](n))))
    end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 15 2013

max = 70; f[x_] := Product[ 1 + Sum[x^(Prime[n]^k), {k, 1, If[n > 4, 1, 6]}], {n, 1, PrimePi[max]}]; CoefficientList[ Series[f[x], {x, 0, max}] , x](* Jean-François Alcover, Sep 12 2012 *)

first(n)=my(x='x,pr=O(x^(n+1))+1); forprime(p=sqrtint(n)+1,n, pr*=1+x^p); forprime(p=2,sqrtint(n), pr*=1+sum(e=1,logint(n,2), x^p^e)); Vec(pr) \\ Charles R Greathouse IV, Jun 25 2017

a(n) = number of m such that A008475(m) = n.

G.f.: Product_{p prime} (1 + Sum_{k >= 1} x^(p^k)).

Better description from David W. Wilson, Apr 19 2000

A356065 Squarefree numbers whose prime indices are all prime-powers.

Original entry on oeis.org

1, 3, 5, 7, 11, 15, 17, 19, 21, 23, 31, 33, 35, 41, 51, 53, 55, 57, 59, 67, 69, 77, 83, 85, 93, 95, 97, 103, 105, 109, 115, 119, 123, 127, 131, 133, 155, 157, 159, 161, 165, 177, 179, 187, 191, 201, 205, 209, 211, 217, 227, 231, 241, 249, 253, 255, 265, 277

Offset: 1

Gus Wiseman, Jul 25 2022

nonn

			105 has prime indices {2,3,4}, all three of which are prime-powers, so 105 is in the sequence.

The multiplicative version (factorizations) is A050361, non-strict A000688.
Heinz numbers of the partitions counted by A054685, with 1's A106244, non-strict A023894, non-strict with 1's A023893.
Counting twice-partitions of this type gives A279786, non-strict A279784.
Counting twice-factorizations gives A295935, non-strict A296131.
These are the odd products of distinct elements of A302493.
Allowing prime index 1 gives A302496, non-strict A302492.
The case of primes (instead of prime-powers) is A302590, non-strict A076610.
These are the squarefree positions of 1's in A355741.
This is the squarefree case of A355743, complement A356066.
A001222 counts prime-power divisors.
A005117 lists the squarefree numbers.
A034699 gives maximal prime-power divisor.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A355742 chooses a prime-power divisor of each prime index.
Cf. A001970, A055887, A063834, A302601, A355731, A355744, A356064.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SquareFreeQ[#]&&And@@PrimePowerQ/@primeMS[#]&]

Intersection of A005117 and A355743.

A111900 Number of partitions of n into distinct squares of primes.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 0

Reinhard Zumkeller, Aug 20 2005

a(n) <= 1 for n < 410.

			G.f. = 1 + x^4 + x^9 + x^13 + x^25 + x^29 + x^34 + x^38 + x^49 + x^53 + x^58 + x^62 + ...
410 = 7^2 + 19^2 = 11^2 + 17^2, therefore a(410)=2.

Alois P. Heinz, Table of n, a(n) for n = 0..20000
Alois P. Heinz, Plot of the first 100000 terms
Index entries for sequences related to sums of squares

Cf. A090677, A001248, A033461, A106244, A111902, A287965.

nmax = 100; CoefficientList[Series[Product[1 + x^Prime[k]^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Ilya Gutkovskiy, Jun 15 2017 *)

{a(n) = if(n < 0, 0, polcoeff( prod(k=1, primepi(sqrtint(n)), 1 + x^prime(k)^2 + x*O(x^n)), n))}; /* Michael Somos, Dec 26 2016 */

G.f.: Product_{k>=1} (1 + x^(prime(k)^2)). - Ilya Gutkovskiy, Dec 26 2016

A354911 Number of factorizations of n into relatively prime prime-powers.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 25 2022

nonn

			The a(n) factorizations for n = 6, 12, 24, 36, 48, 72, 96:
  2*3  3*4    3*8      4*9      3*16       8*9        3*32
       2*2*3  2*3*4    2*2*9    2*3*8      2*4*9      3*4*8
              2*2*2*3  3*3*4    3*4*4      3*3*8      2*3*16
                       2*2*3*3  2*2*3*4    2*2*2*9    2*2*3*8
                                2*2*2*2*3  2*3*3*4    2*3*4*4
                                           2*2*2*3*3  2*2*2*3*4
                                                      2*2*2*2*2*3

Wikipedia, Coprime integers.

This is the relatively prime case of A000688, partitions A023894.
Positions of 0's are A246655 (A000961 includes 1).
For strict instead of relatively prime we have A050361, partitions A054685.
Positions of 1's are A000469 (A120944 excludes 1).
For pairwise coprime instead of relatively prime we have A143731.
The version for partitions instead of factorizations is A356067.
A000005 counts divisors.
A001055 counts factorizations.
A001221 counts distinct prime divisors, with sum A001414.
A001222 counts prime-power divisors.
A289509 lists numbers whose prime indices are relatively prime.
A295935 counts twice-factorizations with constant blocks (type PPR).
A355743 lists numbers with prime-power prime indices, squarefree A356065.
Cf. A000837, A023893, A076610, A085970, A106244, A279784, A318721, A355737, A355742, A356064, A356066.

ufacs[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[ufacs[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Table[Length[Select[ufacs[Select[Divisors[n],PrimePowerQ[#]&],n],GCD@@#<=1&]],{n,100}]

a(n) = A000688(n) if n is nonprime, otherwise a(n) = 0.

cp's OEIS Frontend

A050361 Number of factorizations into distinct prime powers greater than 1.

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A023893 Number of partitions of n into prime power parts (1 included); number of nonisomorphic Abelian subgroups of symmetric group S_n.

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A023894 Number of partitions of n into prime power parts (1 excluded).

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A355743 Numbers whose prime indices are all prime-powers.

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A062051 Number of partitions of n into powers of 3.

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A054685 Number of partitions of n into distinct prime powers (1 not considered a power).

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A051613 a(n) = partitions of n into powers of distinct primes (1 not considered a power).

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