A066882 -id:A066882 - cp's OEIS frontend

A284345 Number of partitions of n into squares dividing n.

1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4, 1, 1, 1, 6, 1, 3, 1, 6, 1, 1, 1, 7, 2, 1, 4, 8, 1, 1, 1, 15, 1, 1, 1, 27, 1, 1, 1, 11, 1, 1, 1, 12, 6, 1, 1, 28, 2, 3, 1, 14, 1, 7, 1, 15, 1, 1, 1, 16, 1, 1, 8, 46, 1, 1, 1, 18, 1, 1, 1, 114, 1, 1, 4, 20, 1, 1, 1, 66, 11, 1, 1, 22, 1, 1, 1, 23, 1, 11, 1, 24, 1, 1, 1, 91, 1, 3, 12, 67

Offset: 0

Ilya Gutkovskiy, Mar 25 2017

nonn

			a(8) = 3 because 8 has 4 divisors {1, 2, 4, 8} among which 2 are squares {1, 4} therefore we have [4, 4], [4, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1].

Cf. A000290, A001156, A018818, A046951, A066882, A161148, A225244, A284289.

with(numtheory):
a:= proc(n) option remember; local b, l; l, b:=
      sort(select(issqr, [divisors(n)[]])),
      proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
        b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
      end; b(n, nops(l))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 30 2017

Join[{1}, Table[d = Divisors[n]; Coefficient[Series[Product[1/(1 - Boole[Mod[DivisorSigma[0, d[[k]]], 2] == 1] x^d[[k]]), {k, Length[d]}], {x, 0, n}], x, n], {n, 1, 100}]]

a(n) = [x^n] Product_{d^2|n} 1/(1 - x^(d^2)).
a(n) = 1 if n is a squarefree.
a(n) = 2 if n is a square of prime.

A014652 Number of partitions of n in its prime divisors with at least one part of size 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 5, 1, 4, 3, 8, 1, 16, 1, 11, 11, 8, 1, 33, 1, 26, 15, 17, 1, 56, 5, 20, 9, 36, 1, 226, 1, 16, 23, 26, 23, 120, 1, 29, 27, 92, 1, 422, 1, 56, 78, 35, 1, 208, 7, 140, 35, 66, 1, 261, 35, 128, 39, 44, 1, 1487, 1, 47, 108, 32, 41, 996, 1, 86, 47, 1062, 1, 456, 1, 56

Offset: 1

Olivier Gérard

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
David A. Corneth, PARI program

Cf. A014648, A014649, A014650, A014651, A066874, A066882, A286852.

\\ This is for computing just a moderate number of terms:
prime_factors_with1_reversed(n) = vecsort(setunion([1],factor(n)[,1]~), , 4);
partitions_into_with_trailing_ones(n,parts,from=1) = if(!n,1, if(#parts<=(from+1), if(#parts == from,1,(1+(n\parts[from]))), my(s=0); for(i=from,#parts,if(parts[i]<=n, s += partitions_into_with_trailing_ones(n-parts[i],parts,i))); (s)));
A014652(n) = partitions_into_with_trailing_ones(n-1,prime_factors_with1_reversed(n)); \\ Antti Karttunen, Sep 10 2018

\\ For an efficient program to compute large numbers of terms, see David A. Corneth's PARI program included in the Links section. - Antti Karttunen, Sep 12 2018

Coefficient of x^(n-1) in expansion of (1/(1-x))*1/Product_{d is prime divisor of n} (1-x^d). - Vladeta Jovovic, Apr 11 2004

A284289 Number of partitions of n into prime power divisors of n (not including 1).

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 7, 1, 2, 2, 10, 1, 7, 1, 10, 2, 2, 1, 34, 2, 2, 5, 13, 1, 21, 1, 36, 2, 2, 2, 72, 1, 2, 2, 73, 1, 28, 1, 19, 13, 2, 1, 249, 2, 10, 2, 22, 1, 50, 2, 127, 2, 2, 1, 419, 1, 2, 17, 202, 2, 42, 1, 28, 2, 43, 1, 1260, 1, 2, 13, 31, 2, 49, 1, 801, 23, 2, 1, 774, 2, 2, 2, 280, 1, 608

Offset: 0

Ilya Gutkovskiy, Mar 24 2017

nonn

			a(8) = 4 because 8 has 4 divisors {1, 2, 4, 8} among which 3 are prime powers {2, 4, 8} therefore we have [8], [4, 4], [4, 2, 2] and [2, 2, 2, 2].

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Prime Power
Index entries for related partition-counting sequences

Cf. A014652, A018818, A023894, A066882, A225244, A211110, A246655.

with(numtheory):
a:= proc(n) option remember; local b, l; l, b:= sort(
      [select(x-> nops(ifactors(x)[2])=1, divisors(n))[]]),
      proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
        b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
      end; b(n, nops(l))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 30 2017

Table[d = Divisors[n]; Coefficient[Series[Product[1/(1 - Boole[PrimePowerQ[d[[k]]]] x^d[[k]]), {k, Length[d]}], {x, 0, n}], x, n], {n, 0, 90}] (* or *)
a[0]=1; a[1]=0; a[n_] := Length@IntegerPartitions[n, All, Join @@ (#[[1]]^Range[#[[2]]] & /@ FactorInteger[n])]; a /@ Range[0, 90] (* Giovanni Resta, Mar 25 2017 *)

a(n) = [x^n] Product_{p^k|n, p prime, k >= 1} 1/(1 - x^(p^k)).
a(n) = 1 if n is a prime.
a(n) = 2 if n is a semiprime.

A284465 Number of compositions (ordered partitions) of n into prime power divisors of n (not including 1).

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 2, 1, 6, 2, 2, 1, 36, 1, 2, 2, 56, 1, 90, 1, 201, 2, 2, 1, 4725, 2, 2, 20, 1085, 1, 15778, 1, 5272, 2, 2, 2, 476355, 1, 2, 2, 270084, 1, 302265, 1, 35324, 3910, 2, 1, 67279595, 2, 14047, 2, 219528, 1, 5863044, 2, 14362998, 2, 2, 1, 47466605656, 1, 2, 35662, 47350056, 2, 119762253, 1, 9479643

Offset: 0

Ilya Gutkovskiy, Mar 27 2017

nonn

			a(8) = 6 because 8 has 4 divisors {1, 2, 4, 8} among which 3 are prime powers > 1 {2, 4, 8} therefore we have [8], [4, 4], [4, 2, 2], [2, 4, 2], [2, 2, 4] and [2, 2, 2, 2].

Robert Israel, Table of n, a(n) for n = 0..5039
Eric Weisstein's World of Mathematics, Prime Power
Index entries for sequences related to compositions

Cf. A066882, A100346, A246655, A280195, A284289.

F:= proc(n) local f,G;
      G:= 1/(1 - add(add(x^(f[1]^j),j=1..f[2]),f = ifactors(n)[2]));
      coeff(series(G,x,n+1),x,n);
end proc:
map(F, [$0..100]); # Robert Israel, Mar 29 2017

Table[d = Divisors[n]; Coefficient[Series[1/(1 - Sum[Boole[PrimePowerQ[d[[k]]]] x^d[[k]], {k, Length[d]}]), {x, 0, n}], x, n], {n, 0, 68}]

from sympy import divisors, primefactors
from sympy.core.cache import cacheit
@cacheit
def a(n):
    l=[x for x in divisors(n) if len(primefactors(x))==1]
    @cacheit
    def b(m): return 1 if m==0 else sum(b(m - j) for j in l if j <= m)
    return b(n)
print([a(n) for n in range(71)]) # Indranil Ghosh, Aug 01 2017

a(n) = [x^n] 1/(1 - Sum_{p^k|n, p prime, k>=1} x^(p^k)).
a(n) = 1 if n is a prime.
a(n) = 2 if n is a semiprime.

A168324 Number of distinct permutations of the list of prime factors of n (with multiplicity), where a(1)=0.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 6, 1, 1, 2, 2, 2, 6, 1, 2, 2, 4, 1, 6, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 12, 1, 2, 3, 1, 2, 6, 1, 3, 2, 6, 1, 10, 1, 2, 3, 3, 2, 6, 1, 5, 1, 2, 1, 12, 2, 2, 2, 4, 1, 12, 2, 3, 2, 2, 2, 6, 1, 3, 3, 6, 1, 6, 1, 4, 6

Offset: 1

Juri-Stepan Gerasimov, Nov 22 2009, May 08 2010

nonn

Apart from a(1) the same as A008480.

			a(18)=3 because 18=2*2*3=2*3*2=3*2*2;
a(24)=4 because 24=2*2*2*3=2*2*3*2=2*3*2*2=3*2*2*2;
a(26)=2 because 26=2*13=13*2;
a(30)=6 because 30=2*3*5=2*5*3=3*2*5=3*5*2=5*2*3=5*3*2.

Cf. A066882, A008480 (same except for initial term).

nn = 105;
f[list_, i] := list[[i]];
a =Table[Boole[PrimeQ[n]], {n, 1, nn}]; Map[Total,Transpose[NestList[Table[
DirichletConvolve[f[#, n], f[a, n], n, m], {m, 1, nn}] &, a,nn]]] (* Geoffrey Critzer, Feb 16 2015 *)

Entries checked by D. S. McNeil, Nov 26 2010

A284463 Number of compositions (ordered partitions) of n into prime divisors of n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 12, 1, 2, 2, 1, 1, 65, 1, 23, 2, 2, 1, 351, 1, 2, 1, 38, 1, 15778, 1, 1, 2, 2, 2, 10252, 1, 2, 2, 1601, 1, 302265, 1, 80, 750, 2, 1, 299426, 1, 13404, 2, 107, 1, 1618192, 2, 5031, 2, 2, 1, 707445067, 1, 2, 2398, 1, 2, 119762253, 1, 173, 2, 39614048, 1, 255418101, 1, 2, 154603

Offset: 0

Ilya Gutkovskiy, Mar 27 2017

nonn

			a(6) = 2 because 6 has 4 divisors {1, 2, 3, 6} among which 2 are primes {2, 3} therefore we have [3, 3] and [2, 2, 2].

Alois P. Heinz, Table of n, a(n) for n = 0..2000
Index entries for sequences related to compositions

Cf. A000040, A014652, A023360, A066882, A100346.

a:= proc(n) option remember; local b, l;
      l, b:= numtheory[factorset](n),
      proc(m) option remember; `if`(m=0, 1,
         add(`if`(j>m, 0, b(m-j)), j=l))
      end; b(n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 28 2017

Table[d = Divisors[n]; Coefficient[Series[1/(1 - Sum[Boole[PrimeQ[d[[k]]]] x^d[[k]], {k, Length[d]}]), {x, 0, n}], x, n], {n, 0, 75}]

from sympy import divisors, isprime
from sympy.core.cache import cacheit
@cacheit
def a(n):
    l=[x for x in divisors(n) if isprime(x)]
    @cacheit
    def b(m): return 1 if m==0 else sum(b(m - j) for j in l if j <= m)
    return b(n)
print([a(n) for n in range(101)]) # Indranil Ghosh, Aug 01 2017, after Maple code

a(n) = [x^n] 1/(1 - Sum_{p|n, p prime} x^p).
a(n) = 1 if n is a prime power > 1.
a(n) = 2 if n is a squarefree semiprime.

A286852 Number of partitions of n into unitary prime divisors of n.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 1, 0, 2, 0, 1, 1, 21, 1, 0, 2, 2, 2, 0, 1, 2, 2, 1, 1, 28, 1, 1, 1, 2, 1, 1, 0, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 5, 1, 2, 1, 0, 2, 42, 1, 1, 2, 43, 1, 0, 1, 2, 1, 1, 2, 49, 1, 1, 0, 2, 1, 5, 2, 2, 2, 1, 1, 10, 2, 1, 2, 2, 2

Offset: 0

Ilya Gutkovskiy, Aug 01 2017

nonn

			a(6) = 2 because 6 has 4 divisors {1, 2, 3, 6} among which 2 are unitary prime divisors {2, 3} therefore we have [3, 3] and [2, 2, 2].

Antti Karttunen, Table of n, a(n) for n = 0..2309
Eric Weisstein's World of Mathematics, Unitary Divisor
Index entries for related partition-counting sequences

Cf. A018818, A055231, A056169, A066874, A066882, A225244, A284289.

Join[{1}, Table[d = Divisors[n]; Coefficient[Series[Product[1/(1 - Boole[GCD[n/d[[k]], d[[k]]] == 1 && PrimeQ[d[[k]]]] x^d[[k]]), {k, Length[d]}], {x, 0, n}], x, n], {n, 1, 95}]]

A055231(n) = {my(f=factor(n)); for (k=1, #f~, if (f[k, 2] > 1, f[k, 2] = 0); ); factorback(f); } \\ From A055231
unitary_prime_factors(n) = { my(ufs = factor(A055231(n))); ufs[,1]~; };
partitions_into(n,parts,from=1) = if(!n,1,my(k = #parts, s=0); for(i=from,k,if(parts[i]<=n, s += partitions_into(n-parts[i],parts,i))); (s));
A286852(n) = if(n<2,1-n,partitions_into(n,vecsort(unitary_prime_factors(n), , 4))); \\ Antti Karttunen, Jul 02 2018

a(n) = [x^n] Product_{p|n, p prime, gcd(p, n/p) = 1} 1/(1 - x^p).

a(n) = 0 if n is a powerful number (A001694).

A107329 Triangle read by rows: T(n,k) gives number of partitions of k, (k=1..n) into the prime factors of n, for n>=1.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Wouter Meeussen, May 22 2005

T(n,n) equals A066882(n).

			T(30,12)=5 counting [2,2,2,2,2,2], [2,2,2,3,3], [3,3,3,3], [2,2,3,5] and [2,5,5].
Triangle begins:
  {0},
  {0, 1},
  {0, 0, 1},
  {0, 1, 0, 1},
  {0, 0, 0, 0, 1},
  {0, 1, 1, 1, 1, 2},
  {0, 0, 0, 0, 0, 0, 1},
  {0, 1, 0, 1, 0, 1, 0, 1},
  {0, 0, 1, 0, 0, 1, 0, 0, 1},
  ...

Alois P. Heinz, rows n = 1..200, flattened

Cf. A066882.

Row sums +1 give A092976.

with(numtheory):
T:= proc(n) local b, l; l:= sort([factorset(n)[]]):
      b:= proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
            b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
          end; forget(b):
      seq(b(k, nops(l)), k=1..n)
    end:
seq(T(n), n=1..20);  # Alois P. Heinz, Oct 28 2021

Table[Rest@CoefficientList[Series[1/Times @@ ((1-x^#)& /@ (First /@ FactorInteger[n])), {x, 0, n}], x], {n, 2, 24}]

T(n,k) is coefficient of x^k in 1/Product(1-x^p_i) with p_i the prime factors of n.

T(1,1) = 0 prepended by Michel Marcus, Oct 28 2021

A284839 Number of compositions (ordered partitions) of n into prime power divisors of n (including 1).

Original entry on oeis.org

1, 1, 2, 2, 6, 2, 24, 2, 56, 20, 128, 2, 1490, 2, 741, 449, 5272, 2, 36901, 2, 81841, 3320, 29966, 2, 4135004, 572, 200389, 26426, 5452795, 2, 110187694, 2, 47350056, 226019, 9262156, 51885, 10783889706, 2, 63346597, 2044894, 14064551462, 2, 109570982403, 2, 35537376325, 470326038, 2972038874, 2

Offset: 0

Ilya Gutkovskiy, Apr 03 2017

nonn

			a(4) = 6 because 4 has 3 divisors {1, 2, 4} and all are prime powers therefore we have [4], [2, 2], [2, 1, 1], [1, 2, 1], [1, 1, 2] and [1, 1, 1, 1].

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Prime Power
Index entries for sequences related to compositions

Cf. A000961, A066882, A100346, A284465.

with(numtheory):
a:= proc(n) local d, b; d, b:= select(x->
      nops(factorset(x))<2, divisors(n)),
      proc(n) option remember; `if`(n=0, 1,
        add(`if`(j>n, 0, b(n-j)), j=d))
      end: b(n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Apr 15 2017

Table[d = Divisors[n]; Coefficient[Series[1/(1 - x - Sum[Boole[PrimePowerQ[d[[k]]]] x^d[[k]], {k, Length[d]}]), {x, 0, n}], x, n], {n, 0, 47}]

a(n) = [x^n] 1/(1 - x - Sum_{p^k|n, p prime, k>=1} x^(p^k)).

a(n) = 2 if n is a prime.

A160810 Numbers k such that the number of partitions of k into prime divisors of k exceeds the number of distinct transpositions of prime factors of k.

Original entry on oeis.org

18, 24, 30, 36, 40, 42, 45, 48, 50, 54, 56, 60, 63, 66, 70, 72, 75, 78, 80, 84, 88, 90, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 117, 120, 126, 130, 132, 135, 136, 138, 140, 144, 147, 150, 152, 153, 154, 156, 160, 162, 165, 168, 170, 171, 174, 175, 176

Offset: 1

Juri-Stepan Gerasimov, Nov 23 2009

nonn

Numbers k such that A066882(k) > A168324(k).

Cf. A066882, A168324.

A066882 := proc(n) gf := 1 ; for d in numtheory[divisors](n) do if isprime(d) then gf := gf/(1-x^d) ; gf := taylor(gf,x=0,n+2) ; end if; end do: coeftayl(gf,x=0,n) ; end proc:
A168324 := proc(n) if n = 1 then 0; else multn := numtheory[bigomega](n) ; multn := factorial(multn) ; for p in ifactors(n)[2] do multn := multn/factorial(op(2,p)) ; end do: multn ; end if; end proc:
for n from 1 to 300 do if A066882(n) > A168324(n) then printf("%d,\n",n) ; end if; end do: # R. J. Mathar, May 21 2010

More terms from R. J. Mathar, May 21 2010

cp's OEIS Frontend

A284345 Number of partitions of n into squares dividing n.

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A014652 Number of partitions of n in its prime divisors with at least one part of size 1.

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PARI

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A284289 Number of partitions of n into prime power divisors of n (not including 1).

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A284465 Number of compositions (ordered partitions) of n into prime power divisors of n (not including 1).

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A168324 Number of distinct permutations of the list of prime factors of n (with multiplicity), where a(1)=0.

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A284463 Number of compositions (ordered partitions) of n into prime divisors of n.

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Maple

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A286852 Number of partitions of n into unitary prime divisors of n.

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A107329 Triangle read by rows: T(n,k) gives number of partitions of k, (k=1..n) into the prime factors of n, for n>=1.

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