A211110 -id:A211110 - cp's OEIS frontend

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

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.

A327642 Number of partitions of n into divisors of n that are at most sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 4, 1, 5, 4, 6, 1, 19, 1, 8, 6, 25, 1, 37, 1, 36, 8, 12, 1, 169, 6, 14, 10, 64, 1, 247, 1, 81, 12, 18, 8, 1072, 1, 20, 14, 405, 1, 512, 1, 144, 82, 24, 1, 2825, 8, 146, 18, 196, 1, 1000, 12, 743, 20, 30, 1, 19858, 1, 32, 112, 969, 14, 1728, 1, 324, 24, 1105

Offset: 0

Ilya Gutkovskiy, Sep 20 2019

a(n) > n if n is in A058080 Union {0}, and, a(n) <= n if n is in A007964; indeed, a(n) = n only for n = 1. - Bernard Schott, Sep 22 2019

			The divisors of 6 are 1, 2, 3, 6 and sqrt(6) = 2.449..., so the possible partitions are 1+1+1+1+1+1 = 1+1+1+1+2 = 1+1+2+2 = 2+2+2; thus a(6) = 4. - _Bernard Schott_, Sep 22 2019

David A. Corneth, Table of n, a(n) for n = 0..9999 (first 5001 terms from Robert Israel)

Cf. A018818, A210442, A211110, A293813.

[1] cat [#RestrictedPartitions(n,{d:d in Divisors(n)| d le Sqrt(n)}):n in [1..70]]; // Marius A. Burtea, Sep 20 2019

f:= proc(n) local x, t, S;
    S:= 1;
    for t in numtheory:-divisors(n) do
      if t^2 <= n then
        S:= series(S/(1-x^t),x,n+1);
      fi
    od;
    coeff(S,x,n);
end proc:
map(f, [$0..100]); # Robert Israel, Sep 22 2019

a[n_] := SeriesCoefficient[Product[1/(1 - Boole[d <= Sqrt[n]] x^d), {d, Divisors[n]}], {x, 0, n}]; Table[a[n], {n, 0, 70}]

a(n) = [x^n] Product_{d|n, d <= sqrt(n)} 1 / (1 - x^d).
a(p) = 1, where p is prime.
a(p*q) = q+1 if p <= q are primes. - Robert Israel, Sep 22 2019

A263432 Number of partitions of n into divisors of n with at most one 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 4, 2, 4, 1, 17, 1, 4, 4, 10, 1, 24, 1, 24, 4, 4, 1, 126, 2, 4, 5, 30, 1, 171, 1, 36, 4, 4, 4, 490, 1, 4, 4, 251, 1, 290, 1, 43, 42, 4, 1, 1822, 2, 50, 4, 50, 1, 462, 4, 421, 4, 4, 1, 13284, 1, 4, 49, 202, 4, 616, 1, 63, 4, 581

Offset: 1

Geoffrey Critzer, Oct 18 2015

a(n) is also the number of ways to partition a group of order n into its center and its nontrivial conjugacy classes. That is, the number of possible sums in the class equation.

			a(15) = 4 because we have: [15], [5,5,5], [5,3,3,3,1], [3,3,3,3,3].

D. S. Dummit and R. M. Foote, Abstract Algebra, Wiley, 3rd edition 2003, page 124.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A018818, A033630, A211110.

with(numtheory):
a:= proc(n) local b, l; l:= sort([(divisors(n) minus {1})[]]):
      b:= proc(m, i) option remember; `if`(m=0, 1, `if`(i<1,
           `if`(m=1, 1, 0), b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
          end; forget(b):
      b(n, nops(l))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 18 2015

Table[d = Drop[Divisors[n], 1];Coefficient[Series[(1 + x)/Product[1 - x^d[[i]], {i, Length[d]}], {x, 0, n}], x,n], {n, 70}]

a(n) is the coefficient of x^n in the expansion of (1 + x)/Product_{d>1,d divides n} (1 - x^d).

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

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A327642 Number of partitions of n into divisors of n that are at most sqrt(n).

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A263432 Number of partitions of n into divisors of n with at most one 1.

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