A318573 -id:A318573 - cp's OEIS frontend

A318574 Denominator of the reciprocal sum of the integer partition with Heinz number n.

1, 1, 2, 1, 3, 2, 4, 1, 1, 3, 5, 2, 6, 4, 6, 1, 7, 1, 8, 3, 4, 5, 9, 2, 3, 6, 2, 4, 10, 6, 11, 1, 10, 7, 12, 1, 12, 8, 3, 3, 13, 4, 14, 5, 3, 9, 15, 2, 2, 3, 14, 6, 16, 2, 15, 4, 8, 10, 17, 6, 18, 11, 4, 1, 2, 10, 19, 7, 18, 12, 20, 1, 21, 12, 6, 8, 20, 3, 22

Offset: 1

Gus Wiseman, Aug 29 2018

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Positions of 1's are A316856.

Cf. A051908, A056239, A058360, A112798, A289506, A289507, A296150, A316854, A316855, A316857, A318573.

Table[Sum[pr[[2]]/PrimePi[pr[[1]]],{pr,If[n==1,{},FactorInteger[n]]}],{n,100}]//Denominator

If n = Product prime(x_i)^y_i is the prime factorization of n, then a(n) is the denominator of Sum y_i/x_i.

A325704 If n = prime(i_1)^j_1 * ... * prime(i_k)^j_k, then a(n) is the numerator of the reciprocal factorial sum j_1/i_1! + ... + j_k/i_k!.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 1, 7, 1, 5, 1, 25, 2, 4, 1, 2, 1, 13, 13, 121, 1, 7, 1, 721, 3, 49, 1, 5, 1, 5, 61, 5041, 5, 3, 1, 40321, 361, 19, 1, 37, 1, 241, 7, 362881, 1, 9, 1, 4, 2521, 1441, 1, 5, 7, 73, 20161, 3628801, 1, 8, 1, 39916801, 25, 6, 121, 181, 1

Offset: 1

Gus Wiseman, May 18 2019

Alternatively, if n = prime(i_1) * ... * prime(i_k), then a(n) is the numerator of 1/i_1! + ... + 1/i_k!.

Factorial numbers: A000142, A002982, A011371, A022559, A071626, A076934, A108731, A325272, A325508, A325709.

Reciprocal sum: A002966, A316855, A316856, A316857, A318573, A318574, A325618, A325619, A325620, A325621, A325622, A325623, A325624, A325703.

Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>k/PrimePi[p]!]],{n,100}]//Numerator

A325704(n) = { my(f=factor(n)); numerator(sum(i=1,#f~,f[i, 2]/(primepi(f[i, 1])!))); }; \\ Antti Karttunen, Nov 17 2019

a(n) = A318573(A325709(n)).

A325703 If n = prime(i_1)^j_1 * ... * prime(i_k)^j_k, then a(n) is the denominator of the reciprocal factorial sum j_1/i_1! + ... + j_k/i_k!.

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 24, 1, 1, 6, 120, 2, 720, 24, 3, 1, 5040, 1, 40320, 6, 24, 120, 362880, 2, 3, 720, 2, 24, 3628800, 3, 39916800, 1, 120, 5040, 24, 1, 479001600, 40320, 720, 6, 6227020800, 24, 87178291200, 120, 6, 362880, 1307674368000, 2, 12, 3, 5040, 720

Offset: 1

Gus Wiseman, May 18 2019

Alternatively, if n = prime(i_1) * ... * prime(i_k), then a(n) is the denominator of 1/i_1! + ... + 1/i_k!.

Robert Israel, Table of n, a(n) for n = 1..3168
Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Factorial numbers: A000142, A002982, A011371, A022559, A071626, A076934, A108731, A325272, A325508, A325709.

Reciprocal sum: A002966, A316855, A316856, A316857, A318573, A318574, A325618, A325619, A325620, A325621, A325622, A325623, A325624, A325704.

f:= proc(n) local F,t;
    F:= ifactors(n)[2];
    denom(add(t[2]/numtheory:-pi(t[1])!,t=F))
end proc:
map(f, [$1..100]); # Robert Israel, Oct 13 2024

Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>k/PrimePi[p]!]],{n,100}]//Denominator

a(n) = A318574(A325709(n)).

A381958 Numerator of the sum of the reciprocals of the indices of distinct prime factors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 3, 1, 5, 5, 1, 1, 3, 1, 4, 3, 6, 1, 3, 1, 7, 1, 5, 1, 11, 1, 1, 7, 8, 7, 3, 1, 9, 2, 4, 1, 7, 1, 6, 5, 10, 1, 3, 1, 4, 9, 7, 1, 3, 8, 5, 5, 11, 1, 11, 1, 12, 3, 1, 1, 17, 1, 8, 11, 19, 1, 3, 1, 13, 5, 9, 9, 5, 1, 4, 1, 14, 1, 7, 10, 15, 3, 6, 1, 11, 5, 10, 13, 16, 11

Offset: 1

Ilya Gutkovskiy, Mar 11 2025

			0, 1, 1/2, 1, 1/3, 3/2, 1/4, 1, 1/2, 4/3, 1/5, 3/2, 1/6, 5/4, 5/6, 1, 1/7, 3/2, 1/8, 4/3, ...

Index entries for sequences computed from indices in prime factorization

Cf. A000720, A028235, A028236, A066328, A083345, A318573, A379141, A381959 (denominators).

Join[{0}, Table[Plus @@ (1/PrimePi[#[[1]]] & /@ FactorInteger[n]), {n, 2, 95}] // Numerator]
nmax = 95; CoefficientList[Series[Sum[x^Prime[k]/(k (1 - x^Prime[k])), {k, 1, nmax}], {x, 0, nmax}], x] // Numerator // Rest

a(n) = my(f=factor(n)); numerator(sum(k=1, #f~, 1/primepi(f[k,1]))); \\ Michel Marcus, Mar 11 2025

If n = Product (p_j^k_j) then a(n) = numerator of Sum (1/pi(p_j)).

G.f. for fractions: Sum_{k>=1} x^prime(k) / (k*(1 - x^prime(k))).

cp's OEIS Frontend

A318574 Denominator of the reciprocal sum of the integer partition with Heinz number n.

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A325704 If n = prime(i_1)^j_1 * ... * prime(i_k)^j_k, then a(n) is the numerator of the reciprocal factorial sum j_1/i_1! + ... + j_k/i_k!.

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A325703 If n = prime(i_1)^j_1 * ... * prime(i_k)^j_k, then a(n) is the denominator of the reciprocal factorial sum j_1/i_1! + ... + j_k/i_k!.

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A381958 Numerator of the sum of the reciprocals of the indices of distinct prime factors of n.

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Programs

Mathematica

PARI

Formula