A325392 -id:A325392 - cp's OEIS frontend

A325390 Heinz number of the negated differences plus one of the integer partition with Heinz number n (with the last part taken to be 0).

1, 3, 5, 6, 7, 9, 11, 12, 10, 15, 13, 18, 17, 21, 15, 24, 19, 18, 23, 30, 25, 33, 29, 36, 14, 39, 20, 42, 31, 27, 37, 48, 35, 51, 21, 36, 41, 57, 55, 60, 43, 45, 47, 66, 30, 69, 53, 72, 22, 30, 65, 78, 59, 36, 35, 84, 85, 87, 61, 54, 67, 93, 50, 96, 49, 63, 71

Offset: 1

Gus Wiseman, May 02 2019

nonn

The Heinz number of a positive integer sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) (with the last part taken to be 0) are (-3,-2,-1).

			The Heinz number of (6,3,1) is 130, and its negated differences plus one are (4,3,2), which has Heinz number 105, so a(130) = 105.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Number of appearances of n is A325392(n).
Positions of squarefree numbers are A325367.
Cf. A007294, A007862, A056239, A112798, A320509, A325324, A325327, A325351, A325352, A325362, A325364, A325460, A325461.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Table[Times@@Prime/@(1-Differences[Append[primeptn[n],0]]),{n,100}]

A325403 Number of permutations of the multiset of prime factors of 2n whose first part is not 2.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 4, 0, 1, 3, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 1, 6, 1, 0, 4, 1, 4, 4, 1, 1, 4, 1, 1, 6, 1, 1, 9, 1, 1, 1, 2, 3, 4, 1, 1, 6, 4, 1, 4, 1, 1, 8, 1, 1, 9, 0, 4, 6, 1, 1, 4, 6, 1, 5, 1, 1, 9, 1, 4, 6, 1, 1, 4, 1, 1, 8, 4, 1, 4, 1, 1, 18, 4, 1, 4, 1, 4, 1, 1, 3, 9, 4, 1, 6, 1, 1, 18

Offset: 1

Gus Wiseman, May 02 2019

nonn

			The a(60) = 8 permutations of {2,2,2,3,5} whose first part is not 2:
  3 2 2 2 5
  3 2 2 5 2
  3 2 5 2 2
  3 5 2 2 2
  5 2 2 2 3
  5 2 2 3 2
  5 2 3 2 2
  5 3 2 2 2

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A008480, A056239, A112798, A325327, A325362, A325364, A325367, A325390, A325392, A325407.

Table[Length[Select[Permutations[Flatten[Table@@@FactorInteger[2*n]]],First[#]!=2&]],{n,100}]

A008480(n) = {my(sig=factor(n)[, 2]); vecsum(sig)!/factorback(apply(k->k!, sig))}; \\ After code in A008480
A325403(n) = (A008480(n+n)-A008480(n)); \\ Antti Karttunen, Dec 06 2021

a(n) = A008480(2n) - A008480(n) = A325392(2n).

Data section extended up to 105 terms by Antti Karttunen, Dec 06 2021

cp's OEIS Frontend

A325390 Heinz number of the negated differences plus one of the integer partition with Heinz number n (with the last part taken to be 0).

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