A390449 - OEIS

A390449

Heinz number of first sums of prime indices of n.

1, 1, 1, 3, 1, 5, 1, 9, 7, 7, 1, 15, 1, 11, 11, 27, 1, 35, 1, 21, 13, 13, 1, 45, 13, 17, 49, 33, 1, 55, 1, 81, 17, 19, 17, 105, 1, 23, 19, 63, 1, 65, 1, 39, 77, 29, 1, 135, 19, 91, 23, 51, 1, 245, 19, 99, 29, 31, 1, 165, 1, 37, 91, 243, 23, 85, 1, 57, 31, 119

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OFFSET

1,4

COMMENTS

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.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

The first sums of a nonempty sequence (a, b, c, d, ...) are (a+b, b+c, c+d, ...).

LINKS

Table of n, a(n) for n=1..70.

EXAMPLE

The prime indices of 18 are (1,2,2), with first sums (3,4), with Heinz number 35, so a(18) = 35.

MATHEMATICA

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

firsums[c_]:=Table[c[[i]]+c[[i+1]], {i, Length[c]-1}];

Table[Times@@Prime/@firsums[prix[n]], {n, 100}]

CROSSREFS

Positions of squarefree numbers are A004709.

For first differences plus 1 instead of first sums we have A325352.

Positions of prime powers are A354144.

These are Heinz numbers of rows of A390307 or A390362, for differences A355536.

The prepended triangle is A390308, for differences A358169, reverse A355534.

The prepended version is A390309, union A390364, complement A390365.

The union is A390448.

For standard compositions we have A390568, triangle A390432.

A112798 lists prime indices, sum A056239, multiplicities A124010.

A342527 counts compositions with all equal first sums.

A390567 counts compositions with all distinct first sums.

Cf. A000040, A001222, A055396, A093641, A174090, A215366, A243055, A286470, A325328, A325351, A355523.

Sequence in context: A155912 A050354 A146434 * A126213 A146935 A360756

Adjacent sequences: A390446 A390447 A390448 * A390450 A390451 A390452

KEYWORD

nonn

AUTHOR

Gus Wiseman, Nov 11 2025

STATUS

approved