A358195 Heinz number of the partial sums plus one of the reversed first differences of the prime indices of n.

1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 9, 1, 7, 3, 8, 1, 6, 1, 25, 5, 11, 1, 27, 2, 13, 4, 49, 1, 15, 1, 16, 7, 17, 3, 18, 1, 19, 11, 125, 1, 35, 1, 121, 9, 23, 1, 81, 2, 10, 13, 169, 1, 12, 5, 343, 17, 29, 1, 75, 1, 31, 25, 32, 7, 77, 1, 289, 19, 21, 1, 54, 1, 37

Offset

1,4

Comments

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.

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 partial sums of first differences of a sequence telescope to "rest minus first", leading to the formula.

Links

Table of n, a(n) for n=1..74.

Formula

If n = Product_{i=1..k} prime(x_i) then a(n) = Product_{i=1..k-1} prime(x_k-x_{k-i}+1).

Example

The prime indices of 36 are (1,1,2,2), differences (0,1,0), reversed (0,1,0), partial sums (0,1,1), plus one (1,2,2), Heinz number 18, so a(36) = 18.

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
osq[q_]:=1+Accumulate[Reverse[Differences[q]]];
Table[Times@@Prime/@osq[primeMS[n]], {n, 20}]

Crossrefs

The even bisection is A241916.

The unreversed version is A246277.

Sum of prime indices of a(n) is A326844(n) + A001222(n) - 1.

A048793 gives partial sums of reversed standard comps, Heinz number A019565.

A112798 list prime indices, sum A056239.

Cf. A005940, A161511, A253565, A358137, A358170.

Sequence in context: A322390 A113308 A325332 * A143862 A115118 A115121

Adjacent sequences: A358192 A358193 A358194 * A358196 A358197 A358198

Keyword

nonn

Author

Gus Wiseman, Dec 23 2022

Status

approved