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

Author

Gus Wiseman, Dec 23 2022

Keywords

• nonn

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.

Examples

			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.
		

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.

Programs

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}]

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).