A358195 - OEIS

%I #5 Dec 24 2022 11:15:30

%S 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,

%T 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,

%U 17,29,1,75,1,31,25,32,7,77,1,289,19,21,1,54,1,37

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

%C 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.

%C 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.

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

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

%e 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.

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

%t osq[q_]:=1+Accumulate[Reverse[Differences[q]]];

%t Table[Times@@Prime/@osq[primeMS[n]],{n,20}]

%Y The even bisection is A241916.

%Y The unreversed version is A246277.

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

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

%Y A112798 list prime indices, sum A056239.

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

%K nonn

%O 1,4

%A _Gus Wiseman_, Dec 23 2022