A266475 Sum of the parts i_1 + i_2 + ... + i_{A001222(n)} of the unique strict partition with encoding n = Product_{j=1..A001222(n)} prime(i_j-j+1).

0, 1, 2, 3, 3, 4, 4, 6, 5, 5, 5, 7, 6, 6, 6, 10, 7, 8, 8, 8, 7, 7, 9, 11, 7, 8, 9, 9, 10, 9, 11, 15, 8, 9, 8, 12, 12, 10, 9, 12, 13, 10, 14, 10, 10, 11, 15, 16, 9, 10, 10, 11, 16, 13, 9, 13, 11, 12, 17, 13, 18, 13, 11, 21, 10, 11, 19, 12, 12, 11, 20, 17, 21

Offset

1,3

Comments

A strict partition is a partition into distinct parts.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Formula

a(n) = Sum_{k=1..A001222(n)} A265146(n,k).

[x^n] Sum_{i>=1} x^a(i) = A000009(n) for n>=0.

Example

n = 12 = 2*2*3 = prime(1)*prime(1)*prime(2) encodes strict partition [1,2,4].  So a(12) = 1+2+4 = 7.  Value a(n) = 7 occurs A000009(7) = 5 times, for n in {12, 17, 21, 22, 25}.

Maple

a:= n-> ((l-> add(l[j]+j-1, j=1..nops(l)))(sort([seq(
        numtheory[pi](i[1])$i[2], i=ifactors(n)[2])]))):
seq(a(n), n=1..100);

Mathematica

a[n_] := Function[l, Sum[l[[j]]+j-1, {j, 1, Length[l]}]][Sort[ Flatten[ Table[ Array[ PrimePi[i[[1]]]&, i[[2]]], {i, FactorInteger[n]}]]]];
Array[a, 100] (* Jean-François Alcover, Mar 23 2017, translated from Maple *)

Crossrefs

Row sums of A265146.

Ordinal transform gives A266476.

Cf. A000009, A001222.

Sequence in context: A204892 A164512 A127434 * A205402 A322007 A226107

Adjacent sequences: A266472 A266473 A266474 * A266476 A266477 A266478

Keyword

nonn

Author

Alois P. Heinz, Dec 29 2015

Status

approved