A379681 Sum plus product of the multiset of prime indices of n.

1, 2, 4, 3, 6, 5, 8, 4, 8, 7, 10, 6, 12, 9, 11, 5, 14, 9, 16, 8, 14, 11, 18, 7, 15, 13, 14, 10, 20, 12, 22, 6, 17, 15, 19, 10, 24, 17, 20, 9, 26, 15, 28, 12, 19, 19, 30, 8, 24, 16, 23, 14, 32, 15, 23, 11, 26, 21, 34, 13, 36, 23, 24, 7, 27, 18, 38, 16, 29, 20

Offset: 1

Author

Gus Wiseman, Jan 05 2025

Keywords

• nonn

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.

Includes all positive integers.

Crossrefs

For prime factors instead of indices we have A075254, subtracted A075255.

Positions of first appearances are A379682.

For minus instead of plus we have A325036, which takes the following values:

- zero: A301987, counted by A001055

- negative: A325037, counted by A114324

- positive: A325038, counted by A096276 shifted right

- negative one: A325041, counted by A028422

- one: A325042, counted by A001055 shifted right

- nonnegative: A325044, counted by A096276

- nonpositive: A379721, counted by A319005

A000040 lists the primes, differences A001223.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798, counted by A001222.

Cf. A000720, A003963, A178503, A175508, A319000, A325032, A325033, A325034, A325035, A325040.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Plus@@prix[n]+Times@@prix[n],{n,30}]

Formula

a(n) = A056239(n) + A003963(n).

The last position of k is 2^(k-1).