A380409 Product of prime indices plus sum of prime factors.

1, 3, 5, 5, 8, 7, 11, 7, 10, 10, 16, 9, 19, 13, 14, 9, 24, 12, 27, 12, 18, 18, 32, 11, 19, 21, 17, 15, 39, 16, 42, 11, 24, 26, 24, 14, 49, 29, 28, 14, 54, 20, 57, 20, 23, 34, 62, 13, 30, 21, 34, 23, 69, 19, 31, 17, 38, 41, 76, 18, 79, 44, 29, 13, 36, 26, 86

Offset

1,2

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, with product A003963.

Links

Table of n, a(n) for n=1..67.

Formula

a(n) = A003963(n) + A001414(n).

Example

72 has prime factors {2,2,2,3,3} and prime indices {1,1,1,2,2}, so a(72) = 12 + 4 = 16.

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Total[Prime/@prix[n]]+Times@@prix[n], {n, 100}]

Crossrefs

For factors instead of indices we have A075254.

For indices instead of factors we have A379681.

For minus instead of plus we have A380344, zeros A331384.

Triangles:

- A027746 = prime factors

- A112798 = prime indices

Statistics:

- A000027 = product of prime factors = row products of A027746

- A001414 = sum of prime factors = row sums of A027746

- A003963 = product of prime indices = row products of A112798

- A056239 = sum of prime indices = row sums of A112798

Combinations:

- A075254 = product of factors + sum of factors = A000027 + A001414

- A075255 = product of factors - sum of factors = A000027 - A001414

- A178503 = product of factors - sum of indices = A000027 - A056239

- A325036 = product of indices - sum of indices = A003963 - A056239

- A379681 = product of indices + sum of indices = A003963 + A056239

- A380344 = product of indices - sum of factors = A003963 - A001414

- A380345 = product of factors + sum of indices = A000027 + A056239

- A380409 = product of indices + sum of factors = A003963 + A001414

A000040 lists the primes, differences A001223.

A001222 counts prime factors with multiplicity.

A055396 gives least prime index, greatest A061395.

Cf. A000720, A175508, A319000, A325032, A325033, A325034, A325035, A325040, A379682, A380220, A380410.

Sequence in context: A019632 A021285 A138575 * A263209 A388193 A101330

Adjacent sequences: A380406 A380407 A380408 * A380410 A380411 A380412

Keyword

nonn

Author

Gus Wiseman, Jan 25 2025

Status

approved