A380344 Product of prime indices minus sum of prime factors of n.

1, -1, -1, -3, -2, -3, -3, -5, -2, -4, -6, -5, -7, -5, -2, -7, -10, -4, -11, -6, -2, -8, -14, -7, -1, -9, -1, -7, -19, -4, -20, -9, -4, -12, 0, -6, -25, -13, -4, -8, -28, -4, -29, -10, 1, -16, -32, -9, 2, -3, -6, -11, -37, -3, -1, -9, -6, -21, -42, -6, -43

Offset

1,4

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..61.

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) = 4 - 12 = -8.

Mathematica

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

Crossrefs

Positions of 0 are A331384.

For plus instead of minus we have A380409.

Positions of positives are A380410.

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.

Sequence in context: A317623 A286244 A230010 * A230847 A234300 A181672

Adjacent sequences: A380341 A380342 A380343 * A380345 A380346 A380347

Keyword

sign

Author

Gus Wiseman, Jan 24 2025

Status

approved