A354512 Number of solutions m >= 2 to m - gpf(m) = n, gpf = A006530.

0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 1, 2, 2, 0, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 2, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 2, 1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 1, 1, 0, 2, 1, 1, 1, 1, 0, 2

Offset

1,6

Comments

Number of primes p such that gpf(n+p) = p (such p must be prime factors of n).

Number of distinct prime factors p of n such that n+p is p-smooth.

Clearly we have a(n) <= omega(n) for all n, omega = A001221. The differences are given by A354527.

Is this sequence unbounded? Note that 4 does not appear until a(1660577).

Links

Jianing Song, Table of n, a(n) for n = 1..10000

Example

a(78) = 2 since the prime factors of 78 are 2,3,13, and we have gpf(78+3) = 3 and gpf(78+13) = 13, so the solutions to m - gpf(m) = 78 are m = 78+3 = 81 or m = 78+13 = 91. Note that gpf(78+2) != 2.
a(12) = 0 since the prime factors of 12 are 2,3, and we have gpf(12+2) != 2 and gpf(12+3) != 3.

Prog

(PARI) gpf(n) = vecmax(factor(n)[, 1]);
a(n) = my(f=factor(n)[, 1]); sum(i=1, #f, gpf(n+f[i])==f[i])

Crossrefs

Cf. A006530, A076563, A001221, A354516 (indices of first occurrence of each number), A354527.

Cf. A354514 (0 together with indices of positive terms), A354515 (indices of 0), A354516, A354525 (indices n for which a(n) reaches omega(n)), A354526 (indices n for which a(n) is smaller than omega(n)).

Sequence in context: A016353 A016398 A024359 * A088705 A065712 A153172

Adjacent sequences: A354509 A354510 A354511 * A354513 A354514 A354515

Keyword

nonn,easy

Author

Jianing Song, Aug 16 2022

Status

approved