A354514 -id:A354514 - cp's OEIS frontend

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

Jianing Song, Aug 16 2022

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).

			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.

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

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)).

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

A354515 Numbers k such that m - gpf(m) = k has no solution m >= 2, gpf = A006530.

Original entry on oeis.org

1, 4, 8, 12, 16, 18, 27, 32, 36, 48, 50, 54, 60, 64, 72, 80, 81, 84, 90, 96, 100, 108, 112, 125, 128, 132, 135, 144, 147, 150, 160, 162, 176, 180, 192, 196, 198, 200, 208, 210, 216, 224, 225, 234, 242, 243, 250, 252, 256, 270, 275, 280, 288, 294, 300, 306, 320, 324

Offset: 1

Jianing Song, Aug 16 2022

Numbers k such that there is no prime p such that gpf(k+p) = p.

Numbers k such that there is no prime factor p of k such that k+p is p-smooth.

			12 is a term since the prime factors of 12 are 2,3, and we have gpf(12+2) != 2 and gpf(12+3) != 3.

Jianing Song, Table of n, a(n) for n = 1..9826 (all terms <= 80000)

Cf. A006530, A076563.

Indices of 0 in A354512. Complement of A354514.

gpf(n) = vecmax(factor(n)[, 1]);
isA354515(n) = if(n, my(f=factor(n)[, 1]); for(i=1, #f, if(gpf(n+f[i])==f[i], return(0))); 1, 0)

cp's OEIS Frontend

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

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