A354514 Numbers k such that m - gpf(m) = k has solutions m >= 2, gpf = A006530.

0, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88

Offset: 1

Author

Jianing Song, Aug 16 2022

Keywords

• nonn
• easy

Comments

Numbers k such that there is a prime p such that gpf(k+p) = p (such p must be a prime factor of n).

Numbers k such that there is a prime factor p of k such that k+p is p-smooth.

A076563 sorted and duplicates removed.

Examples

			0 is a term because 0 = p - gpf(p) for every prime p.
if k/gpf(k) <= nextprime(gpf(k)) - 2, where nextprime = A151800, then k is a term since k+gpf(k) <= gpf(k)*(nextprime(gpf(k)) - 1) implies gpf(k+gpf(k)) = gpf(k).
		

Links

• Jianing Song, Table of n, a(n) for n = 1..8650 (all terms <= 10000)

Crossrefs

Cf. A006530, A076563, A151800.

0 together with indices of positive terms in A354512. Complement of A354515.

Programs

PARI

gpf(n) = vecmax(factor(n)[, 1]);
isA354514(n) = if(n, my(f=factor(n)[, 1]); for(i=1, #f, if(gpf(n+f[i])==f[i], return(1))); 0, 1)