A354516 Smallest k such that m - gpf(m) = k has exactly n solutions m >= 2, gpf = A006530; or -1 if no such k exists.

1, 2, 6, 483, 1660577

Offset: 0

Author

Jianing Song, Aug 16 2022

Keywords

• nonn
• hard
• more

Comments

Smallest k such that there are exactly n primes p such that gpf(k+p) = p (such p must be prime factors of k).

Smallest k having exactly n distinct prime factors p such that k+p is p-smooth.

Conjectures (if no term equals -1): (Start)

(1) Sequence is strictly increasing.

(2) All terms are squarefree.

(3) All terms are in A354525. (End)

Examples

			a(4) = 1660577: 1660577 = 17*23*31*127, and we have 1660577+17 = 2*13^2*17^3 is 17-smooth, 1660577+23 = 2^3*5^2*19^2*23 is 23-smooth, 1660577+31 = 2^6*3^3*31^2 is 31-smooth, 1660577+137 = 2*11*19*29*137, so m - gpf(m) = 1660577 has 4 solutions m = 1660577+17 = 1660594, 1660577+23 = 1660600, 1660577+31 = 1660608, and 1660577+137 = 1660714.
		

Crossrefs

Cf. A006530, A076563, A354512, A354525.

Programs

PARI

gpf(n) = vecmax(factor(n)[, 1]);
A354512(n) = my(f=factor(n)[, 1]); sum(i=1, #f, gpf(n+f[i])==f[i]);
a(n) = my(k=1); while(omega(k)A354512(k) != n, k++); return(k)