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

%I A354512 #25 Aug 17 2022 05:07:13
%S A354512 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,
%T A354512 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,
%U A354512 2,1,1,0,1,1,1,1,2,2,1,0,0,1,1,0,2,1,1,1,1,0,2
%N A354512 Number of solutions m >= 2 to m - gpf(m) = n, gpf = A006530.
%C A354512 Number of primes p such that gpf(n+p) = p (such p must be prime factors of n).
%C A354512 Number of distinct prime factors p of n such that n+p is p-smooth.
%C A354512 Clearly we have a(n) <= omega(n) for all n, omega = A001221. The differences are given by A354527.
%C A354512 Is this sequence unbounded? Note that 4 does not appear until a(1660577).
%H A354512 Jianing Song, <a href="/A354512/b354512.txt">Table of n, a(n) for n = 1..10000</a>
%e A354512 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.
%e A354512 a(12) = 0 since the prime factors of 12 are 2,3, and we have gpf(12+2) != 2 and gpf(12+3) != 3.
%o A354512 (PARI) gpf(n) = vecmax(factor(n)[, 1]);
%o A354512 a(n) = my(f=factor(n)[, 1]); sum(i=1, #f, gpf(n+f[i])==f[i])
%Y A354512 Cf. A006530, A076563, A001221, A354516 (indices of first occurrence of each number), A354527.
%Y A354512 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)).
%K A354512 nonn,easy
%O A354512 1,6
%A A354512 _Jianing Song_, Aug 16 2022