A354514 - OEIS

%I #17 Aug 17 2022 05:07:33

%S 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,

%T 33,34,35,37,38,39,40,41,42,43,44,45,46,47,49,51,52,53,55,56,57,58,59,

%U 61,62,63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,82,83,85,86,87,88

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

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

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

%C A076563 sorted and duplicates removed.

%H Jianing Song, <a href="/A354514/b354514.txt">Table of n, a(n) for n = 1..8650</a> (all terms <= 10000)

%e 0 is a term because 0 = p - gpf(p) for every prime p.

%e 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).

%o (PARI) gpf(n) = vecmax(factor(n)[, 1]);

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

%Y Cf. A006530, A076563, A151800.

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

%K nonn,easy

%O 1,2

%A _Jianing Song_, Aug 16 2022