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A263569 Number of distinct prime divisors p of 2*n such that lpf(2*n - p) = p, where lpf = least prime factor (A020639).

%I A263569 #17 Nov 30 2021 13:27:37
%S A263569 0,1,2,1,2,2,2,1,2,1,2,2,2,1,3,1,2,2,2,2,2,1,2,2,1,1,2,2,2,3,2,1,2,1,
%T A263569 2,2,2,1,2,1,2,3,2,1,3,1,2,2,2,2,2,1,2,2,1,1,2,1,2,3,2,1,3,1,2,3,2,1,
%U A263569 2,2,2,2,2,1,3,1,2,2,2,2,2,1,2,3,1,1,2,1,2,3,2,1,2,1,2,2,2,1,3,1,2,2,2,1,4
%N A263569 Number of distinct prime divisors p of 2*n such that lpf(2*n - p) = p, where lpf = least prime factor (A020639).
%C A263569 a(n) gives the number of times 2*n occurs in A061228.
%H A263569 Antti Karttunen, <a href="/A263569/b263569.txt">Table of n, a(n) for n = 1..20000</a>
%e A263569 a(10) = 1 since the distinct prime divisors of 2*10 = 20 are 2 and 5, A020639(20 - 2) = 2 and A020639(20 - 5) = 3, so only prime 2 is to be considered.
%e A263569 a(15) = 3 since the distinct prime divisors of 2*15 = 30 are 2, 3 and 5, A020639(30 - 2) = 2 and A020639(30 - 3) = 3 and A020639(30 - 5) = 5, so all three prime 2, 3 and 5 are to be considered.
%t A263569 f[n_] := Select[First /@ FactorInteger[2 n], FactorInteger[2 n - #][[1, 1]] == # &]; Length /@ Table[f@ n, {n, 2, 105}] (* _Michael De Vlieger_, Oct 22 2015 *)
%o A263569 (PARI) a(n) = {my(f=factor(2*n)); sum(k=1, #f~, p=f[k,1]; p == factor(2*n-p)[1,1]);} \\ _Michel Marcus_, Oct 31 2015
%Y A263569 Cf. A020639, A061228.
%K A263569 nonn
%O A263569 1,3
%A A263569 _Gionata Neri_, Oct 21 2015