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A226448 Composite squarefree numbers k such that the ratios (k - 1/2)/(p - 1/2) are integers for each prime p dividing k.

%I A226448 #26 Jul 16 2021 20:01:51
%S A226448 260054438,597892523,1200695738,3287998643,3423456563,10524308498,
%T A226448 13292859563,15646705718,19441707170,33309521438,38848586123,
%U A226448 43312628678,61899936935,72422400713,75439031063,85338414662,112419230963,132624705038,136084511063,141236121758
%N A226448 Composite squarefree numbers k such that the ratios (k - 1/2)/(p - 1/2) are integers for each prime p dividing k.
%C A226448 Also composite squarefree numbers k such that (2p - 1) | (2k - 1).
%H A226448 Giovanni Resta, <a href="/A226448/b226448.txt">Table of n, a(n) for n = 1..45</a> (terms < 10^12)
%e A226448 3287998643 is a term since it is equal to 743*787*5623 and 3287998643-1/2 divided by 743-1/2, 787-1/2 and 5623-1/2 gives 3 integers, namely 4428281, 4180545 and 584793.
%p A226448 with(numtheory); ListA226448:=proc(i, j) local c, d, n, ok, p;
%p A226448 for n from 2 to i do if not isprime(n) then p:=ifactors(n)[2]; ok:=1;
%p A226448 for d from 1 to nops(p) do if p[d][2]>1 or not type((n-j)/(p[d][1]-j), integer) then ok:=0; break; fi; od;
%p A226448 if ok=1 then print(n); fi; fi; od; end: ListA226448(10^9, 1/2); # _Paolo P. Lava_, Oct 06 2013
%o A226448 (PARI) is(n, P)=n=2*n-1; for(i=1, #P, if(n%(2*P[i]-1), return(0))); 1
%o A226448 list(lim, P=[], n=1, mx=lim\2)=my(v=[], t); if(#P>1&&is(n, P), v=[n]); P=concat(P, 0); forprime(p=2, min(lim, mx), P[#P]=p; t=list(lim\p, P, n*p, p-1); if(#t, v=concat(v, t))); v \\ _Charles R Greathouse IV_, Jun 07 2013
%Y A226448 Cf. A208728, A225702-A225720, A226020, A226111-A226114, A226364.
%K A226448 nonn
%O A226448 1,1
%A A226448 _Paolo P. Lava_ and _Giovanni Resta_, Jun 07 2013