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A378664 Greatest divisor d of n such that sigma(d) <= 2*d < A003961(d), or 1 if no such divisor exists, where A003961 is fully multiplicative with a(p) = nextprime(p).

%I A378664 #20 Dec 12 2024 15:14:32
%S A378664 1,1,1,4,1,6,1,8,9,10,1,6,1,14,15,16,1,9,1,10,21,1,1,8,1,1,27,28,1,15,
%T A378664 1,32,1,1,35,9,1,1,39,10,1,21,1,44,45,1,1,16,49,50,1,52,1,27,1,28,57,
%U A378664 1,1,15,1,1,63,64,1,6,1,68,69,35,1,9,1,1,75,76,1,39,1,16,81,1,1,28,1,1,1,44,1,45,91,92
%N A378664 Greatest divisor d of n such that sigma(d) <= 2*d < A003961(d), or 1 if no such divisor exists, where A003961 is fully multiplicative with a(p) = nextprime(p).
%C A378664 Largest term of {1} U A341614 that divides n.
%H A378664 Antti Karttunen, <a href="/A378664/b378664.txt">Table of n, a(n) for n = 1..65537</a>
%H A378664 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>
%H A378664 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A378664 a(n) <= A378665(n).
%t A378664 Table[If[Length[#] == 0, 1, Max[#]] &@ Select[Divisors[n], DivisorSigma[1, #] <= 2 # < (Times @@ Map[Power @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi[p] + 1], e}] - Boole[# == 1]) &], {n, 92}] (* _Michael De Vlieger_, Dec 06 2024 *)
%o A378664 (PARI)
%o A378664 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A378664 A341612(n) = ((sigma(n)<=(2*n))&&((2*n)<A003961(n)));
%o A378664 A378664(n) = { fordiv(n,d,if(A341612(n/d), return(n/d))); (1); };
%Y A378664 Cf. A000203, A003961, A337372, A341612, A341614, A378662, A378665, A378736 [= a(A005101(n))], A378737, A378738, A378739, A378740, A378741, A378742.
%Y A378664 Positions of fixed points (where a(n)=n) is given by {1} U A341614.
%Y A378664 Cf. A246281 (positions of 1's), A246282 (of terms > 0), A005101 (of terms that are neither 1 nor fixed points).
%K A378664 nonn
%O A378664 1,4
%A A378664 _Antti Karttunen_, Dec 06 2024