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A387166 Numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)) > 1, and that satisfy Euler's condition for odd perfect numbers (A228058).

%I A387166 #9 Aug 28 2025 17:26:06
%S A387166 14157,33525,101025,118825,129605,281025,300713,301725,335405,348525,
%T A387166 358925,438525,573525,618525,686025,688205,696725,742577,776025,
%U A387166 838125,909225,911025,978525,1046025,1079225,1099805,1226025,1293525,1316025,1322893,1428889,1451025,1529045,1563525,1698525,1721025,1788525,1991025,2036025
%N A387166 Numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)) > 1, and that satisfy Euler's condition for odd perfect numbers (A228058).
%H A387166 Antti Karttunen, <a href="/A387166/b387166.txt">Table of n, a(n) for n = 1..20000</a>
%H A387166 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A387166 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%o A387166 (PARI)
%o A387166 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A387166 isA228058(n) = if(!(n%2)||(omega(n)<2), 0, my(f=factor(n), y=0); for(i=1, #f~, if(1==(f[i, 2]%4), if((1==y)||(1!=(f[i, 1]%4)), return(0), y=1), if(f[i, 2]%2, return(0)))); (y));
%o A387166 isA349176(n) = if(!(n%2),0,my(u=A003961(n),t=gcd(u,n)); (t>1)&&(gcd(u,sigma(n))==t));
%o A387166 isA387166(n) = (isA228058(n) && isA349176(n));
%Y A387166 Intersection of A228058 and A349176.
%Y A387166 Intersection of A387164 and A104210, or equally, intersection of A387164 and A349166.
%Y A387166 Setwise difference A387164 \ A387167.
%Y A387166 Cf. A000203, A003961.
%K A387166 nonn,new
%O A387166 1,1
%A A387166 _Antti Karttunen_, Aug 28 2025