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A387167 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 A387167 #7 Aug 28 2025 17:26:19
%S A387167 117,153,333,369,425,477,549,637,657,845,873,909,925,1017,1053,1233,
%T A387167 1325,1377,1413,1421,1445,1525,1557,1629,1737,1773,1805,1813,1825,
%U A387167 2009,2097,2169,2225,2313,2493,2525,2529,2597,2637,2725,2817,2825,2853,2989,2997,3033,3177,3321,3357,3425,3509,3573,3577,3609,3725
%N A387167 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).
%C A387167 Terms k of A228058 for which k and A003961(k) are relatively prime, and also sigma(k) and A003961(k) are coprime.
%H A387167 Antti Karttunen, <a href="/A387167/b387167.txt">Table of n, a(n) for n = 1..20000</a>
%H A387167 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A387167 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%o A387167 (PARI)
%o A387167 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A387167 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 A387167 isA349177(n) = if(!(n%2),0,my(u=A003961(n),t=gcd(u,n)); (1==t)&&(gcd(u,sigma(n))==t));
%o A387167 isA387167(n) = (isA228058(n) && isA349177(n));
%Y A387167 Intersection of A228058 and A349177.
%Y A387167 Intersection of A387164 and A319630, or equally, intersection of A387164 and A349165.
%Y A387167 Setwise difference A387164 \ A387166.
%Y A387167 Cf. A000203, A003961.
%Y A387167 Subsequence of A387164 from which this differs for the first time at n=201, where a(201) = 14225, while A387164(201) = 14157.
%K A387167 nonn,new
%O A387167 1,1
%A A387167 _Antti Karttunen_, Aug 28 2025