cp's OEIS Frontend

A387164 Numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)), and that satisfy Euler's condition for odd perfect numbers (A228058).

%I A387164 #8 Aug 28 2025 17:26:13
%S A387164 117,153,333,369,425,477,549,637,657,845,873,909,925,1017,1053,1233,
%T A387164 1325,1377,1413,1421,1445,1525,1557,1629,1737,1773,1805,1813,1825,
%U A387164 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 A387164 Numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)), and that satisfy Euler's condition for odd perfect numbers (A228058).
%C A387164 Terms k of A228058 for which A322361(k) = A342671(k), or equally, such that A319626(k) = A349164(k).
%H A387164 Antti Karttunen, <a href="/A387164/b387164.txt">Table of n, a(n) for n = 1..20000</a>
%H A387164 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A387164 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%H A387164 <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>
%o A387164 (PARI)
%o A387164 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A387164 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 A387164 isA349174(n) = if(!(n%2), 0, my(u=A003961(n)); gcd(u, sigma(n))==gcd(u, n));
%o A387164 isA387164(n) = (isA228058(n) && isA349174(n));
%Y A387164 Intersection of A228058 and A349174.
%Y A387164 Union of A387166 and A387167.
%Y A387164 Differs from its subsequence A387167 for the first time at n=201, where a(201) = 14157, while A387167(201) = 14225.
%Y A387164 Cf. A000203, A003961, A319626, A322361, A342671, A349164.
%Y A387164 Cf. also A371082.
%K A387164 nonn,new
%O A387164 1,1
%A A387164 _Antti Karttunen_, Aug 28 2025