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A386429 Odd composites k such that A342926(k) is even and A342926(2*k) is a multiple of 3 and which satisfy Euler's condition for odd perfect numbers (A228058).

%I A386429 #12 Aug 27 2025 18:22:07
%S A386429 45,153,261,325,369,405,477,801,909,925,1017,1233,1341,1377,1525,1557,
%T A386429 1573,1773,1825,2097,2205,2313,2349,2421,2425,2529,2637,2725,2853,
%U A386429 3177,3321,3501,3609,3645,3757,3825,3925,4041,4149,4293,4477,4525,4581,4689,4825,5013,5121,5337,5445,5553,5725,5733,5769,5877,6025
%N A386429 Odd composites k such that A342926(k) is even and A342926(2*k) is a multiple of 3 and which satisfy Euler's condition for odd perfect numbers (A228058).
%C A386429 Sequence contains also some terms of A386428: 28125, 253125, 1378125, 2278125, 3341637, 3403125, 4753125, etc.
%H A386429 Antti Karttunen, <a href="/A386429/b386429.txt">Table of n, a(n) for n = 1..10000</a>
%H A386429 <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>
%H A386429 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%o A386429 (PARI)
%o A386429 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A386429 A342926(n) = (A003415(sigma(n))-n);
%o A386429 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 A386429 isA347874(n) = ((n%2)&&!isprime(n)&&!(A342926(n)%2)&&!(A342926(2*n)%3));
%o A386429 isA386429(n) = (isA228058(n) && isA347874(n));
%Y A386429 Intersection of A228058 and A347874.
%Y A386429 Conjectured to be also the intersection of A228058 and A349751.
%Y A386429 Setwise difference A228058 \ A351574.
%Y A386429 Cf. also A349755, A387162.
%K A386429 nonn,new
%O A386429 1,1
%A A386429 _Antti Karttunen_, Aug 18 2025