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q[n_] := Module[{f = FactorInteger[n]}, p = f[[;; , 1]]; e = f[[;; , 2]]; odde = Select[e, OddQ]; Length[e] > 1 && Length[odde] == 1 && Divisible[odde[[1]] - 1, 4] && Divisible[p[[Position[e, odde[[1]]][[1, 1]]]] - 1, 4]]; f[2, e_] := 1; f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[1, 10000, 2], q[#] && s[DivisorSigma[1, #]] > s[#] &] (* Amiram Eldar, Nov 04 2021 *)

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A064989(n) = { my(f = factor(n)); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f) };

Odd numbers k for which A326042(A064989(sigma(k)) > A064989(k), and which are of the form p^(1+4k) * r^2, where p is prime of the form 1+4m, r > 1, and gcd(p,r) = 1.

Any Obviously, any hypothetical odd perfect number would be neither in this sequence nor in A348938.

<a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

<a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

A003961A064989(n) = { my(f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = nextprimeprecprime(f[i, 1]+-1)); factorback(f) }; \\ From A003961

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f)};

A326042(n) = A064989(sigma(A003961(n)));

isA348749(n) = ((n%2)&&(A326042(A064989(sigma(n)) > A064989(n)));

Cf. A000203, A064989, A326042, A348938.

Any hypothetical odd perfect number would be neither in this sequence nor in A348938.

allocated Odd numbers k for Antti Karttunenwhich A326042(A064989(k)) > A064989(k), and which are of the form p^(1+4k) * r^2, where p is prime of the form 1+4m, r > 1, and gcd(p,r) = 1.

45, 117, 325, 333, 405, 549, 605, 657, 925, 1053, 1413, 1445, 1525, 1737, 1825, 2205, 2493, 2817, 2825, 2925, 2997, 3033, 3573, 3645, 3789, 3825, 3925, 4113, 4825, 4869, 4941, 5445, 5517, 5733, 5913, 5949, 6057, 6425, 6525, 6597, 6813, 6925, 7025, 7497, 7605, 7825, 7893, 8125, 8325, 8425, 8973, 9225, 9477, 9837, 9925

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(PARI)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f)};

A326042(n) = A064989(sigma(A003961(n)));

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));

isA348749(n) = (n%2&&(A326042(A064989(n)) > A064989(n)));

isA348939(n) = (isA228058(n)&&isA348749(n));

Intersection of A228058 and A348749.

Cf. A064989, A326042, A348938.

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Antti Karttunen, Nov 04 2021

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