A348939 Odd numbers k for which 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.

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

Offset: 1

Author

Antti Karttunen, Nov 04 2021

Keywords

• nonn

Comments

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

Links

• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Intersection of A228058 and A348749.

Cf. A000203, A064989, A326042, A348938.

Programs

Mathematica

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 *)

PARI

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) };
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)&&(A064989(sigma(n)) > A064989(n)));
isA348939(n) = (isA228058(n)&&isA348749(n));