A348938 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.

153, 245, 261, 369, 425, 477, 637, 725, 801, 833, 845, 873, 909, 981, 1017, 1025, 1233, 1325, 1341, 1377, 1421, 1557, 1573, 1629, 1773, 1805, 1813, 2009, 2057, 2061, 2097, 2169, 2225, 2313, 2349, 2421, 2425, 2525, 2529, 2597, 2637, 2645, 2725, 2853, 2873, 2989, 3141, 3177, 3321, 3357, 3425, 3501, 3509, 3577, 3609, 3681

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 A348939.

Of the numbers in range 1..2^20, 9644 reside in this sequence and 3865 in A348939. Of the numbers <= 2^25, 229480 are in this sequence, and 88270 in A348939.

Links

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

Crossrefs

Intersection of A228058 and A348748.

Cf. A064989, A326042, A348939.

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, 4000, 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));
isA348748(n) = ((n%2)&&(A064989(sigma(n)) < A064989(n)));
isA348938(n) = (isA228058(n)&&isA348748(n));