A348753 Numbers k congruent to 1 or 5 mod 6, for which A064989(A064989(sigma(k))) < A064989(A064989(k)), where A064989 shifts the prime factorization one step towards lower primes, and sigma is the sum of divisors function.

5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 37, 41, 43, 47, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 125, 127, 131, 133, 137, 139, 143, 145, 149, 151, 155, 157, 161, 163, 167, 173, 179, 181, 185, 187, 191, 193, 197, 199, 203, 205, 209, 211, 215, 217, 221, 223, 227

Offset: 1

Author

Antti Karttunen, Nov 04 2021

Keywords

• nonn

Comments

Sequence A003961(A003961(A348751(n))), n>=1, sorted into ascending order.

a(38) = 125 is the first term not in A276378.

Not a subsequence of A348748. The first terms that occur here but not there are: 529, 605, 2825, 6425, 7025, 8425, 10825, 15425, 16025, 16325, 16925, 17689, ...

The first squares in this sequence are: 361, 529, 961, 1369, 1849, 2209, 2809, 3721, etc., see A348935 for their square roots.

Of the natural numbers < 2^20, 347712 are in this sequence and only 1812 in A348754.

Links

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

Crossrefs

Cf. A003961, A007310, A064989, A276378, A348750, A348751, A348754.

Cf. also A348748, A348931, A348935.

Programs

Mathematica

f[2, e_] := 1; f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[250], MemberQ[{1, 5}, Mod[#, 6]] && s[s[DivisorSigma[1, #]]] < s[s[#]] &] (* Amiram Eldar, Nov 04 2021 *)

PARI

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)};
isA348753(n) = ((n%2)&&(n%3)&&(A064989(A064989(sigma(n))) < A064989(A064989(n))));