A348935 Numbers k congruent to 1 or 5 mod 6, for which A064989(A064989(sigma(k^2))) < A064989(A064989(k^2)), where A064989 shifts the prime factorization one step towards lower primes, and sigma is the sum of divisors function.

19, 23, 31, 37, 43, 47, 53, 61, 67, 73, 79, 83, 97, 103, 107, 109, 113, 127, 133, 137, 139, 149, 151, 157, 163, 179, 181, 191, 193, 197, 199, 211, 217, 223, 229, 233, 239, 241, 247, 251, 257, 259, 263, 269, 271, 277, 281, 283, 299, 301, 307, 311, 313, 317, 331, 335, 337, 341, 347, 349, 353, 359, 367, 371, 373, 379

Offset: 1

Author

Antti Karttunen, Nov 04 2021

Keywords

• nonn

Comments

Square roots of squares present in A348753.

Any hypothetical odd term y of A005820 must by necessity be a square. If y is also a nonmultiple of 3, then the square root x = A000196(y) of such a number y must satisfy the condition that for all nontrivial unitary divisor pairs d and x/d [with gcd(d,x/d) = 1, 1 < d < x], the other unitary divisor (d) should reside in this sequence, and the other divisor (x/d) in A348936. The explanation is similar to the one given in A348738. See also comments in A348933.

In range 1..2^20, there are 256143 numbers in this sequence and 93381 numbers in A348936.

The composites in this sequence are: 133, 217, 247, 259, 299, 301, 335, 341, 371, etc.

Links

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

Crossrefs

Cf. A000196, A007310, A348750, A348753, A348933, A348936.

Programs

Mathematica

f[2, e_] := 1; f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[400], MemberQ[{1, 5}, Mod[#, 6]] && s[s[DivisorSigma[1, #^2]]] < s[s[#^2]] &] (* 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)};
isA348935(n) = ((n%2)&&(n%3)&&(A064989(A064989(sigma(n^2))) < A064989(A064989(n^2))));