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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.
6
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
1,1
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.
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 *)
PROG
(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))));
KEYWORD
nonn
AUTHOR
Antti Karttunen, Nov 04 2021
STATUS
approved