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A348935
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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.
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3
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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MATHEMATICA
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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 *)
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PROG
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(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)};
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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