A348935 -id:A348935 - cp's OEIS frontend

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

Antti Karttunen, Nov 04 2021

nonn

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.

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

Cf. also A348748, A348931, A348935.

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 *)

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))));

A348933 Numbers k congruent to 1 or 5 mod 6, for which A348930(k^2) < k^2.

Original entry on oeis.org

7, 13, 19, 31, 35, 37, 43, 61, 65, 67, 73, 77, 79, 91, 95, 97, 103, 109, 119, 127, 133, 139, 143, 151, 155, 157, 161, 163, 175, 181, 185, 193, 199, 203, 209, 211, 215, 217, 221, 223, 229, 241, 247, 259, 271, 277, 283, 287, 299, 301, 305, 307, 313, 323, 325, 329, 331, 335, 337, 341, 349, 365, 367, 371, 373, 377, 379

Offset: 1

Antti Karttunen, Nov 04 2021

nonn

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 divisor should reside in this sequence, and the other divisor in A348934. The explanation is similar to the one given in A348738. See also comments in A348935.

Index entries for sequences related to sigma(n)

Cf. A000196, A005820, A348738, A348930, A348931, A348934, A348935.

s[n_] := n / 3^IntegerExponent[n, 3]; Select[Range[400], MemberQ[{1, 5}, Mod[#, 6]] && s[DivisorSigma[1, #^2]] < #^2 &] (* Amiram Eldar, Nov 04 2021 *)

A038502(n) = (n/3^valuation(n, 3));
A348930(n) = A038502(sigma(n));
isA348933(n) = ((n%2)&&(n%3)&&(A348930(n^2)<(n^2)));

A348936 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.

Original entry on oeis.org

5, 7, 11, 13, 17, 25, 29, 35, 41, 49, 55, 59, 65, 71, 77, 85, 89, 91, 95, 101, 115, 119, 121, 125, 131, 143, 145, 155, 161, 167, 169, 173, 175, 185, 187, 203, 205, 209, 215, 221, 227, 235, 245, 253, 265, 275, 287, 289, 293, 295, 305, 319, 323, 325, 329, 343, 355, 361, 365, 377, 383, 385, 391, 413, 415, 425, 445, 451

Offset: 1

Antti Karttunen, Nov 04 2021

nonn

Square roots of squares present in A348754.

See comments in A348935.

Cf. A000203, A003961, A007310, A064989, A348750, A348754, A348934, A348935.

f[2, e_] := 1; f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[450], MemberQ[{1, 5}, Mod[#, 6]] && s[s[DivisorSigma[1, #^2]]] > s[s[#^2]] &] (* Amiram Eldar, Nov 04 2021 *)

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)};
isA348936(n) = ((n%2)&&(n%3)&&(A064989(A064989(sigma(n^2))) > A064989(A064989(n^2))));

cp's OEIS Frontend

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.

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A348933 Numbers k congruent to 1 or 5 mod 6, for which A348930(k^2) < k^2.

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A348936 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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