A348752 -id:A348752 - cp's OEIS frontend

A348750 a(n) = A064989(A064989(sigma(A003961(A003961(n))))), where A003961 shifts the prime factorization of n one step towards larger primes, and A064989 shifts it back towards smaller primes.

1, 1, 1, 23, 1, 1, 3, 7, 13, 1, 1, 23, 2, 3, 1, 305, 1, 13, 2, 23, 3, 1, 1, 7, 39, 2, 4, 69, 13, 1, 3, 69, 1, 1, 3, 299, 5, 2, 2, 7, 1, 3, 1, 23, 13, 1, 2, 305, 53, 39, 1, 46, 23, 4, 1, 21, 2, 13, 11, 23, 1, 3, 39, 19501, 2, 1, 29, 23, 1, 3, 2, 91, 3, 5, 39, 46, 3, 2, 2, 305, 2791, 1, 9, 69, 1, 1, 13, 7, 11, 13, 6

Offset: 1

Antti Karttunen, Nov 02 2021

Cf. A000203, A003961, A003973, A064989, A326042, A348751 (a(n) < n), A348752 (a(n) > n).

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
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)};
A348750(n) = A064989(A064989(sigma(A003961(A003961(n)))));

a(n) = A064989(A326042(A003961(n))).

Multiplicative with a(p^e) = A064989(A064989((q^(e+1)-1)/(q-1))), where q = nextPrime(nextPrime(p)).

A348754 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.

Original entry on oeis.org

25, 49, 121, 169, 175, 275, 289, 325, 625, 841, 925, 1225, 1445, 1525, 1675, 1681, 1825, 2401, 3025, 3125, 3481, 3757, 3925, 4075, 4225, 4375, 4825, 5041, 5275, 5929, 6125, 6875, 6925, 7075, 7225, 7825, 7921, 8125, 8275, 8281, 9025, 9925, 10201, 10525, 10625, 10693, 11425, 11875, 12005, 12025, 13075, 13225, 13475

Offset: 1

Antti Karttunen, Nov 04 2021

nonn

Sequence A003961(A003961(A348752(n))), n=1.., sorted into ascending order.

Not a subsequence of A348749. The first terms that occur here but not there are: 169, 175, 275, 1675, 3757, 4075, 5275, 7075, 8275, 10693, 12025, ...

Cf. A003961, A007310, A064989, A348750, A348752, A348753.

Cf. also A348749, A348932, A348936 (square roots of squares present).

f[2, e_] := 1; f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[15000], 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)};
isA348754(n) = ((n%2)&&(n%3)&&(A064989(A064989(sigma(n))) > A064989(A064989(n))));

cp's OEIS Frontend

A348750 a(n) = A064989(A064989(sigma(A003961(A003961(n))))), where A003961 shifts the prime factorization of n one step towards larger primes, and A064989 shifts it back towards smaller primes.

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