A115559 -id:A115559 - cp's OEIS frontend

A115558 a(n) is the square root of A115557(n).

1, 7, 11, 13, 19, 23, 29, 31, 43, 47, 53, 73, 83, 97, 103, 113, 127, 157, 179, 197, 199, 223, 227, 233, 239, 241, 251, 257, 271, 281, 311, 316, 317, 333, 353, 389, 401, 409, 419, 421, 443, 449, 461, 467, 479, 491, 503, 509, 549, 563, 587, 593, 599, 617, 641

Offset: 1

Labos Elemer, Jan 25 2006

nonn

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A076360, A076361, A062068, A062069, A115557, A115559, A115560.

Select[Range[1000], DivisorSigma[0, DivisorSigma[1, #^2]] == DivisorSigma[1, DivisorSigma[0, #^2]] &] (* Amiram Eldar, Jan 28 2025 *)

isok(k) = numdiv(sigma(k^2)) == sigma(numdiv(k^2)); \\ Amiram Eldar, Jan 28 2025

The commutator [sigma, tau] is zero and a(n) is the square root of solutions. Both prime and composite numbers.

A115560 Twin prime pairs k-1 and k+1 such that the squares of both are present in A115557.

Original entry on oeis.org

11, 13, 29, 31, 197, 199, 239, 241, 419, 421, 659, 661, 881, 883, 1019, 1021, 1061, 1063, 1481, 1483, 1877, 1879, 3167, 3169, 3821, 3823, 4019, 4021, 4049, 4051, 4787, 4789, 6359, 6361, 7589, 7591, 9437, 9439, 13691, 13693, 14447, 14449, 14627, 14629, 16451, 16453

Offset: 1

Labos Elemer, Jan 25 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A076360, A076361, A062068, A062069, A115557, A115558, A115559.

ta={{0}};tb={{0}}; Do[s=DivisorSigma[1,DivisorSigma[0,n]]; s1=DivisorSigma[0,DivisorSigma[1,n]]; If[Equal[s-s1,0]&&IntegerQ[Sqrt[n]&&PrimeQ[Sqrt[n]]],Print[n]; ta=Append[ta,n];tb=Append[tb,Sqrt[n]]],{n,1,100000000}] ta=Delete[ta,1];tb=Delete[tb,1];ni=Intersection[tb,2+tb]; Union[ni,ni-2]

isok(n) = issquare(n) && (sigma(numdiv(n)) == numdiv(sigma(n))); \\ A115557
lista(nn) = {forprime(p=2, nn, if (isprime(p+2) && isok(p^2) && isok((p+2)^2), print1(p, ", ", p+2, ", ")););} \\ Michel Marcus, Jul 17 2019

The commutator [sigma, tau] is zero and a(n) is the square root of special prime solutions. These solutions are twin primes. Both twins are displayed.

More terms from Amiram Eldar, Jul 17 2019

cp's OEIS Frontend

A115558 a(n) is the square root of A115557(n).

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