A324214 Sequence lists numbers k > 1 such that k^2 == phi(k) (mod sigma(k)), where phi = A000010 and sigma = A000203.

2, 4, 8, 12, 16, 32, 64, 76, 114, 128, 170, 256, 512, 1024, 1824, 2048, 2166, 4096, 7752, 8192, 16384, 16514, 28896, 32768, 41154, 59400, 65536, 68526, 90914, 131072, 177714, 230280, 262144, 276002, 428064, 524288, 722400, 781926, 1048576, 1299674, 1414272, 1546488

Offset

1,1

Comments

All powers of 2 (A000079) are part of the sequence.

Yes, proof: if k = 2^q with q >= 1, we have: k^2 = 2^(2q), phi(k) = 2^(q-1) and sigma(k) = 2^(q+1) - 1. Then, k^2 - phi(k) = 2^(q-1) * sigma(k). - Bernard Schott, Feb 20 2019

Contains 6*19^k for k >= 1. - David A. Corneth, May 29 2019

Links

Giovanni Resta, Table of n, a(n) for n = 1..180 (first 80 terms from Robert Israel)

Formula

Solutions of k^2 mod sigma(k) = phi(k).

Example

sigma(114) = 240 and 114^2 mod 240 = 36 = phi(114).

Maple

with(numtheory): op(select(n->n^2 mod sigma(n)=phi(n), [$1..1546488]));

Mathematica

Select[Range[10^5], Mod[#^2, DivisorSigma[1, #]] == EulerPhi[#] &] (* Giovanni Resta, May 29 2019 *)

Prog

(PARI) isok(k) = (k > 1) && ((k^2 % sigma(k)) == eulerphi(k)); \\ Michel Marcus, Feb 20 2019

Crossrefs

Cf. A000010, A000079, A000203, A324215, A324216.

Sequence in context: A152768 A266047 A375599 * A371732 A368507 A273109

Adjacent sequences: A324211 A324212 A324213 * A324215 A324216 A324217

Keyword

nonn

Author

Paolo P. Lava, Feb 18 2019

Status

approved