A349760 Numbers k such that d(k) = A000005(k), sigma(k) = A000203(k) and phi(k) = A000010(k) are all abundant numbers (A005101).

84, 90, 108, 126, 132, 140, 150, 156, 180, 198, 220, 224, 228, 234, 252, 260, 276, 294, 300, 306, 308, 315, 336, 342, 348, 350, 352, 360, 364, 372, 380, 396, 414, 416, 420, 432, 444, 460, 476, 486, 490, 492, 495, 500, 504, 516, 522, 525, 528, 532, 540, 550, 558

Offset

1,1

Comments

Sándor (2005) proved that this sequence is infinite by showing that it includes all the numbers of the form 11 * p^11 * k where p != 11 is a prime and k is any number coprime to 11*p.

Links

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

József Sándor, Selected Chapters of Geometry, Analysis and Number Theory, 2005, pp. 132-134.

Example

84 is a term since d(84) = 12, sigma(84) = 224 and phi(84) = 24 are all abundant numbers: sigma(12) = 28 > 2*12 = 24, sigma(224) = 504 > 2*224 = 448 and sigma(24) = 60 > 2*24 = 48.

Mathematica

abQ[n_] := DivisorSigma[1, n] > 2*n; q[n_] := And @@ abQ /@ Join[DivisorSigma[{0, 1}, n], {EulerPhi[n]}]; Select[Range[500], q]

Prog

(PARI) isab(k) = sigma(k) > 2*k; \\ A005101
isok(k) = my(f=factor(k)); isab(numdiv(f)) && isab(sigma(f)) && isab(eulerphi(f)); \\ Michel Marcus, Dec 03 2021

Crossrefs

Cf. A000005, A000010, A000203, A005101.

Subsequence of A349758.

Sequence in context: A128873 A095607 A068405 * A045569 A219183 A289218

Adjacent sequences: A349757 A349758 A349759 * A349761 A349762 A349763

Keyword

nonn

Author

Amiram Eldar, Nov 29 2021

Status

approved