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

Author

Amiram Eldar, Nov 29 2021

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A000005, A000010, A000203, A005101.

Subsequence of A349758.

Programs

Mathematica

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

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