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Numbers k such that sigma(k) = A000203(k) is an abundant number (A005101) and phi(k) = A000010(k) is a deficient number (A005100).
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%I #10 Dec 06 2021 03:12:24

%S 6,10,11,15,17,20,22,23,24,30,34,40,46,47,51,53,59,60,68,69,80,83,85,

%T 92,94,96,102,106,107,118,120,131,136,137,138,141,149,160,166,167,170,

%U 173,177,179,188,191,204,214,227,233,235,236,239,240,249,251,255,257

%N Numbers k such that sigma(k) = A000203(k) is an abundant number (A005101) and phi(k) = A000010(k) is a deficient number (A005100).

%C Sándor (2005) proved that this sequence is infinite by showing that it includes all the numbers of the form 3 * 2^k where k == 11 (mod 12). If gcd((k+1)/12, 6) = 1, then this number is also nobly abundant (A349758).

%H Amiram Eldar, <a href="/A349761/b349761.txt">Table of n, a(n) for n = 1..10000</a>

%H József Sándor, <a href="https://blngcc.files.wordpress.com/2008/11/jozsel-sandor-selected-chaters-of-geometry-analysis-and-number-theory.pdf">Selected Chapters of Geometry, Analysis and Number Theory</a>, 2005, pp. 132-134.

%e 6 is a term since sigma(6) = 12 is an abundant number, sigma(12) = 28 > 2*12 = 24, and phi(6) = 2 is a deficient number, sigma(2) = 3 < 2*2 = 4.

%t abQ[n_] := DivisorSigma[1, n] > 2*n; defQ[n_] := DivisorSigma[1, n] < 2*n; q[n_] := abQ[DivisorSigma[1, n]] && defQ[EulerPhi[n]]; Select[Range[250], q]

%Y Cf. A000010, A000203, A005100, A005101, A349758, A349759, A349762.

%K nonn

%O 1,1

%A _Amiram Eldar_, Nov 29 2021