This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A115405 #22 Feb 16 2025 08:33:00
%S A115405 1,2,3,4,5,7,8,9,11,13,15,16,17,19,21,23,25,27,29,31,32,33,35,37,39,
%T A115405 41,43,45,47,49,51,53,55,57,59,61,63,64,65,67,69,71,73,75,77,79,81,83,
%U A115405 85,87,89,91,93,95,97,99,101,103,107,109,111,113,115,117,119,121,123,125
%N A115405 Numbers n such that n^k is deficient for all k>0.
%C A115405 Formerly called colossally deficient numbers, but this is not a good name.
%C A115405 This sequence includes, but is not limited to, all prime numbers and powers of prime numbers. The only even numbers in this sequence are the powers of 2. The first odd number not in this sequence is 105. 105 is deficient but 105^2 (11025) is not. The first deficient number not in this sequence is 10.
%C A115405 Laatsch shows that if a number n has prime factors p1, p2,..., then the least upper bound of the sequence sigma(n^k)/n^k is p1/(p1-1) p2/(p2-1)... This equals n/phi(n), where phi is Euler's totient function. Hence n is in this sequence if 2 phi(n) >= n, which is the complement of A054741. - _T. D. Noe_, May 08 2006
%H A115405 G. C. Greubel, <a href="/A115405/b115405.txt">Table of n, a(n) for n = 1..1000</a>
%H A115405 Richard Laatsch, <a href="http://www.jstor.org/stable/2690424">Measuring the abundancy of integers</a>, Mathematics Magazine 59 (2) (1986) 84-92.
%H A115405 Eric W. Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DeficientNumber.html">Deficient</a>.
%e A115405 Let x be a deficient number (A005100, sigma(n) < 2n). Then x is colossally deficient if for every integer k > 0, x^k is also deficient.
%e A115405 E.g. 3 is in the sequence because 3 is deficient and also are the powers of 3 (9, 27, 81...) 22 is not in the sequence even though 22 is deficient since 22^3 = 10648 is abundant
%t A115405 fQ[n_] := Block[{k = 1}, While[k < 100 && DivisorSigma[1, n^k] < 2n^k, k++ ]; If[k == 100, True, False]]; Select[Range@ 126, fQ@ # &] (* _Robert G. Wilson v_, May 01 2006 *)
%t A115405 Select[Range[200], 2*EulerPhi[ # ]>=#&] (* _T. D. Noe_, May 08 2006 *)
%o A115405 (PARI) is(n)=2*eulerphi(n)>=n \\ _Charles R Greathouse IV_, May 30 2013
%Y A115405 Cf. A005100, A005101, A087244, A000203, A083254, A089684. Complement: A054741.
%K A115405 nonn
%O A115405 1,2
%A A115405 _Sergio Pimentel_, Mar 08 2006
%E A115405 More terms from _Robert G. Wilson v_, May 01 2006
%E A115405 Better description from _T. D. Noe_, May 08 2006