1,2

Formerly called colossally deficient numbers, but this is not a good name.

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.

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 (noe(AT)sspectra.com), May 08 2006

Richard Laatsch, Measuring the abundancy of integers, Math. Mag. 59 (1986), no. 2, 84-92.

MathWorld, Deficient.

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.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

fQ[n_] := Block[{k = 1}, While[k < 100 && DivisorSigma[1, n^k] < 2n^k, k++ ]; If[k == 100, True, False]]; Select[Range@ 126, fQ@ # &] (* RGWV *)

Select[Range[200], 2*EulerPhi[ # ]>=#&] - T. D. Noe (noe(AT)sspectra.com), May 08 2006

Cf. A005100, A005101, A087244, A000203, A083254, A089684. Complement: A054741.

Sequence in context: A078175 A078174 A088948 this_sequence A056867 A062491 A087092

Adjacent sequences: A115402 A115403 A115404 this_sequence A115406 A115407 A115408

nonn

Sergio Pimentel (ferdiego(AT)cox.net), Mar 08 2006

More terms from Robert G. Wilson v (rgwv(at)rgwv.com), May 01 2006

Better description from T. D. Noe (noe(AT)sspectra.com), May 08 2006