A115405 Numbers n such that n^k is deficient for all k>0.

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, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125

Offset: 1

Author

Sergio Pimentel, Mar 08 2006

Keywords

• nonn

Comments

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, May 08 2006

Examples

			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
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Richard Laatsch, Measuring the abundancy of integers, Mathematics Magazine 59 (2) (1986) 84-92.
• Eric W. Weisstein's World of Mathematics, Deficient.

Crossrefs

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

Programs

Mathematica

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

PARI

is(n)=2*eulerphi(n)>=n \\ Charles R Greathouse IV, May 30 2013

Extensions

More terms from Robert G. Wilson v, May 01 2006

Better description from T. D. Noe, May 08 2006