

A115405


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


3



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
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OFFSET

1,2


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/(p11) p2/(p21)... 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


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) 8492.
Eric W. Weisstein's World of Mathematics, Deficient.


EXAMPLE

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


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 *)


PROG

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


CROSSREFS

Cf. A005100, A005101, A087244, A000203, A083254, A089684. Complement: A054741.
Sequence in context: A174894 A275616 A088948 * A257144 A316476 A056867
Adjacent sequences: A115402 A115403 A115404 * A115406 A115407 A115408


KEYWORD

nonn


AUTHOR

Sergio Pimentel, Mar 08 2006


EXTENSIONS

More terms from Robert G. Wilson v, May 01 2006
Better description from T. D. Noe, May 08 2006


STATUS

approved



