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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 (list; graph; refs; listen; history; text; internal format)
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/(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

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.

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

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Last modified March 21 01:18 EDT 2019. Contains 321356 sequences. (Running on oeis4.)