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A115405
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Numbers n such that n^k is deficient for all k>0.
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3
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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
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1,2
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COMMENTS
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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
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LINKS
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Eric W. Weisstein's World of Mathematics, Deficient.
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Better description from T. D. Noe, May 08 2006
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STATUS
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approved
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