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A051772 Distended numbers. 5
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91, 92, 93 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Let 1 = d_1 < d_2 < ... < d_k = n be the k distinct divisors of n. Then n is said to be distended if and only if d_1+d_2+...+d_m < d_(m+1) for all 0 < m < k.

By definition, all distended numbers are deficient (A005100). For this, it suffices to consider the case for m = k-1, then the sum of divisors = d_1+d_2+...+d_(k-1) < d_(m+1) = d_k = n. - Jaycob Coleman, Michel Marcus, Oct 24 2013

If n is distended and defined as above, then 2^r(d_(m-r)+d_(m-r-1)+...+d_1) < d_(m+1) for all 0 < m < k and 0 <= r < m, which for r = 0 is the definition above. When r = m-1 the inequality reduces to 2^(m-1) < d_(m+1) for all 0 < m < k. In particular, 2^k < 4n. - Jaycob Coleman, Oct 29 2013

Contains A000961. - Robert Israel, Dec 20 2015

LINKS

Ivan Neretin, Table of n, a(n) for n = 1..10000

MAPLE

filter:= proc(n) local F, L;

  F:= sort(convert(numtheory:-divisors(n), list));

  L:= ListTools:-PartialSums(F);

  max(L[1..-2] - F[2..-1]) < 0;

end proc:

select(filter, [$1..1000]); # Robert Israel, Dec 20 2015

MATHEMATICA

Select[Range[93], Min[Rest[d = Divisors[#]] - Most[Accumulate[d]]] > 0 &] (* Ivan Neretin, Dec 19 2015 *)

PROG

(PARI) isok(n) = {d = divisors(n); k = #d; for (m = 1, k-1, if (sum(j = 1, m, d[j]) >= d[m+1], return (0)); ); return (1); } \\ Michel Marcus, Sep 04 2013

CROSSREFS

Cf. A000961, A005100, A051773, A051774.

Sequence in context: A094520 A136447 A005100 * A049093 A098901 A098767

Adjacent sequences:  A051769 A051770 A051771 * A051773 A051774 A051775

KEYWORD

nonn

AUTHOR

Alexander Benjamin Schwartz (QBOB(AT)aol.com), Dec 08 1999

STATUS

approved

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Last modified October 26 00:28 EDT 2020. Contains 338026 sequences. (Running on oeis4.)