A051772 Distended numbers.

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

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