A253266 Numbers such that the minimum distance between divisors of n occurs only between composite numbers.

14399, 34595, 82943, 89999, 100793, 116963, 158389, 172975, 224675, 244783, 245021, 266255, 278783, 281957, 285155, 304703, 331177, 338723, 343387, 380545, 417571, 446369, 447557, 484415, 497021, 532763, 580601, 585221, 588115, 590359, 608399, 619157, 627263, 629993

Offset

1,1

Comments

All members of the sequence are either a square minus a small square, or a multiple of a smaller member of the sequence. The square root of the small square must be less than half the fourth root of the number.

245021 is the smallest member of this sequence that is neither 1 less than a square nor a multiple of a smaller member of the sequence. It is 4 less than a square.

Links

Table of n, a(n) for n=1..34.

Franklin T. Adams-Watters, Proof of assertion

Example

The divisors of 14399 = 119*121 are 1, 7, 11, 17, 77, 119, 121, 187, 847, 1309, 2057, 14399. The minimum difference between divisors is 2, which occurs only between 119 and 121, both of which are composite; so 14399 is in the sequence.
The divisors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. The minimum difference is 1, which does occur between two composites - 8, 9 - but it also occurs between pairs not both composite (e.g. 1, 2 or 3, 4), so 72 is not in the sequence.

Prog

(PARI) isa(n) = local(ds=divisors(n), diff, mind, dcomp); mind=n; for(k=2, #ds, diff=ds[k]-ds[k-1]; if(diff<=mind, if(diff<mind, mind=diff; dcomp=1); if(isprime(ds[k])||isprime(ds[k-1]), dcomp=0))); dcomp

Crossrefs

Cf. A060680 (minimum difference), A033676 and A033677 (central divisors).

Sequence in context: A234678 A237788 A106725 * A190469 A237372 A250851

Adjacent sequences: A253263 A253264 A253265 * A253267 A253268 A253269

Keyword

nonn

Author

Franklin T. Adams-Watters, May 01 2015

Status

approved