A145832 Numbers k such that for each divisor d of k, d + k/d is "round" ("square-root smooth").

3, 7, 11, 15, 17, 23, 29, 31, 35, 39, 47, 53, 55, 59, 63, 71, 79, 83, 89, 95, 97, 107, 111, 119, 125, 127, 131, 139, 143, 146, 149, 159, 161, 164, 167, 175, 179, 181, 191, 197, 199, 207, 209, 215, 223, 233, 239, 241, 251, 263, 269, 279, 287, 293, 299, 307, 311

Offset

1,1

Comments

A necessary condition is that the number be one less than a round number; if this number is prime it's in the sequence.

Even composites in this sequence seem rare (see examples below for more details).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Round Number

Example

The first term is a prime one less than the round number 4.
The first composite number in this sequence is 15, with divisor-pair sum 3+5 = 8.
Another such composite is 63, with divisor-pair sums: 3+21 = 24, 7+9 = 16.
There are only five even composites among the first 100 terms of this sequence.
The first such is 146, with divisor-pair sum 2+73 = 75. The second is 164, with divisor-pair sums 2+82 = 84 and 4+41 = 45. The remaining three are 458, 524 and 584.

Mathematica

smQ[n_] := FactorInteger[n][[-1, 1]]^2 <= n; seqQ[n_] := AllTrue[Divisors[n], smQ[# + n/#] &]; Select[Range[320], seqQ] (* Amiram Eldar, Jun 13 2020 *)

Prog

(Magma) [ n: n in [1..310] | forall{ k: k in [ Integers()!(d+n/d): d in [ D[j]: j in [1..a] ] ] | k ge (IsEmpty(T) select 1 else Max(T) where T is [ x[1]: x in Factorization(k) ])^2 } where a is IsOdd(#D) select (#D+1)/2 else #D/2 where D is Divisors(n) ]; // Klaus Brockhaus, Oct 24 2008

Crossrefs

Cf. A004809, A048098, A080715.

Sequence in context: A285497 A079710 A373415 * A292637 A279608 A322408

Adjacent sequences: A145829 A145830 A145831 * A145833 A145834 A145835

Keyword

nonn

Author

Dan Sonnenschein (dans(AT)portal.ca), Oct 20 2008

Extensions

Wrong term 305 removed by Amiram Eldar, Jun 13 2020

Status

approved