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

Author

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

Keywords

• nonn

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).

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Round Number

Crossrefs

Cf. A004809, A048098, A080715.

Programs

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

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 *)

Extensions

Wrong term 305 removed by Amiram Eldar, Jun 13 2020