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A145832 Numbers k such that for each divisor d of k, d + k/d is "round" ("square-root smooth"). 3
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 (list; graph; refs; listen; history; text; internal format)
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: A196079 A285497 A079710 * 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

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Last modified December 2 05:20 EST 2022. Contains 358485 sequences. (Running on oeis4.)