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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
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
Sequence in context: A196079 A285497 A079710 * A292637 A279608 A322408
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 March 28 11:59 EDT 2024. Contains 371254 sequences. (Running on oeis4.)