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
Keywords
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
Programs
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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
Comments