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

%I A145832 #15 Feb 16 2025 08:33:09
%S A145832 3,7,11,15,17,23,29,31,35,39,47,53,55,59,63,71,79,83,89,95,97,107,111,
%T A145832 119,125,127,131,139,143,146,149,159,161,164,167,175,179,181,191,197,
%U A145832 199,207,209,215,223,233,239,241,251,263,269,279,287,293,299,307,311
%N A145832 Numbers k such that for each divisor d of k, d + k/d is "round" ("square-root smooth").
%C A145832 A necessary condition is that the number be one less than a round number; if this number is prime it's in the sequence.
%C A145832 Even composites in this sequence seem rare (see examples below for more details).
%H A145832 Amiram Eldar, <a href="/A145832/b145832.txt">Table of n, a(n) for n = 1..10000</a>
%H A145832 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RoundNumber.html">Round Number</a>
%e A145832 The first term is a prime one less than the round number 4.
%e A145832 The first composite number in this sequence is 15, with divisor-pair sum 3+5 = 8.
%e A145832 Another such composite is 63, with divisor-pair sums: 3+21 = 24, 7+9 = 16.
%e A145832 There are only five even composites among the first 100 terms of this sequence.
%e A145832 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.
%t A145832 smQ[n_] := FactorInteger[n][[-1, 1]]^2 <= n; seqQ[n_] := AllTrue[Divisors[n], smQ[# + n/#] &]; Select[Range[320], seqQ] (* _Amiram Eldar_, Jun 13 2020 *)
%o A145832 (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
%Y A145832 Cf. A004809, A048098, A080715.
%K A145832 nonn
%O A145832 1,1
%A A145832 Dan Sonnenschein (dans(AT)portal.ca), Oct 20 2008
%E A145832 Wrong term 305 removed by _Amiram Eldar_, Jun 13 2020