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%I #20 Jun 09 2022 19:33:04
%S 1,2,3,5,6,7,11,12,13,17,18,19,23,28,29,31,37,41,43,45,47,48,53,56,59,
%T 61,67,71,72,73,79,80,83,89,96,97,101,103,107,109,113,117,127,131,137,
%U 139,149,151,157,163,167,173,179,181,191,193,196,197,199,211,223,227
%N Numbers k for which the sum of all divisors of k <= sqrt(k) is a divisor of k.
%C Includes all prime numbers. Includes all even perfect numbers. Includes no power of 2 > 2. Includes no number of the form 2p where p is a prime number greater than 3.
%C 43681 is the first term after 1 which would not be there if the inequality in the definition were strict (i.e., it is divisible by the sum of divisors <= sqrt(k), but not by the sum of those < sqrt(k)). - _Ivan Neretin_, Dec 21 2017
%H Ivan Neretin, <a href="/A145739/b145739.txt">Table of n, a(n) for n = 1..10000</a>
%F {k: A066839(k) | k}. - _R. J. Mathar_, Nov 02 2008
%e 4 is not a term of this sequence because the divisors of 4 <= sqrt(4) are 1 and 2 and 1+2 = 3 and 3 is not a divisor of 4.
%e 12 is in the sequence because the divisors of 12 <= sqrt(12) are 1, 2 and 3 and 1+2+3 = 6 is a divisor of 12. - _Emeric Deutsch_, Oct 27 2008
%p with(numtheory): a:=proc(n) local div, s, j: div:=divisors(n): s:=0: for j while div[j] <= evalf(sqrt(n)) do s:=s+div[j] end do: if type(n/s,integer) = true then n else end if end proc: 1, seq(a(n), n=2..250); # _Emeric Deutsch_, Oct 27 2008
%p A066839 := proc(n) local a,d ; a := 0 ; for d in numtheory[divisors](n) do if d^2 <= n then a := a+d ; fi; od: a ; end: A145739 := proc(n) option remember ; local a; if n = 1 then 1; else for a from procname(n-1)+1 do if a mod A066839(a) = 0 then RETURN(a) ; fi; od: fi; end: for n from 1 to 300 do printf("%d,",A145739(n)) ; od: # _R. J. Mathar_, Nov 02 2008
%t Select[Range[230], Divisible[#, Total@Take[d = Divisors[#], Ceiling[Length[d]/2]]] &] (* _Ivan Neretin_, Dec 21 2017 *)
%o (PARI) isok(n) = (n % sumdiv(n, d, d*(d^2<=n))) == 0; \\ _Michel Marcus_, Dec 22 2017
%Y Cf. A066839.
%K nonn
%O 1,2
%A _J. Lowell_, Oct 17 2008
%E More terms from _R. J. Mathar_ and _Emeric Deutsch_, Nov 01 2008
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