

A145739


Numbers k for which the sum of all divisors of k <= sqrt(k) is a divisor of k.


1



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, 61, 67, 71, 72, 73, 79, 80, 83, 89, 96, 97, 101, 103, 107, 109, 113, 117, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 196, 197, 199, 211, 223, 227
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OFFSET

1,2


COMMENTS

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.
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


LINKS

Ivan Neretin, Table of n, a(n) for n = 1..10000


FORMULA

{k: A066839(k)  k}.  R. J. Mathar, Nov 02 2008


EXAMPLE

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.
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


MAPLE

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
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(n1)+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


MATHEMATICA

Select[Range[230], Divisible[#, Total@Take[d = Divisors[#], Ceiling[Length[d]/2]]] &] (* Ivan Neretin, Dec 21 2017 *)


PROG

(PARI) isok(n) = (n % sumdiv(n, d, d*(d^2<=n))) == 0; \\ Michel Marcus, Dec 22 2017


CROSSREFS

Cf. A066839.
Sequence in context: A164922 A205523 A343027 * A198191 A243058 A348440
Adjacent sequences: A145736 A145737 A145738 * A145740 A145741 A145742


KEYWORD

nonn


AUTHOR

J. Lowell, Oct 17 2008


EXTENSIONS

More terms from R. J. Mathar and Emeric Deutsch, Nov 01 2008


STATUS

approved



