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A145739 Numbers n for which the sum of all divisors of n <= sqrt(n) is a divisor of n. 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 (list; graph; refs; listen; history; text; internal format)
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(n), but not by the sum of those < sqrt(n)). - Ivan Neretin, Dec 21 2017

LINKS

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

FORMULA

{n: A066839(n) | n}. - R. J. Mathar, Nov 02 2008

EXAMPLE

4 does not qualify for 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(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

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

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Last modified December 8 14:41 EST 2021. Contains 349596 sequences. (Running on oeis4.)