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A356635 Numbers k that can be written as the sum of 7 divisors of k (not necessarily distinct).

%I A356635 #22 Aug 08 2023 03:22:21
%S A356635 7,8,9,10,12,14,15,16,18,20,21,22,24,27,28,30,32,33,35,36,39,40,42,44,
%T A356635 45,48,49,50,52,54,55,56,60,63,64,66,68,70,72,75,77,78,80,81,84,88,90,
%U A356635 91,96,98,99,100,102,104,105,108,110,112,114,117,119,120,126,128,130
%N A356635 Numbers k that can be written as the sum of 7 divisors of k (not necessarily distinct).
%C A356635 If k is in the sequence then so is k*m for positive m. - _David A. Corneth_, Aug 19 2022
%C A356635 Numbers that are divisible by at least one of 7, 8, 9, 10, 12, 15, 22, 33, 39, 52, 55, 68, 102, 114, 138.  For proof, see link. - _Robert Israel_, Sep 02 2022
%C A356635 The asymptotic density of this sequence is 17819629/37182145 = 0.479252... . - _Amiram Eldar_, Aug 08 2023
%H A356635 Robert Israel, <a href="/A356635/b356635.txt">Table of n, a(n) for n = 1..10000</a>
%H A356635 Robert Israel, <a href="/A356635/a356635.pdf">Proof that A356635 consists of all numbers divisible by at least one of 7, 8, 9, 10, 12, 15, 22, 33, 39, 52, 55, 68, 102, 114, 138</a>.
%e A356635 10 is in the sequence since 10 = 2+2+2+1+1+1+1, where each summand divides 10.
%p A356635 filter:= n -> ormap(t -> n mod t = 0, [7, 8, 9, 10, 12, 15, 22, 33, 39, 52, 55, 68, 102, 114, 138]):
%p A356635 select(filter, [$1..200]); # _Robert Israel_, Sep 02 2022
%t A356635 q[n_, k_] := AnyTrue[Tuples[Divisors[n], k], Total[#] == n &]; Select[Range[130], q[#, 7] &] (* _Amiram Eldar_, Aug 19 2022 *)
%o A356635 (PARI) isok(k) = my(d=divisors(k)); forpart(p=k, if (setintersect(d, Set(p)) == Set(p), return(1)), , [7,7]); \\ _Michel Marcus_, Aug 19 2022
%Y A356635 Numbers k that can be written as the sum of j divisors of k (not necessarily distinct) for j=1..10: A000027 (j=1), A299174 (j=2), A355200 (j=3), A354591 (j=4), A355641 (j=5), A356609 (j=6), this sequence (j=7), A356657 (j=8), A356659 (j=9), A356660 (j=10).
%K A356635 nonn
%O A356635 1,1
%A A356635 _Wesley Ivan Hurt_, Aug 18 2022