A265646 - OEIS

%I #31 Aug 23 2023 08:37:27

%S 24,84,120,270,672,1080,1170,1488,1638,2160,3360,3960,5400,7392,10920,

%T 13104,22932,24384,30240,32640,64260,167580,199584,241920,290160,

%U 293760,333312,351000,411264,435708,445536,510720,523776,524160,594048,725400,760320,804384

%N Numbers k such that k = Sum_{i=1..j} (d_i mod d), where d_i are their aliquot parts and d is one of them.

%C All terms are multiples of 6.

%C The sequence is a subset of A273124.

%H Paolo P. Lava, <a href="/A265646/a265646.txt">First 50 terms and their value of d</a>

%e Aliquot parts of 24 are 1, 2, 3, 4, 6, 8, 12 and 1 mod 12 + 2 mod 12 + 3 mod 12 + 4 mod 12 + 6 mod 12 + 8 mod 12 + 12 mod 12 = 1 + 2 + 3 + 4 + 6 + 8 + 0 = 24.

%e Aliquot parts of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60. In this case we can choose two different values: 24 and 30. In fact the sum of divisors from 1 to 20 is 86. Then 24 mod 24 = 0, 30 mod 24 = 6, 40 mod 24 = 16, 60 mod 24 = 12 and 86 + 0 + 6 + 16 + 12 = 120. Again, 24 mod 30 = 24, 30 mod 30 = 0, 40 mod 30 = 10, 60 mod 30 = 0 and 86 + 24 + 0 + 10 + 0 = 120.

%p with(numtheory): P:= proc(q) local a,b,j,k,n;

%p for n from 1 to q do a:=sort([op(divisors(n))]);

%p for k from 1 to nops(a)-1 do b:=0; for j from 1 to nops(a)-1 do b:=b+(a[j] mod a[k]); od;

%p if b=n then print(n); break; fi; od; od; end: P(10^9);

%t Select[6 Range[10^5], MemberQ[Function[d, Total@ Mod[d, #] & /@ d]@ Most@ Divisors@ #, #] &] (* _Michael De Vlieger_, May 17 2016 *)

%Y Cf. A273124.

%K nonn,easy

%O 1,1

%A _Paolo P. Lava_, May 17 2016