A377001 - OEIS

%I #11 Oct 16 2024 21:30:54

%S 4,8,32,72,94,118,128,144,147,204,284,1017,1102,1210,1462,1968,2294,

%T 2342,2457,2486,2670,2924,5564,6128,6368,7008,8192,10856,12216,12914,

%U 14066,14595,16694,18416,18825,19668,21870,22401,22713,23388,26234,26966,29038,31806

%N Integers k equal to the sum over A000203(t) mod t, for some steps, starting with t = k and then using the result to feed the next calculation.

%C Up to 10^7, the longest process takes place with 2813292 which needs 23 steps.

%C Numbers of the form 2^A000043(n) or 1+A000668(n) are a subsequence.

%C If we multiply instead of adding A000203(t) mod t, we get the twice even perfect numbers (A139256).

%C E.g. k = 12 -> sigma(12) mod 12 = 4; sigma(4) mod 4 = 3 and 4 * 3 = 12.

%e k = 72 (2 steps):

%e sigma(72) mod 72 = 51;

%e sigma(51) mod 51 = 21 and 51 + 21 = 72.

%e k = 147  (6 steps):

%e sigma(147) mod 147 = 81;

%e sigma(81) mod 81 = 40;

%e sigma(40) mod 40 = 10;

%e sigma(10) mod 10 = 8;

%e sigma(8) mod 8 = 7;

%e sigma(7) mod 7 = 1 and 81 + 40 + 10 + 8 + 7 + 1 = 147.

%p with(numtheory): P:=proc(q) local a,b,n,v; v:=[];

%p for n from 1 to q do a:=0; b:=n; while a<n do b:=(sigma(b) mod b); if b=0 then break;

%p else a:=a+b; fi; od; if a=n then v:=[op(v),n]; fi; od; op(v); end: P(10^5);

%Y Cf. A000043, A000203, A000668, A139256, A377002.

%K nonn,easy

%O 1,1

%A _Paolo P. Lava_, Oct 12 2024