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%I #19 Jan 03 2024 10:01:48
%S 39,147,413,1268,1550,3964,9987,137097,238268,285993,2139783,4866838,
%T 74523325,131135109
%N Consider a number of k digits n = d_(k)*10^(k-1) + d_(k-1)*10^(k-2) + … + d_(2)*10 + d_(1). Sequence lists the numbers n such that sigma(n)-n = Sum_{i=1..k-1}{sigma(Sum_{j=1..i}{d_(j)*10^(j-1)})} + Sum_{i=1..k-1}{sigma(Sum_{j=1..i}{d_(k-j+1)*10^(i-j)})} (see example below)
%C a(15) > 10^10. - _Giovanni Resta_, May 23 2016
%e If n = 238268, starting from the least significant digit, let us cut the number into the set 8, 68, 268, 8268, 38268. We have:
%e sigma(8) = 15;
%e sigma(68) = 126;
%e sigma(268) = 476;
%e sigma(8268) = 21168;
%e sigma(38268) = 96824.
%e Then, starting from the most significant digit, let us cut the number into the set 2, 23, 238, 2382, 23826. We have:
%e sigma(2) = 3;
%e sigma(23) = 24;
%e sigma(238) = 432;
%e sigma(2382) = 4776;
%e sigma(23826) = 54864.
%e Finally, 15 + 126 + 476 + 21168 + 96824 + 3 + 24 + 432 + 4776 + 54864 = 178708 = sigma(238268) - 238268.
%p with(numtheory); P:=proc(q) local a,b,k,n;
%p for n from 2 to q do a:=0; k:=1; while trunc(n/10^k)>0 do
%p a:=a+sigma(trunc(n/10^k)); k:=k+1; od; b:=0; k:=1;
%p while (n mod 10^k)<n do b:=b+sigma(n mod 10^k); k:=k+1; od;
%p if sigma(n)-n=a+b then print(n); fi; od; end: P(10^9);
%o (PARI) isok(n) = {sa = 0; k = 1; while (x=n\10^k, sa += sigma(x); k++;); sb = 0; k = 1; while ((x=n % 10^k) < n, if (x, sb += sigma(x)); k++;); sigma(n)-n == sa+sb;} \\ _Michel Marcus_, Jun 19 2015
%Y Cf. A000203, A240894-A240901.
%K nonn,base,more
%O 1,1
%A _Paolo P. Lava_, Apr 16 2014
%E a(11)-a(12) from _Michel Marcus_, Jun 19 2015
%E a(13)-a(14) from _Giovanni Resta_, May 23 2016
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