%I #11 May 07 2016 10:41:47
%S 10,15,24,136,196,1266,5217,8236,8695,46338,98826,181000,387145,
%T 705250,1226175,1291122,3809269,8778718,9294985,37478179,49945002,
%U 63158635,159342696,175624512,419753094,4606837030,4939169059,10229566834
%N Numbers n such that n*(n+1)/2 - sigma(n) = concat(s,t) and n = s + t, where sigma(n) is the sum of the divisors of n.
%e 10*11/2 - sigma(10) = 55 - 18 = 37 and 3 + 7 = 10;
%e 5217*5218/2 - sigma(5217) = 13611153 - 7296 = 13603857 and 1360 + 3857 = 5217.
%p with(numtheory): P:=proc(q) local a, b, c, i, n;
%p for n from 1 to q do c:=n*(n+1)/2-sigma(n); for i from 1 to ilog10(c) do
%p a:=trunc(c/10^i); b:=c-a*10^i; if a+b=n then print(n); break;
%p fi; od; od; end: P(10^9);
%t Select[Range[10^5], Function[n, Total@ Boole@ Function[k, n == First@ # + Last@ # & /@ Map[FromDigits /@ TakeDrop[IntegerDigits@ k, #] &, Range[IntegerLength@ k - 1]]][n (n + 1)/2 - DivisorSigma[1, n]] > 0]] (* _Michael De Vlieger_, May 07 2016, Version 10.2 *)
%t ok[t_, n_] := Catch@ Block[{p=10}, While[p<t, If[n == Mod[t,p] + Floor[t/p], Throw@ True, p *= 10]]; False]; Select[ Range[10^5], ok[# (# + 1)/2 - DivisorSigma[1, #], #] &] (* _Giovanni Resta_, May 07 2016, older Mma, faster *)
%Y Cf. A000203, A024816, A272778.
%K nonn,base
%O 1,1
%A _Paolo P. Lava_, May 06 2016
%E a(15)-a(28) from _Giovanni Resta_, May 07 2016