A239686 Numbers n such that if n = a U b (where U denotes concatenation) then sigma*(a) + sigma*(b) = abs(sigma*(n) - n), where sigma*(n) is the sum of the anti-divisors of n.

47, 118, 205, 846, 898, 1219, 4181, 4236, 4701, 4929, 6014, 6516, 13276, 30445, 59956, 61916, 63216, 67314, 72066, 79554, 90674, 106316, 128998, 129179, 136816, 142486, 143396, 180448, 229914, 284894, 357841, 421318, 483286, 486721, 487618, 500218, 642445

Offset: 1

Author

Paolo P. Lava, Mar 24 2014

Keywords

• nonn
• base
• hard

Comments

Neither a or b minor than 2 are considered because numbers 1 and 2 have no anti-divisors.

Similar to A239562 but using anti-divisors instead of divisors.

Examples

			Anti-divisors of 4701 are 2, 6, 7, 17, 79, 119, 1343, 553, 3134 and their sum is 5260. Consider 4701 as 4 U 701. Anti-divisors of 4 is 3 and of 701 are 2, 3, 23, 61, 467 whose sum is 556. At the end we have that 5260 - 4701 = 559 = 3 + 556.
		

Crossrefs

Cf. A066272, A066417, A239687.

Programs

Maple

with(numtheory);
T:=proc(t) local w, x, y; x:=t; y:=0; while x>0 do x:=trunc(x/10); y:=y+1; od; end:
P:=proc(q) local a, b, c, d, f, g, i, j, k,n;
for n from 1 to q do b:=T(n); k:=0; j:=n; while j mod 2<>1 do k:=k+1; j:=j/2; od;
a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;
for i from 1 to b-1 do c:=trunc(n/10^i); d:=n-c*10^i; if c>2 and d>2 then
k:=0; j:=c; while j mod 2<>1 do k:=k+1; j:=j/2; od;
f:=sigma(2*c+1)+sigma(2*c-1)+sigma(c/2^k)*2^(k+1)-6*c-2;
k:=0; j:=d; while j mod 2<>1 do k:=k+1; j:=j/2; od;
g:=sigma(2*d+1)+sigma(2*d-1)+sigma(d/2^k)*2^(k+1)-6*d-2;
if f+g=a-n then print(n); break; fi; fi; od; od; end: P(10^9);