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

1,1

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.

Links

Table of n, a(n) for n=1..37.

Example

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.

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);

Crossrefs

Cf. A066272, A066417, A239687.

Sequence in context: A255149 A255142 A114646 * A118091 A235176 A044298

Adjacent sequences: A239683 A239684 A239685 * A239687 A239688 A239689

Keyword

nonn,base,hard

Author

Paolo P. Lava, Mar 24 2014

Status

approved