A239687 Numbers n such that if n = a U b (where U denotes concatenation) then abs(sigma*(a) - sigma*(b)) = abs(sigma*(n) - n), where sigma*(n) is the sum of the anti-divisors of n.

54, 436, 2014, 2466, 3365, 4143, 4965, 7922, 9332, 15426, 17554, 24006, 32874, 33574, 39476, 44296, 49976, 54118, 83726, 116174, 137635, 163964, 164824, 177546, 203514, 220789, 235434, 379096, 420716, 476475, 597741, 600354, 604986, 680266, 736306, 748966

Offset

1,1

Comments

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

Similar to A239563 but using anti-divisors instead of divisors.

Links

Table of n, a(n) for n=1..36.

Example

Anti-divisors of 4143 are 2, 5, 6, 1657, 2762 and their sum is 4432. Consider 4143 as 4 U 143. Anti-divisors of 4 is 3 and of 143 are 2, 3, 5, 7, 15, 19, 22, 26, 41, 57, 95 whose sum is 292. At the end we have that 4432 - 4143 = 289 = 292 - 3.

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 abs(f-g)=abs(a-n) then print(n); break; fi; fi; od; od; end: P(10^9);

Crossrefs

Cf. A066272, A066417, A239686.

Sequence in context: A232511 A262853 A230921 * A192216 A238908 A233364

Adjacent sequences: A239684 A239685 A239686 * A239688 A239689 A239690

Keyword

nonn,base

Author

Paolo P. Lava, Mar 24 2014

Status

approved