A239687 - OEIS

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

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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);