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
Examples
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.
Programs
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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);
Comments