A305196 a(n) is the smallest number k such that tau(k + n) = tau(k) + n where tau(n) is the number of divisors of n (A000005).

1, 1, 10, 9, 26, 25, 74, 29, 82, 441, 170, 133, 348, 131, 166, 3025, 344, 559, 1602, 557, 820, 9979, 986, 4333, 1236, 9191, 694, 3249, 1652, 3481, 9378, 34969, 3118, 249967, 5636, 36829, 3324, 51947, 3994, 6561, 5000, 15835, 16806, 3557, 6436, 119025, 6254, 589777, 7512, 1768851

Offset: 0

Author

Michel Marcus, May 27 2018

Keywords

• nonn

Examples

			10 and 12 have respectively 4 and 6 divisors, that is, 12-10 = 6-4, so a(2)=10.
9 and 12 have respectively 3 and 6 divisors, that is, 12-9 = 6-3, so a(3)=9.
		

Links

• Michel Marcus and Giovanni Resta, Table of n, a(n) for n = 0..244 (terms up to a(106) from Michel Marcus)

Crossrefs

Cf. A000005, A099642, A015886 (similar, with sigma).

Programs

Maple

f:= proc(n) local k;
  for k from 1 do
      if numtheory:-tau(k+n)=numtheory:-tau(k)+n then return k fi
    od
end proc:
map(f, [$0..50]); # Robert Israel, May 28 2018

Mathematica

Array[Block[{k = 1}, While[DivisorSigma[0, k + #] != DivisorSigma[0, k] + #, k++]; k] &, 40, 0] (* Michael De Vlieger, May 27 2018 *)

PARI

a(n) = {my(k=1); while(numdiv(k+n) != numdiv(k) + n, k++); k;}