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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).

%I A305196 #17 May 06 2022 13:13:51
%S A305196 1,1,10,9,26,25,74,29,82,441,170,133,348,131,166,3025,344,559,1602,
%T A305196 557,820,9979,986,4333,1236,9191,694,3249,1652,3481,9378,34969,3118,
%U A305196 249967,5636,36829,3324,51947,3994,6561,5000,15835,16806,3557,6436,119025,6254,589777,7512,1768851
%N 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).
%H A305196 Michel Marcus and Giovanni Resta, <a href="/A305196/b305196.txt">Table of n, a(n) for n = 0..244</a> (terms up to a(106) from Michel Marcus)
%e A305196 10 and 12 have respectively 4 and 6 divisors, that is, 12-10 = 6-4, so a(2)=10.
%e A305196 9 and 12 have respectively 3 and 6 divisors, that is, 12-9 = 6-3, so a(3)=9.
%p A305196 f:= proc(n) local k;
%p A305196   for k from 1 do
%p A305196       if numtheory:-tau(k+n)=numtheory:-tau(k)+n then return k fi
%p A305196     od
%p A305196 end proc:
%p A305196 map(f, [$0..50]); # _Robert Israel_, May 28 2018
%t A305196 Array[Block[{k = 1}, While[DivisorSigma[0, k + #] != DivisorSigma[0, k] + #, k++]; k] &, 40, 0] (* _Michael De Vlieger_, May 27 2018 *)
%o A305196 (PARI) a(n) = {my(k=1); while(numdiv(k+n) != numdiv(k) + n, k++); k;}
%Y A305196 Cf. A000005, A099642, A015886 (similar, with sigma).
%K A305196 nonn
%O A305196 0,3
%A A305196 _Michel Marcus_, May 27 2018