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A348837 a(n) is the smallest k such that the sum of the number of divisors of the n numbers from k to k+n-1 equals tau(k+n).

%I A348837 #30 Dec 17 2021 20:30:05
%S A348837 2,10,93,236,355,2634,2873,5032,11331,20150,18889,80628,55427,207886,
%T A348837 205905,371264,369583,617742,166301,1436380,720699,2227658,1081057,
%U A348837 831576,4633175,3326374,2633373,5045012,11850271,6683010,11642369,9979168,9424767,8648606,24418765
%N A348837 a(n) is the smallest k such that the sum of the number of divisors of the n numbers from k to k+n-1 equals tau(k+n).
%H A348837 David A. Corneth, <a href="/A348837/b348837.txt">Table of n, a(n) for n = 1..102</a>
%H A348837 David A. Corneth, <a href="/A348837/a348837.gp.txt">Upper bounds on some terms <= 10^13</a>
%e A348837 a(1) = 2 because tau(2) = tau(3) = 2; a(1) = A005237(1).
%e A348837 a(2) = 10 because tau(10) + tau(11) = 4 + 2 =  6, the same as tau(12) = 6.
%e A348837 a(3) = 93 because tau(93) + tau(94) + tau(95) = 4 + 4 + 4 = 12, the same as tau(96) = 12.
%t A348837 a[n_] := Module[{div = DivisorSigma[0, Range[n]], k = n + 1}, While[(d = DivisorSigma[0, k]) != Plus @@ div, div = Join[Drop[div, 1], {d}]; k++]; k - n]; Array[a, 20] (* _Amiram Eldar_, Nov 01 2021 *)
%o A348837 (PARI) a(n) = my(k=1); while (sum(i=k, k+n-1, numdiv(i)) != numdiv(k+n), k++); k; \\ _Michel Marcus_, Nov 01 2021
%o A348837 (PARI) a(n)=my(v=vector(n,k,numdiv(k)),s=vecsum(v),t,i=n); forfactored(k=n+1,2^63-1, t=numdiv(k); if(s==t, return(k[1]-n)); if(i++>n,i=1); s+=t-v[i]; v[i]=t) \\ _Charles R Greathouse IV_, Nov 01 2021
%Y A348837 Cf. A000005, A005237, A348335.
%K A348837 nonn
%O A348837 1,1
%A A348837 _Metin Sariyar_, Nov 01 2021
%E A348837 a(21)-a(26) from _Michel Marcus_, Nov 01 2021
%E A348837 a(27)-a(35) from _Amiram Eldar_, Nov 01 2021