A353032 - OEIS

%I #54 May 21 2022 14:53:09

%S 0,0,4,8,81,0,0,0,441,6723,0,0,0,0,767495140624,2024,

%T 665416609183179841,0,0,0,2050624,263168,0,0,670801950625,0,

%U 10871232294189453124,532899,0,0,0,0,67634176,0,55471075527984793933106579132930662929175947116953798971172816083061185149078369140624

%N a(n) is the smallest number m with n divisors such that m+1 has n-1 divisors, or 0 if no such number exists.

%C For n > 33, a(64) = 6890624 is the only positive term <= 10^8.

%C There is no number m <= 10^10 that is the first start of run of 3 consecutive integers m, m+1 and m+2 with triplet [tau(m), tau(m+1), tau(m+2)] = [tau(m), tau(m) - 1, tau(m) - 2].

%C If a(11) > 0 then a(11) > 10^100. - _Charles R Greathouse IV_, Apr 20 2022

%C a(36) = 1626347583, a(40) = 1173953168, a(49) = 304006671424, a(65) = 25221297570561, a(81) = 15579533124, a(96) = 68195356770303, a(100) = 1698353697680, a(136) = 28528257204224, a(256) = 334435516415. - _Jon E. Schoenfield_, Apr 24 2022

%C From _Jon E. Schoenfield_, May 01 2022: (Start)

%C a(35) is the smallest m such that m = 16*p^6 = q^16*r - 1 where p, q, and r are odd primes; a(35) <= 16*123024356097427^6 (an 86-digit number).

%C a(37) = a(38) = 0;

%C a(39) <= 1134572901070399771884918212890624;

%C a(41) <= 350847983^40 (a 342-digit number). (End)

%e For n = 5; a(5) = 81 because 81 is the smallest number m such that tau(m) = tau(81) = 5 and tau(82) = tau(m) - 1 = 4.

%o (Magma) Ax:=func<n|exists(r){m: m in[1..10^6] | #Divisors(m) eq #Divisors(m + 1) + 1 and #Divisors(m) eq n} select r else 0>; [Ax(n): n in [1..5]]

%Y Cf. A000005, A068208, A343144.

%K nonn

%O 1,3

%A _Jaroslav Krizek_, Apr 18 2022

%E a(6)-a(8) from _Jon E. Schoenfield_, Apr 20 2022

%E a(9)-a(10), a(16), a(21)-a(22), a(28), a(33) from _Jaroslav Krizek_, Apr 20 2022

%E Remaining terms through a(34) from _Jon E. Schoenfield_, Apr 30 2022

%E a(35) from _Jinyuan Wang_, May 21 2022