A353032 - OEIS

A353032

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.

0, 0, 4, 8, 81, 0, 0, 0, 441, 6723, 0, 0, 0, 0, 767495140624, 2024, 665416609183179841, 0, 0, 0, 2050624, 263168, 0, 0, 670801950625, 0, 10871232294189453124, 532899, 0, 0, 0, 0, 67634176, 0, 55471075527984793933106579132930662929175947116953798971172816083061185149078369140624

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OFFSET

1,3

COMMENTS

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

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].

If a(11) > 0 then a(11) > 10^100. - Charles R Greathouse IV, Apr 20 2022

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

From Jon E. Schoenfield, May 01 2022: (Start)

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).

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

a(39) <= 1134572901070399771884918212890624;

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

LINKS

Table of n, a(n) for n=1..35.

EXAMPLE

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.

PROG

(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]]

CROSSREFS