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

Offset: 1

Author

Jaroslav Krizek, Apr 18 2022

Keywords

• nonn

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)

Examples

			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.
		

Crossrefs

Cf. A000005, A068208, A343144.

Programs

Magma

Ax:=func; [Ax(n): n in [1..5]]

Extensions

a(6)-a(8) from Jon E. Schoenfield, Apr 20 2022

a(9)-a(10), a(16), a(21)-a(22), a(28), a(33) from Jaroslav Krizek, Apr 20 2022

Remaining terms through a(34) from Jon E. Schoenfield, Apr 30 2022

a(35) from Jinyuan Wang, May 21 2022