A350132 a(n) is the smallest number m such that tau(m-1) = tau(m+1) = tau(m)^n, where tau(m) is the number of divisors of m (A000005).

34, 7, 41, 919, 18089, 446081, 27033161, 663929729, 74335064959, 6132592231039

Offset: 1

Author

Jaroslav Krizek, Dec 15 2021

Keywords

• nonn
• more

Comments

Triples of [tau(a(n) - 1), tau(a(n)), tau(a(n) + 1)] = [tau(a(n))^n, tau(a(n)), tau(a(n))^n]: [4, 4, 4], [4, 2, 4], [8, 2, 8], [16, 2, 16], [32, 2, 32], [64, 2, 64], [128, 2, 128], ...

Conjecture: a(n) is prime for all n >= 2, i.e., the sequence {a(n)} without the first term is the sequence of smallest primes p such that tau(p-1) = tau(p+1) = 2^n for n >= 2.

a(10) <= 6132592231039. - Jon E. Schoenfield, Jan 19 2022

From David A. Corneth, Jan 21 2022: (Start)

a(11) <= 864808145605249.

a(12) <= 246846832951283839.

a(13) <= 14552217960448488319. (End)

Examples

			34 is the 1st term of A169834, so a(1) = 34.
		

Crossrefs

Cf. A000005, A169834, A343020, A343087.

Programs

Magma

Ax:=func; [Ax(n): n in [1..6]];

PARI

isok(m, n) = my(nd1=numdiv(m-1)); (nd1 == numdiv(m)^n) && (nd1 == numdiv(m+1));
a(n) = {my(m=2); while (!isok(m, n), m++); m;} \\ Michel Marcus, Dec 16 2021

Extensions

a(8) from Jon E. Schoenfield and David A. Corneth, Dec 15 2021

a(9) from David A. Corneth and Martin Ehrenstein, Jan 14 2022

a(10) verified by Martin Ehrenstein, Jan 21 2022