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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).
3
34, 7, 41, 919, 18089, 446081, 27033161, 663929729, 74335064959, 6132592231039
OFFSET
1,1
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)
EXAMPLE
34 is the 1st term of A169834, so a(1) = 34.
PROG
(Magma) Ax:=func<n|exists(r){m: m in[2..10^6] | #Divisors(m - 1) eq #Divisors(m) ^ n and #Divisors(m + 1) eq #Divisors(m) ^ n} select r else 0>; [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
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Jaroslav Krizek, Dec 15 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
STATUS
approved