A343020 a(n) is the smallest prime p such that tau(p+1) = 2^n.

2, 5, 23, 167, 839, 7559, 128519, 1081079, 20540519, 397837439, 8031343319, 188972783999, 3212537327999, 125568306863999, 2888071057871999, 190487121512687999, 4381203794791823999, 215961289494494543999, 13283916764437951631999, 540119185025730854543999, 26465840066260811872655999, 1356699703068812438127791999

Offset: 1

Author

Jaroslav Krizek, Apr 02 2021

Keywords

• nonn

Comments

tau(m) = the number of divisors of m (A000005).

Sequences of primes p such that tau(p+1) = 2^n for 2 <= n <= 5:

n = 2: 5, 7, 13, 37, 61, 73, 157, 193, 277, 313, 397, 421, ...

n = 3: 23, 29, 41, 53, 101, 103, 109, 113, 127, 137, 151, ...

n = 4: 167, 263, 269, 311, 383, 389, 439, 461, 509, 569, ...

n = 5: 839, 1319, 1511, 1559, 1847, 1889, 2039, 2309, 2687, ...

Conjecture: a(n) is also the smallest number m such that tau(m+1) = tau(m)^n.

Examples

			a(4) = 167 because 167 is the smallest prime p such that tau(p+1) = 16 = 2^4.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..31

Crossrefs

Cf. A037992, A080371, A080372, A340799, A343018, A343019.

Programs

Magma

Ax:=func; [Ax(n): n in [1..7]]

Mathematica

Do[p = 1; While[DivisorSigma[0, Prime[p] + 1] != 2^n, p++]; Print[n, " ", Prime[p]], {n, 1, 9}] (* Vaclav Kotesovec, Apr 03 2021 *)

PARI

a(n) = my(t=2^n); forprime(p=2, oo, if(numdiv(p+1)==t, return(p))); \\ Jinyuan Wang, Apr 02 2021

Python

from sympy import isprime,nextprime
primes=[2]
def solve(v,k,i,j):
    global record,stack,primes
    if k==0:
        if isprime(v-1):
            record=v
        return True
    sizeok=False
    cnt=True
    while cnt:
        if i>=len(primes):
            primes.append(nextprime(primes[-1]))
        if jBert Dobbelaere, Apr 11 2021

Extensions

a(11) from Jinyuan Wang, Apr 02 2021

More terms from David A. Corneth, Apr 09 2021