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A343087 a(n) is the smallest prime p such that tau(p-1) = 2^n.

%I A343087 #19 Sep 08 2022 08:46:26
%S A343087 3,7,31,211,1321,7561,120121,1580041,24864841,328648321,7558911361,
%T A343087 162023621761,5022732274561,93163582512001,4083134943888001,
%U A343087 151075992923856001,5236072827921936001,188391763176048432001,8854412869274276304001,469283882071536644112001,29844457947060064452144001,1917963226026370264485744001
%N A343087 a(n) is the smallest prime p such that tau(p-1) = 2^n.
%C A343087 tau(m) = the number of divisors of m (A000005).
%C A343087 Sequences of primes p such that tau(p-1) = 2^n for 2 <= n <= 5:
%C A343087 n = 2: 7, 11, 23, 47, 59, 83, 107, 167, 179, ... (A005385(k) for k >= 2).
%C A343087 n = 3: 31, 41, 43, 67, 71, 79, 89, 103, 131, 137, 139, 191, ...
%C A343087 n = 4: 211, 271, 281, 313, 331, 379, 409, 457, 463, 521, 547, ...
%C A343087 n = 5: 1321, 2281, 2311, 2377, 2689, 2731, 2857, 2971, 3001, ...
%C A343087 Conjecture: a(n) is also the smallest number m such that tau(m-1) = tau(m)^n.
%H A343087 Bert Dobbelaere, <a href="/A343087/b343087.txt">Table of n, a(n) for n = 1..60</a>
%e A343087 For n = 4; a(4) = 211 because 211 is the smallest prime p such that tau(p - 1) = 2^4; tau(210) = 16.
%o A343087 (Magma) Ax:=func<n|exists(r){m: m in[1..10^8] | IsPrime(m) and #Divisors(m - 1) eq 2 ^ n} select r else 0>; [Ax(n): n in [1..9]]
%o A343087 (Python)
%o A343087 from sympy import isprime,nextprime
%o A343087 primes=[2]
%o A343087 def solve(v,k,i,j):
%o A343087     global record,stack,primes
%o A343087     if k==0:
%o A343087         if isprime(v+1):
%o A343087             record=v
%o A343087         return
%o A343087     while True:
%o A343087         if i>=len(primes):
%o A343087             primes.append(nextprime(primes[-1]))
%o A343087         if j<len(stack) and stack[j]<primes[i]:
%o A343087             f=stack[j] ; j+=1
%o A343087         else:
%o A343087             f=primes[i] ; i+=1
%o A343087         if record==None or v * f**k < record:
%o A343087             stack.append(f**2)
%o A343087             solve(v*f,k-1,i,j)
%o A343087             stack.pop()
%o A343087         else:
%o A343087             return
%o A343087 def a343087(n):
%o A343087     global record,stack
%o A343087     record,stack = None,[]
%o A343087     solve(1,n,0,0)
%o A343087     return record+1
%o A343087 # _Bert Dobbelaere_, Apr 11 2021
%Y A343087 Cf. A000005, A000079, A037992, A343020.
%K A343087 nonn
%O A343087 1,1
%A A343087 _Jaroslav Krizek_, Apr 04 2021 (following a suggestion of _Vaclav Kotesovec_)
%E A343087 a(11) from _Vaclav Kotesovec_, Apr 05 2021
%E A343087 More terms from _Bert Dobbelaere_, Apr 11 2021