cp's OEIS Frontend

A357258 a(n) is the smallest prime p such that the minimum number of divisors among the numbers between p and NextPrime(p) is n, or -1 if no such prime exists.

%I A357258 #10 Sep 24 2022 15:40:42
%S A357258 3,5,12117359,11,7212549413159,29,42433,7207,
%T A357258 51110866676606486280448872612994247,59,
%U A357258 494606165132621236223919803061511452090140639,191,149767,269,14634848452286682176094429263857838452673635407760468708604736302749897919,179
%N A357258 a(n) is the smallest prime p such that the minimum number of divisors among the numbers between p and NextPrime(p) is n, or -1 if no such prime exists.
%C A357258 If n is prime, then one of the integers in the interval [a(n)+1, NextPrime(a(n))-1] is the (n-1)st power of a prime, and that number has fewer divisors than any other integer in the interval:
%C A357258    a(3)  = PreviousPrime(2^(3-1)),
%C A357258    a(5)  = PreviousPrime(59^(5-1)),
%C A357258    a(7)  = PreviousPrime(139^(7-1)),
%C A357258    a(11) = PreviousPrime(2957^(11-1)),
%C A357258    a(13) = PreviousPrime(5303^(13-1)),
%C A357258    a(17) = PreviousPrime(37397^(17-1)),
%C A357258    a(19) = PreviousPrime(23563^(19-1)).
%C A357258 For larger prime values of n, the prime q whose (n-1)st power lies in the interval [a(n)+1, NextPrime(a(n))-1] tends to be large because the (n-1)st powers of primes grow rapidly and the intervals between large consecutive primes tend to include at least one composite number with fewer than n divisors. E.g., for each prime q < 37397, the interval between the two consecutive primes between which q^16 falls contains at least one integer with fewer than 17 divisors.
%C A357258 Do there exist values of n such that a(n) = -1?
%e A357258 All integers between successive primes are composite numbers, and every composite number has at least 3 divisors, so the sequence begins with a(3).
%e A357258 p=2 is not a term of the sequence because there are no integers between 2 and NextPrime(2)=3.
%e A357258 p=3 is a(3): NextPrime(3)=5, and the only integer between 3 and 5 is 4 = 2^2, which has 3 divisors, and no prime < 3 has this property.
%e A357258 p=5 is a(4): NextPrime(5)=7, and the only integer between 5 and 7 is 6 = 2*3, which has 4 divisors, and no prime < 5 has this property.
%e A357258 Since 5 is a prime, every number that has exactly 5 divisors is the 4th power of a prime, so a(5) is a prime p such that p < q^4 < NextPrime(p), where q is also a prime. q=2 will not work; it would require p < 2^4 = 16 < NextPrime(p), so p=13 and NextPrime(p)=17, but 5 is not the minimum value of tau(k) for 13 < k < 17 (e.g., tau(16)=4 < 5). Similarly, no prime q < 59 will work; for each such prime q, q^4 lies in an interval between consecutive primes that also includes at least one other composite number that has only 4 divisors. But q=59 gives q^4 = 12117361, which lies between consecutive primes p=12117359 and NextPrime(p)=12117367, and for the composite numbers k in the interval between them, the values of tau(k) are 80, 5, 16, 8, 24, 16, 24, the minimum of which is 5, so a(5)=12117359.
%Y A357258 Cf. A061112 (minimum tau(k) for prime(n) < k < prime(n+1)).
%Y A357258 Other sequences defined in terms of the minimum tau(k) for p < k < NextPrime(p):
%Y A357258 A357170 (primes p such that minimum tau(k) is a prime power);
%Y A357258 A356833 (primes p such that minimum tau(k) is a square);
%Y A357258 A357175 (primes p such that minimum tau(k) is a cube).
%K A357258 nonn
%O A357258 3,1
%A A357258 _Jon E. Schoenfield_, Sep 20 2022