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A357258
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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.
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0
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3, 5, 12117359, 11, 7212549413159, 29, 42433, 7207, 51110866676606486280448872612994247, 59, 494606165132621236223919803061511452090140639, 191, 149767, 269, 14634848452286682176094429263857838452673635407760468708604736302749897919, 179
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OFFSET
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3,1
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COMMENTS
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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:
a(3) = PreviousPrime(2^(3-1)),
a(5) = PreviousPrime(59^(5-1)),
a(7) = PreviousPrime(139^(7-1)),
a(11) = PreviousPrime(2957^(11-1)),
a(13) = PreviousPrime(5303^(13-1)),
a(17) = PreviousPrime(37397^(17-1)),
a(19) = PreviousPrime(23563^(19-1)).
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.
Do there exist values of n such that a(n) = -1?
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LINKS
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EXAMPLE
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All integers between successive primes are composite numbers, and every composite number has at least 3 divisors, so the sequence begins with a(3).
p=2 is not a term of the sequence because there are no integers between 2 and NextPrime(2)=3.
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.
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.
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.
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CROSSREFS
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Cf. A061112 (minimum tau(k) for prime(n) < k < prime(n+1)).
Other sequences defined in terms of the minimum tau(k) for p < k < NextPrime(p):
A357170 (primes p such that minimum tau(k) is a prime power);
A356833 (primes p such that minimum tau(k) is a square);
A357175 (primes p such that minimum tau(k) is a cube).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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