cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A357170 Primes p such that the minimum number of divisors among the numbers between p and NextPrime(p) is a prime power.

Original entry on oeis.org

3, 5, 7, 13, 19, 23, 29, 31, 37, 41, 43, 47, 53, 61, 67, 73, 79, 83, 89, 101, 103, 109, 113, 127, 131, 137, 139, 151, 157, 163, 167, 173, 181, 193, 199, 211, 223, 229, 233, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 349, 353, 359, 367, 373
Offset: 1

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Author

Claude H. R. Dequatre, Sep 16 2022

Keywords

Examples

			19 is a term because up to the next prime 23, tau(20) = 6, tau(21) = 4, tau(22) = 4, thus the smallest tau(k) is 4 and 4 is a prime power (2^2).
97 is prime but not a term because up to the next prime 101, tau(98) = 6, tau(99) = 6, tau(100) = 9, thus the smallest tau(k) is 6 and 6 is not a prime power.

Crossrefs

Cf. A000005, A000040, A061112, A246655.
Cf. A353284, A353285, A353286, A356833, A357175.

Programs

  • PARI
    isok(p)=isprimepower(vecmin(apply(numdiv, [p+1..nextprime(p+1)-1])));
forprime(p=3, 2000, if(isok(p), print1(p", ")))

A357175 Primes p such that the minimum of the number of divisors among the numbers between p and NextPrime(p) is a cube.

Original entry on oeis.org

29, 41, 101, 137, 229, 281, 349, 439, 617, 641, 643, 739, 821, 823, 853, 967, 1087, 1423, 1429, 1447, 1549, 1579, 1597, 1693, 1697, 1783, 1877, 1999, 2081, 2131, 2237, 2239, 2293, 2377, 2381, 2539, 2617, 2657, 2683, 2693, 2713, 2749, 2791, 2801, 3079, 3319
Offset: 1

Views

Author

Claude H. R. Dequatre, Sep 16 2022

Keywords

Examples

			349 is a term because up to the next prime 353, tau(350) = 12, tau(351) = 8, tau(352) = 12, thus the smallest tau(k) = 8 and 8 is a cube (2^3).
379 is prime but not a term because up to the next prime 383, tau(380) = 12, tau(381) = 4, tau(382) = 4, thus the smallest tau(k) is 4 and 4 is not a cube.

Crossrefs

Cf. A000005, A000040, A000578. A061112.
Cf. A353284, A353285, A353286, A356833, A357170.

Programs

  • PARI
    isok(p)=ispower(vecmin(apply(numdiv, [p+1..nextprime(p+1)-1])), 3);
forprime(p=3, 10000, if(isok(p), print1(p", ")))

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.

Original entry on oeis.org

3, 5, 12117359, 11, 7212549413159, 29, 42433, 7207, 51110866676606486280448872612994247, 59, 494606165132621236223919803061511452090140639, 191, 149767, 269, 14634848452286682176094429263857838452673635407760468708604736302749897919, 179
Offset: 3

Views

Author

Jon E. Schoenfield, Sep 20 2022

Keywords

Comments

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?

Examples

			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.

Crossrefs

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).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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