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.

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A280257 Numbers k such that tau(k^(k-1)) is a prime.

Original entry on oeis.org

2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229
Offset: 1

Views

Author

Jaroslav Krizek, Mar 07 2017

Keywords

Comments

tau(k) is the number of positive divisors of k (A000005).
Numbers k such that A000005(A000169(k)) is a prime.
All primes (A000040) are terms. If p is prime then tau(p^(p-1)) = p.
Sequence of composite terms c: 4, 9, 16, 27, 49, 64, 121, 125, 169, 289, ...; (tau(c^(c-1)): 7, 17, 61, 79, 97, 379, 241, 373, 337, 577, ...).
All terms are powers of primes (A000961). - Robert Israel, Mar 07 2017

Examples

			tau(4^3) = tau(64) = 7 (prime).

Links

Crossrefs

Cf. A000005, A000040, A000169, A000961, A280255, A280256.

Programs

  • Magma
    [n: n in [1..100] | IsPrime(NumberOfDivisors(n^(n-1)))]
  • Maple
    N:= 5000: # to get all terms <= N
Primes:= select(isprime, {2,seq(i,i=3..N,2)}):
sort([seq(seq(`if`(isprime(k*(p^k-1)+1),p^k,NULL), k=1..floor(log[p](N))), p=Primes)]); # Robert Israel, Mar 07 2017
  • Mathematica
    Select[Range@ 230, PrimeQ@ DivisorSigma[0, #^(# - 1)] &] (* Michael De Vlieger, Mar 07 2017 *)
  • PARI
    isok(n) = isprime(numdiv(n^(n-1))); \\ Michel Marcus, Mar 07 2017
  • PARI
    list(lim)=my(v=List(primes([2,lim\=1]))); for(e=2,logint(lim,2), forprime(p=2,sqrtnint(lim,e), if(ispseudoprime(e*(p^e-1)+1), listput(v,p^e)))); Set(v) \\ Charles R Greathouse IV, Mar 07 2017

Formula

a(n) ~ n log n. - Charles R Greathouse IV, Mar 07 2017

A280255 Numbers k such that tau(k^(k+1)) is a prime.

Original entry on oeis.org

3, 4, 5, 11, 17, 25, 29, 41, 49, 59, 71, 101, 107, 125, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 343, 347, 419, 431, 461, 521, 529, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319
Offset: 1

Views

Author

Jaroslav Krizek, Mar 07 2017

Keywords

Comments

tau(k) is the number of positive divisors of k (A000005).
Numbers k such that A000005(A007778(k)) is a prime.
Lesser of twin primes (A001359) are terms. If p is lesser of twin primes then tau(p^(p+1)) = p + 2 (see A006512).
Sequence of composite terms c: 4, 25, 49, 125, 343, 529, 1369, ...; (tau(c^(c+1)): 11, 53, 101, 379, 1033, 1061, 2741, ...).
Numbers of the form p^k where p is prime and 1 + k * (1 + p^k) is prime. - Robert Israel, Sep 02 2024

Examples

			tau(4^5) = tau(1024) = 11 (prime).

Links

Crossrefs

Cf. A000005, A001359, A006512, A007778, A280256, A280257.

Programs

  • Magma
    [n: n in [1..500] | IsPrime(NumberOfDivisors(n^(n+1)))];
  • Maple
    N:= 10000: # for terms <= N
P:= select(isprime,[2,seq(i,i=3..N,2)]):
R:= {}:
for p in P do
  Qs:= select(q -> isprime(1 + q + q*p^q), {$1..ilog[p](N)});
  R:= R union map(q -> p^q, Qs)
od:
sort(convert(R,list)); # Robert Israel, Sep 02 2024
  • Mathematica
    Select[Range[1319], PrimeQ@DivisorSigma[0, #^(# + 1)] &] (* Giovanni Resta, Mar 07 2017 *)
  • PARI
    isok(n) = isprime(numdiv(n^(n+1))); \\ Michel Marcus, Mar 07 2017
Showing 1-2 of 2 results.

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