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 1-5 of 5 results.

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

A280256 Numbers k such that tau(k^k) is a prime.

Original entry on oeis.org

2, 9, 6561, 25937424601, 1853020188851841, 58149737003040059690390169, 54116956037952111668959660849, 2787593149816327892691964784081045188247552, 2465034704958067503996131453373943813074726512397600969, 285273917723723876056171083405292782327767461712708093041
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(A000312(k)) = A062319(k) is a prime.
Corresponding values of primes: 3, 19, 52489, ...
All the terms are prime powers.

Examples

			tau(9^9) = tau(387420489) = 19 (prime).

Links

Crossrefs

Cf. A000005, A000312, A062319, A280255, A280257.

Programs

  • Magma
    [n: n in [1..500] | IsPrime(NumberOfDivisors(n^n))];
  • Mathematica
    mx = 10^200; Union@ Flatten@ Reap[ Sow[2^ Select[ Range@ Log2[mx], PrimeQ[1 + # 2^#] &]]; Do[ If[ PrimeQ[1 + q p^q], Sow[p^q]], {p, Prime@ Range@ PrimePi@ 34}, {q, 2, Log[p, mx], 2}]; Do[ Sow@ (Select[ Prime@ Range[2, PrimePi[ mx^(1/e)]], PrimeQ[1 + e #^e] &]^e), {e, 34, Floor@Log[31, mx], 2}]][[2, 1]] (* all the 231 terms < 10^200, Giovanni Resta, Mar 07 2017 *)
  • PARI
    isok(n) = isprime(numdiv(n^n)); \\ Michel Marcus, Mar 07 2017

Extensions

a(4)-a(10) from Giovanni Resta, Mar 07 2017

A283450 Least prime p such that n*(p^n-1)+1 is prime.

Original entry on oeis.org

2, 2, 3, 2, 19, 2, 5, 17, 13, 7, 1129, 59, 47, 7, 19, 7, 31, 79, 11, 37, 199, 5, 907, 43, 5, 43, 3, 13, 919, 2, 13, 2, 263, 127, 241, 3, 131, 71, 11, 421, 223, 2, 31, 3, 7, 89, 3673, 61, 293, 5, 131, 919, 3, 3, 349, 457, 1091, 461, 67, 7, 331, 7177, 571, 43, 1621, 109, 2521, 3, 1061, 5, 967, 1093, 1423
Offset: 1

Views

Author

Robert Israel, Mar 07 2017

Keywords

Comments

a(n) is the least prime p such that p^n is in A280257.
The generalized Dickson conjecture would imply that a(n) exists for all n.
a(n) = 2 for n in A046847.

Examples

			For n=5, 5*(19^5-1)+1 = 12380491 is prime, but 5*(p^5-1)+1 is not prime for primes p < 19, so a(5)=19.

Links

Crossrefs

Cf. A046847, A280257.

Programs

  • Maple
    f:= proc(n) local p;
     p:= 2:
     while not isprime(n*(p^n-1)+1) do p:= nextprime(p) od;
     p
end proc:
map(f, [$1..100]);
  • Mathematica
    Table[p=2; While[!PrimeQ[n (p^n-1)+1], p=NextPrime@p]; p, {n, 100}] (* Vincenzo Librandi, Oct 11 2017 *)
  • PARI
    a(n)=forprime(p=2,, if(ispseudoprime(n*(p^n-1)+1), return(p))) \\ Charles R Greathouse IV, Mar 07 2017

A283549 Composite numbers k such that tau(k^(k-1)) is a prime.

Original entry on oeis.org

4, 9, 16, 27, 49, 64, 121, 125, 169, 289, 1681, 1849, 2401, 3481, 4913, 5329, 11881, 12769, 16129, 18769, 24649, 29791, 32041, 32761, 38809, 39601, 44521, 63001, 69169, 76729, 78125, 79507, 85849, 96721, 124609, 130321, 134689, 143641, 167281, 175561, 187489, 237169, 316969, 326041, 332929, 380689, 383161, 434281, 491401
Offset: 1

Views

Author

Robert G. Wilson v, Mar 10 2017

Keywords

Comments

A proper subset of A280257 and of A025475.

Links

Crossrefs

Cf. A280257, A025475.

Programs

  • Mathematica
    k = 1; lst = {}; While[k < 100001, If [ !PrimeQ@ k && PrimeQ[ DivisorSigma[0, k^(k -1)]], AppendTo[lst, k]]; k++]; lst (* or *)
mx = 10^6; Union@ Flatten@ Reap[ Do[ Sow@ Select[ Prime[ Range[ PrimePi[ mx^(1/e) ]]]^e, PrimeQ[1 + e (#-1)] &], {e, 2, Log2[mx]}]][[2, 1]] (* Giovanni Resta, Mar 10 2017 *)
  • PARI
    is(n)=!isprime(n) && ispseudoprime(numdiv(n^(n-1))) \\ Charles R Greathouse IV, Mar 10 2017

A283515 Numbers k such that sigma(k^(k-1)) is a prime.

Original entry on oeis.org

2, 3, 4, 16, 19, 31, 7547
Offset: 1

Views

Author

Jaroslav Krizek, Mar 10 2017

Keywords

Comments

sigma(k) is the sum of the divisors of k (A000203).
Numbers k such that A000203(A000169(k)) is a prime.
a(8) > 10^4.
Corresponding values of k^(k-1): 2, 9, 64, 1152921504606846976, ...
Corresponding values of sigma(k^(k-1)): 3, 13, 127, 2305843009213693951, ...
Subsequence of A280257 (numbers k such that tau(k^(k-1)) is prime).
Prime terms are in A088790.
For k < 1000, sigma(k^(k+1)) is prime only for k = 5: sigma(5^6) = sigma(15625) = 19531 (prime).

Examples

			sigma(4^3) = sigma(64) = 127 (prime).

Crossrefs

Cf. A000169, A000203, A088790, A280257.

Programs

  • Magma
    [n: n in [1..500] | IsPrime(SumOfDivisors(n^(n-1)))];
  • Mathematica
    fQ[n_] := PrimeQ[DivisorSigma[1, n^(n - 1)]]; Select[Range@1000, fQ] (* Robert G. Wilson v, Mar 10 2017 *)
  • PARI
    isok(n) = isprime(sigma(n^(n-1))); \\ Michel Marcus, Mar 10 2017

Extensions

a(7) from Giovanni Resta, Mar 10 2017
Showing 1-5 of 5 results.

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