A030514 a(n) = prime(n)^4.

16, 81, 625, 2401, 14641, 28561, 83521, 130321, 279841, 707281, 923521, 1874161, 2825761, 3418801, 4879681, 7890481, 12117361, 13845841, 20151121, 25411681, 28398241, 38950081, 47458321, 62742241, 88529281, 104060401, 112550881, 131079601, 141158161

Offset: 1

Author

Jeff Burch

Keywords

• nonn
• easy

Comments

Numbers with 5 divisors (1, p, p^2, p^3, p^4, where p is the n-th prime). - Alexandre Wajnberg, Jan 15 2006

Subsequence of A036967. - Reinhard Zumkeller, Feb 05 2008

The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. - Omar E. Pol, May 06 2008

The general product formula for even s is: product_{p = A000040} (p^s-1)/(p^s+1) = 2*Bernoulli(2s)/( binomial(2s, s)*Bernoulli^2(s)), where the infinite product is over all primes. Here, with s = 4, product_{n = 1, 2, ...} (a(n)-1)/(a(n)+1) = 6/7. In A030516, where s = 6, the product of the ratios is 691/715. For s = 8, the 8th row in A120458, the corresponding product of ratios is 7234/7293. - R. J. Mathar, Feb 01 2009

Except for the first three terms, all others are congruent to 1 mod 240. - Robert Israel, Aug 29 2014

Links

• R. J. Mathar, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Prime Power.
• OEIS Wiki, Index entries for number of divisors
• Index to sequences related to prime signature

Crossrefs

Cf. A030078, A085964, A131991, A131992, A000005, A000040, A001248, A157290, A215267.

Cf. A258601.

Programs

Haskell

a030514 = (^ 4) . a000040
a030514_list = map (^ 4) a000040_list
-- Reinhard Zumkeller, Jun 03 2015

Magma

[NthPrime(n)^4: n in [1..100] ]; // Vincenzo Librandi, Apr 22 2011

Maple

map(p -> p^4, select(isprime,[2,seq(2*i+1,i=1..100)])); # Robert Israel, Aug 29 2014

Mathematica

Array[Prime[#]^4 &, 5!] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)

PARI

a(n)=prime(n)^4 \\ Charles R Greathouse IV, Mar 21 2013

Sage

[p**4 for p in prime_range(100)]
# Zerinvary Lajos, May 15 2007

Formula

a(n) = A000040(n)^(5-1) = A000040(n)^4, where 5 is the number of divisors of a(n). - Omar E. Pol, May 06 2008

A000005(a(n)) = 5. - Alexandre Wajnberg, Jan 15 2006

A056595(a(n)) = 2. - Reinhard Zumkeller, Aug 15 2011

Sum_{n>=1} 1/a(n) = P(4) = 0.0769931397... (A085964). - Amiram Eldar, Jul 27 2020

From Amiram Eldar, Jan 23 2021: (Start)

Product_{n>=1} (1 + 1/a(n)) = zeta(4)/zeta(8) = 105/Pi^4 (A157290).

Product_{n>=1} (1 - 1/a(n)) = 1/zeta(4) = 90/Pi^4 (A215267). (End)

Extensions

Description corrected by Eric W. Weisstein