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A030514 4th powers of primes. 50
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

Solutions of the equation n' = 4*n^(3/4), where n' is the arithmetic derivative of n. - Paolo P. Lava, Oct 31 2012

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

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. - Juri-Stepan Gerasimov, Oct 10 2009

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

PROG

(Sage) BB = primes_first_n(36) list = [] for i in range(36): list.append(BB[i]^4) list # Zerinvary Lajos, May 15 2007

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

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

CROSSREFS

Cf. A030078, A131991, A131992, A000005, A000040, A001248.

Sequence in context: A153157 A113849 A046453 * A056571 A053909 A151502

Adjacent sequences:  A030511 A030512 A030513 * A030515 A030516 A030517

KEYWORD

nonn,easy,changed

AUTHOR

Jeff Burch

EXTENSIONS

Description corrected by Eric W. Weisstein

STATUS

approved

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Last modified September 2 03:04 EDT 2014. Contains 246321 sequences.