

A030514


4th powers of primes.


68



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
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OFFSET

1,1


COMMENTS

Numbers with 5 divisors (1, p, p^2, p^3, p^4, where p is the nth prime).  Alexandre Wajnberg, Jan 15 2006
Subsequence of A036967.  Reinhard Zumkeller, Feb 05 2008
The nth number with p divisors is equal to the nth prime raised to power p1, where p is prime.  Omar E. Pol, May 06 2008
The general product formula for even s is: product_{p = A000040} (p^s1)/(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
Index to sequences related to prime signature


FORMULA

a(n) = A000040(n)^(51) = 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


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


CROSSREFS

Cf. A030078, A131991, A131992, A000005, A000040, A001248.
Cf. A258601.
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



