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A030514
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a(n) = prime(n)^4.
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86
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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
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1,1
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COMMENTS
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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
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LINKS
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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
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FORMULA
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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)
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MAPLE
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map(p -> p^4, select(isprime, [2, seq(2*i+1, i=1..100)])); # Robert Israel, Aug 29 2014
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MATHEMATICA
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Array[Prime[#]^4 &, 5!] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)
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PROG
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(Sage)
[p**4 for p in prime_range(100)]
# 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
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CROSSREFS
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Cf. A030078, A085964, A131991, A131992, A000005, A000040, A001248, A157290, A215267.
Cf. A258601.
Sequence in context: A153157 A113849 A046453 * A056571 A053909 A151502
Adjacent sequences: A030511 A030512 A030513 * A030515 A030516 A030517
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KEYWORD
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nonn,easy,changed
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AUTHOR
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Jeff Burch
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EXTENSIONS
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Description corrected by Eric W. Weisstein
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STATUS
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approved
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