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%I #91 Feb 04 2024 01:12:29
%S 16,81,625,2401,14641,28561,83521,130321,279841,707281,923521,1874161,
%T 2825761,3418801,4879681,7890481,12117361,13845841,20151121,25411681,
%U 28398241,38950081,47458321,62742241,88529281,104060401,112550881,131079601,141158161
%N a(n) = prime(n)^4.
%C Numbers with 5 divisors (1, p, p^2, p^3, p^4, where p is the n-th prime). - _Alexandre Wajnberg_, Jan 15 2006
%C Subsequence of A036967. - _Reinhard Zumkeller_, Feb 05 2008
%C 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
%C 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
%C Except for the first three terms, all others are congruent to 1 mod 240. - _Robert Israel_, Aug 29 2014
%H R. J. Mathar, <a href="/A030514/b030514.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimePower.html">Prime Power</a>.
%H OEIS Wiki, <a href="https://oeis.org/wiki/Index_entries_for_number_of_divisors">Index entries for number of divisors</a>
%H <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%F 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
%F A000005(a(n)) = 5. - _Alexandre Wajnberg_, Jan 15 2006
%F A056595(a(n)) = 2. - _Reinhard Zumkeller_, Aug 15 2011
%F Sum_{n>=1} 1/a(n) = P(4) = 0.0769931397... (A085964). - _Amiram Eldar_, Jul 27 2020
%F From _Amiram Eldar_, Jan 23 2021: (Start)
%F Product_{n>=1} (1 + 1/a(n)) = zeta(4)/zeta(8) = 105/Pi^4 (A157290).
%F Product_{n>=1} (1 - 1/a(n)) = 1/zeta(4) = 90/Pi^4 (A215267). (End)
%p map(p -> p^4, select(isprime,[2,seq(2*i+1,i=1..100)])); # _Robert Israel_, Aug 29 2014
%t Array[Prime[#]^4 &, 5!] (* _Vladimir Joseph Stephan Orlovsky_, Sep 01 2008 *)
%o (Sage)
%o [p**4 for p in prime_range(100)]
%o # _Zerinvary Lajos_, May 15 2007
%o (Magma) [NthPrime(n)^4: n in [1..100] ]; // _Vincenzo Librandi_, Apr 22 2011
%o (PARI) a(n)=prime(n)^4 \\ _Charles R Greathouse IV_, Mar 21 2013
%o (Haskell)
%o a030514 = (^ 4) . a000040
%o a030514_list = map (^ 4) a000040_list
%o -- _Reinhard Zumkeller_, Jun 03 2015
%Y Cf. A030078, A085964, A131991, A131992, A000005, A000040, A001248, A157290, A215267.
%Y Cf. A258601.
%K nonn,easy
%O 1,1
%A _Jeff Burch_
%E Description corrected by _Eric W. Weisstein_
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