cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A066885 a(n) = (prime(n)^2 + 1)/2.

Original entry on oeis.org

5, 13, 25, 61, 85, 145, 181, 265, 421, 481, 685, 841, 925, 1105, 1405, 1741, 1861, 2245, 2521, 2665, 3121, 3445, 3961, 4705, 5101, 5305, 5725, 5941, 6385, 8065, 8581, 9385, 9661, 11101, 11401, 12325, 13285, 13945, 14965, 16021, 16381, 18241, 18625
Offset: 2

Views

Author

Enoch Haga, Jan 22 2002

Keywords

Comments

a(n) is the average of the numbers from 1 to prime(n)^2. It's also the average of the primes in a prime(n) X prime(n) example of Haga's conjecture (see link below).
If a(n) is a square c^2, then prime(n) is an NSW prime (A088165) and a prime RMS number (A140480). - Ctibor O. Zizka, Aug 26 2008
The sequence starts with a(2) = (3^2 + 1)/2 = 5 since a(1) would be (2^2 + 1)/2 = 5/2. - Michael B. Porter, Dec 14 2009

Links

Crossrefs

Cf. A066883, A066886.
Cf. A088165, A140480.
Cf. A084921.
Partial sums of A124434.

Programs

Formula

a(n) = 1 + A084921(n). - R. J. Mathar, Sep 30 2011
a(n) mod 4 = 1. - Altug Alkan, Apr 08 2016
Product_{n>=2} (1 - 1/a(n)) = 2/3. - Amiram Eldar, Jun 03 2022

Extensions

Edited by Dean Hickerson, Jun 08 2002

A075892 Average of squares of successive primes: a(n) = (prime(n+1)^2 + prime(n)^2)/2, with n >= 2.

Original entry on oeis.org

17, 37, 85, 145, 229, 325, 445, 685, 901, 1165, 1525, 1765, 2029, 2509, 3145, 3601, 4105, 4765, 5185, 5785, 6565, 7405, 8665, 9805, 10405, 11029, 11665, 12325, 14449, 16645, 17965, 19045, 20761, 22501, 23725, 25609, 27229, 28909, 30985, 32401
Offset: 2

Views

Author

Zak Seidov, Oct 17 2002

Keywords

Comments

a(n) is prime for n in A240749. - Robert Israel, Jul 06 2017
If p and q are primes such that p > q > 3, then ((p^2 - q^2)/2, p*q, (p^2 + q^2)/2) is a primitive Pythagorean triple. - César Aguilera, Jun 02 2022

Examples

			a(2)=17 because (prime(3)^2 + prime(2)^2)/2 = (5^2 + 3^2)/2 = 17.

Links

Crossrefs

Cf. A006094, A124434, A143850, A240749.

Programs

  • Magma
    [(NthPrime(n+1)^2+NthPrime(n)^2)/2: n in [2..50]]; // Vincenzo Librandi, Mar 07 2015
  • Maple
    seq((ithprime(i)^2 + ithprime(i+1)^2)/2, i=2..100); # Robert Israel, Jul 06 2017
  • Mathematica
    Table[(Prime[n + 1]^2 + Prime[n]^2)/2, {n, 2, 50}] (* Vincenzo Librandi, Mar 07 2015 *)
p=2;q=3;Table[p=q;q=NextPrime[q];(q^2+p^2)/2,{100}] (* Zak Seidov, Jul 06 2017 *)
  • PARI
    a(n) = (prime(n+1)^2+prime(n)^2)/2; \\ Michel Marcus, Oct 03 2013

Formula

a(n)^2 = A124434(n)^2 + A006094(n)^2. - César Aguilera, Jun 02 2022
Showing 1-2 of 2 results.

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