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 1-3 of 3 results.

A207337 Primes of the form (m^2+1)/10.

Original entry on oeis.org

5, 17, 29, 53, 73, 109, 137, 281, 397, 449, 593, 757, 941, 1061, 1277, 1613, 1877, 2161, 2341, 2657, 2789, 3881, 4973, 5153, 6101, 6917, 7129, 7673, 8009, 8237, 8821, 9181, 10433, 12041, 13177, 13469, 13913, 14669, 15761, 17389, 18233, 18749
Offset: 1

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Author

Wolfdieter Lang, Feb 27 2012

Keywords

Comments

Equivalently, primes of the form (K^2 + (K+1)^2)/5. The connection to the primes of the form (m^2+1)/10 is given by m=2*K+1 (m is necessarily odd). The corresponsding m=m(n) values are given in A002733(n).
Equivalently, primes of the form (4*T(K)+1)/5, with the corresponding triangular numbers T(K):=A000217(K), for K(n)=(m(n)-1)/2, given in A207339(n).
For n>=2 the smallest positive representative of the class of nontrivial solutions of the congruence x^2==1 (Modd a(n)) is x=m(n). The trivial solution is the class with representative x=1, which also includes -1. For the prime a(1)=5 the smallest positive nontrivial solution is 3 (see A027862(1) with A002731(1)). Such a nontrivial smallest positive representative exists for each unique class of solutions of this congruence Modd p for any prime p of the form 4*k+1, given in A002144. Here the subset with k=k(n)=(a(n)-1)/4 appears, namely 1, 4, 7, 13, 18, 27, 34, 70,... For Modd n see a comment on A203571.

Examples

			a(3)=29, m(3)=A002733(3)=17. T(K(3))=A000217((17-1)/2)= A000217(8)=A207339(3)=36. (8^2+9^2)/5 = 29 = (4*36+1)/5.

Links

Crossrefs

Cf. A207339, A129307, A027862, A002731.
Cf. A010051, A002522.

Programs

  • Haskell
    a207337 n = a207337_list !! (n-1)
a207337_list = f a002522_list where
   f (x:xs) | m == 0 && a010051 y == 1 = y : f xs
            | otherwise                = f xs
            where (y,m) = divMod x 10
-- Reinhard Zumkeller, Apr 06 2012

Formula

a(n) is the n-th member of the increasingly ordered list of primes of the form (m^2+1)/10, where m=m(n) is necessarily an odd integer, namely A002733(n).

A207339 Triangular numbers T from A000217 such that (4*T+1)/5 is prime.

Original entry on oeis.org

6, 21, 36, 66, 91, 136, 171, 351, 496, 561, 741, 946, 1176, 1326, 1596, 2016, 2346, 2701, 2926, 3321, 3486, 4851, 6216, 6441, 7626, 8646, 8911, 9591, 10011, 10296, 11026, 11476, 13041, 15051, 16471, 16836, 17391, 18336, 19701, 21736
Offset: 1

Views

Author

Wolfdieter Lang, Feb 27 2012

Keywords

Comments

The corresponding primes are gven in A207337, where also equivalent formulations are found.
The indices of these triangular numbers are given by (A002733(n)-1)/2.

Examples

			a(3) = 36 = T((17-1)/2) = T(8)=A000217(8). (4*36+1)/5 = 29 = A207337(3).

Links

Crossrefs

Cf. A207337, A129307, A027862, A002731.

Programs

  • Mathematica
    Select[Accumulate[Range[300]],PrimeQ[(4#+1)/5]&] (* Harvey P. Dale, Sep 18 2019 *)

Formula

a(n) = T(K(n)):= A000217(K(n)) with K(n)=(m(n)-1)/2, and m(n) given in A002733(n).

A208292 Primes of the form (n^2+1)/26.

Original entry on oeis.org

17, 37, 457, 601, 701, 877, 997, 2017, 3037, 3257, 4957, 5237, 5701, 10601, 11257, 11677, 14737, 15217, 16001, 17317, 17837, 21577, 22157, 24677, 29717, 34057, 39157, 39937, 41201, 50777, 52201, 53101, 75277, 78101, 79201, 89917, 91097, 93001, 94201, 96137
Offset: 1

Views

Author

Wolfdieter Lang, Feb 27 2012

Keywords

Comments

Equivalently, primes of the form (K^2 + (K+1)^2)/13. The
connection to the primes of the form (m^2+1)/26 is given by m=2*K+1 (m is necessarily odd).
The corresponding m=m(n) values are given in A208293(n).
Equivalently, primes of the form (4*T(K)+1)/13, with the
corresponding triangular numbers T(K):=A000217(K), for
K=K(n)=(m(n)-1)/2, given in A208294(n).
For n>=2 the smallest positive representative of the class of
nontrivial solutions of the congruence x^2==1 (Modd a(n)) is
x=m(n). The trivial solution is the class with representative x=1, which also includes -1. For the prime
a(1)=17 the nontrivial solution is 13 (see A002733(2)). Unique nontrivial smallest positive representatives exist for the solutions for any prime of the form 4*k+1, given in A002144. Here the subset with k=k(n)=(a(n)-1)/4 appears, namely 4,9,114,150,175,219,.... For Modd n see a comment on A203571.
These primes with corresponding m values are such that floor(m(n)^2/p(n)) = 5^2, n>=1.

Examples

			a(3)=457, m(3)=A208293(3)=109. T(K(3))=A000217((109-1)/2)=
  A000217(54)=A208294(3)=1485.

Links

Crossrefs

Cf. A207337, A207339 (case floor(m^2/p)=3^2); A129307, A027862, A002731 (case floor(m^2/p)=1^2).

Programs

  • Mathematica
    Select[(Range[2000]^2 + 1)/26, PrimeQ] (* T. D. Noe, Feb 28 2012 *)

Formula

a(n) is the n-th member of the increasingly ordered list of primes of the form (m^2+1)/10, where m=m(n) is necessarily an odd integer, the positive one is A208293(n).
Showing 1-3 of 3 results.

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