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.

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A372113 Numbers k for which (k-1)/2 and 2*k+1 are both primes.

Original entry on oeis.org

5, 11, 15, 23, 35, 39, 63, 75, 83, 95, 119, 135, 179, 215, 219, 299, 303, 315, 359, 363, 455, 459, 483, 515, 543, 615, 663, 699, 719, 735, 779, 803, 879, 915, 923, 935, 975, 999, 1019, 1043, 1143, 1155, 1175, 1199, 1295, 1323, 1355, 1383, 1439, 1539, 1595, 1659, 1679, 1755, 1763, 1815, 1859, 1883
Offset: 1

Views

Author

Alexandre Herrera, Apr 19 2024

Keywords

Comments

Intersection of A072055 and A104635.

Examples

			5 is a term because (5-1)/2 = 2 is prime and 2*5+1 = 11 is prime.

Links

Crossrefs

Cf. A072055, A104635.
Cf. A023213, A126330.

Programs

  • Mathematica
    Select[Range[1, 2000, 2], AllTrue[{(# - 1)/2, 2 # + 1}, PrimeQ] &] (* Michael De Vlieger, Apr 19 2024 *)
  • Python
    from sympy import isprime
def a(n): return n%2 == 1 and isprime((n-1)>>1) and isprime(2*n+1)
print([n for n in range(2, 1900) if a(n)])

Formula

a(n) = 2*A023213(n) + 1.
a(n) = (A126330(n)-1)/2.

A177083 A006093(k)-fold repetition of A001248(k), k=1,2,3,..

Original entry on oeis.org

4, 9, 9, 25, 25, 25, 25, 49, 49, 49, 49, 49, 49, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 169, 169, 169, 169, 169, 169, 169, 169, 169, 169, 169, 169
Offset: 1

Views

Author

Paul Curtz, Dec 09 2010

Keywords

Comments

Consider the initial terms of numerator sequences (dropping initial zeros) of
3; A005563=N(1) ,
5,3; A061037=N(2) ,
7,16,1; A061039=N(3) ,
9,5,33,3; A061041=N(4) ,
11,24,39,56,3; A061043=N(5) ,
13,7,5,4,85,1; A061045=N(6) ,
15,32,51,72,95,120,3; A061047=N(7) ,
17,9,57,5,105,33,161,3; A061049=N(8) ,
19,40,7,88,115,16,175,208,1; N(9),
21,11,69,6,1,39,189,14,261,3; N(10),
23,48,75,104,135,168,203,240,279,320,3; N(11)
One must add the following associated (minimum) squares (taken from squared entries in A172038) to these values to reach the next possible square not larger than the entry itself:
1; N(1)
4,1; N(2)
9,9,0; N(3)
16,4,16,1; N(4)
25,25,25,25,1; N(5)
36,9,4,0,36,0; N(6)
49,49,49,49,49,49,1; N(7)
64,16,64,4,64,16,64,1, ; N(8)
Only if the index of N(.) is a prime we obtain a string of equal consecutive terms in these complementary rows: 4, 9, 25, 49, 121, 169..
The current sequence lists the consecutive complementary squares, A001248, in the rows with prime index, including their multiplicity (which is A006093).
This generates a link between the primes and the Rydberg-Ritz spectrum of the hydrogen atom.

Crossrefs

Cf. A072055, A135177.

A225874 Primes of the form 5*p^2+5*p+1, where p is a prime.

Original entry on oeis.org

31, 61, 151, 281, 661, 911, 1531, 1901, 9461, 18911, 25561, 27011, 31601, 51511, 57781, 59951, 81281, 86461, 94531, 97301, 111751, 114761, 140281, 183361, 187211, 286801, 347161, 363151, 401861, 485161, 603781, 697511, 720101, 758551, 806011, 939611, 965801
Offset: 1

Views

Author

Jayanta Basu, May 19 2013

Keywords

Comments

Primes generated in A175063.
The square roots of (4*a(n)+1)/5 are in A072055. [Bruno Berselli, May 19 2013]

Crossrefs

Cf. A175063.

Programs

  • Magma
    [a: p in PrimesUpTo(500) | IsPrime(a) where a is 5*p^2+5*p+1]; // Bruno Berselli, May 19 2013
  • Mathematica
    Select[Table[p=Prime[n]; 5*p^2+5*p+1, {n,85}], PrimeQ]
Previous Showing 21-23 of 23 results.

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