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 1-5 of 5 results.

A115760 Slowest growing sequence of numbers having the prime-pairwise-average property: if i

Original entry on oeis.org

3, 7, 19, 55, 139, 859, 2119, 112999, 333679, 10040119, 15363619, 548687179, 16632374359, 5733638351299, 14360489685499, 433098704482699, 44258681327079259, 5009018648920510999
Offset: 1

Views

Author

Zak Seidov, Jan 30 2006

Keywords

Comments

Inspired by A113875 (case of prime numbers). See A113832 minimal sets of primes having the P-P-A property, A115782 primes in A115760.
Equals 2*A103828(n) + 1. - N. J. A. Sloane, Apr 28 2007. This sequence is surely infinite - see comments in A103828.
After a(4), terms are == 19 mod 60. The sequence may also be defined by "a(1)=3 and for n>1, a(n) is the smallest number of the form 4k+3, a(n)>a(n-1) such that the pairwise sums of all elements are semiprimes." - Don Reble, Aug 17 2021

Examples

			The pairwise averages of {3,7,19} are the primes {5,11,13}.

Crossrefs

Cf. A113832, A113875, A115782, A175532.

Formula

a(n) == 19 (mod 60) for n>4 [consequence of mod 30 congruence of A103828(n).] - Don Reble, Aug 17 2021

Extensions

More terms from Don Reble and Giovanni Resta, Feb 15 2006
More terms from Don Reble, Aug 17 2021

A119751 a(1) = 1; a(n) = first odd number greater than a(n-1) such that a(n) + a(i) + 1 is prime for all i=1,2,...,n.

Original entry on oeis.org

1, 3, 9, 69, 429, 4089, 86529, 513099, 913569, 7914339, 6593621379, 9366241599, 456246278469, 4565283812559
Offset: 1

Views

Author

Walter Kehowski, Jun 17 2006

Keywords

Crossrefs

Cf. A113875, A119752.

Programs

  • Mathematica
    Table[If[n == 1, a[1] = 1, j = a[n - 1] + 2; While[a[n] = j; !
    AllTrue[Table[a[i] + a[n] + 1, {i, 1, n}], PrimeQ], j += 2]; j]
, {n, 1, 7}] (* Robert Price, Apr 03 2019 *)
  • PARI
    isok(va, k, n) = if (isprime(2*k+1), for (i=1, n-1, if (! isprime(va[i]+k+1), return(0))); return(1));
lista(nn) = my(va=vector(nn)); va[1]=1; for (n=2, nn, my(k=va[n-1]+2); while (!isok(va, k, n), k+=2); va[n] = k); va; \\ Michel Marcus, Mar 07 2023

Formula

2*a(n)+1 = A113875(n). - Don Reble, Aug 17 2021

Extensions

a(13)-a(14) from Donovan Johnson, Mar 23 2008
Corrected and edited by Walter Kehowski, Oct 18 2008

A115782 Primes generated by pairwise averages of terms in A115760.

Original entry on oeis.org

5, 11, 13, 29, 31, 37, 71, 73, 79, 97, 431, 433, 439, 457, 499, 1061, 1063, 1069, 1087, 1129, 1489, 56501, 56503, 56509, 56527, 56569, 56929, 57559, 166841, 166843, 166849, 166867, 166909, 167269, 167899, 223339, 5020061, 5020063, 5020069
Offset: 1

Views

Author

Zak Seidov, Jan 30 2006

Keywords

Comments

Also primes arising from A103828. - N. J. A. Sloane, Apr 28 2007

Examples

			The pairwise averages of first three terms in A115760, {3,7,19} produce the set of primes {5,11,13}.

Crossrefs

Cf. A113832, A113875, A115760.

A114845 Slowest growing sequence of semiprimes having the semiprime-pairwise-average property: for any i,j, (a(i)+a(j))/2 is semiprime.

Original entry on oeis.org

4, 14, 38, 134, 254, 13238, 252254, 691958, 952814, 3316238, 30364918838, 210339665174, 575167942574
Offset: 1

Views

Author

Jonathan Vos Post, Feb 20 2006

Keywords

Comments

Semiprime analog of A113875.

Examples

			The pairwise average of the semiprimes {4 = 2^2, 14 = 2*7} is {9 = 3^2}.
The pairwise averages of the semiprimes {4, 14, 38} are {9, 21, 26}.
The pairwise averages of the semiprimes {4, 14, 38, 134} are {9, 21, 26, 69, 74, 86}.
The pairwise averages of the semiprimes {4, 14, 38, 134, 254} are {9, 21, 26, 69, 74, 86, 129, 134, 146, 194}.

Crossrefs

Cf. A001358, A113832, A113875, A115760, A164979.

Formula

a(n) = 2*A164979(n).

Extensions

More terms from Zak Seidov, Feb 21 2006
Corrected and extended by Zak Seidov, Sep 03 2009
a(11)-a(12) from Amiram Eldar, Jun 27 2024
a(13) from Jinyuan Wang, May 29 2025

A335752 The lexicographically earliest sequence of primes, each two of which sum to a triprime (A014612).

Original entry on oeis.org

3, 5, 47, 239, 3023, 27983, 53231, 118343, 198959, 452831, 580871, 9364871, 26386463, 67876751, 662675543, 2952903983, 22614861263
Offset: 1

Views

Author

Zak Seidov, Jul 19 2020

Keywords

Examples

			3 + 5 = 8 = 2*2*2, 3 + 47 = 50 = 2*5*5, 5 + 47 = 52 = 2*2*13.

Crossrefs

Cf. A000040, A014612, A113875.

Programs

  • Mathematica
    ps = {3};  Do[p = NextPrime[ps[[-1]]]; While[! AllTrue[ps + p, PrimeOmega[#] == 3 &], p = NextPrime[p]]; AppendTo[ps, p], {10}]; ps (* Amiram Eldar, Jul 22 2020 *)

Extensions

a(17) from Amiram Eldar, Jul 22 2020
Name edited by Jon E. Schoenfield, Oct 04 2021
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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