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.

A266950 Indices of record values in A266948: least prime p such that p-2 and 6n-p are also prime.

Original entry on oeis.org

0, 2, 5, 21, 37, 93, 133, 463, 1759, 2676, 4328, 8931, 10919, 17929, 22663, 43451, 45355, 55533, 60551, 92233, 112723, 190371, 485648, 791123, 1072318, 1458993, 2617393, 4161343, 7605593, 8632704, 14214953, 22690938, 24886878, 40268266, 75544661, 106544643, 236451678
Offset: 1

Views

Author

M. F. Hasler, Jan 06 2016

Keywords

Crossrefs

Goldbach conjecture related: see A266951 (the record values) and A266948 for further information.

Extensions

a(28)-a(37) from Pontus von Brömssen, Jan 06 2024

A266951 Record values in A266948: least prime p such that p-2 and 6n-p are also prime.

Original entry on oeis.org

0, 5, 7, 13, 31, 109, 139, 199, 241, 283, 601, 619, 643, 823, 859, 883, 1021, 1051, 1093, 1291, 1621, 1873, 1879, 1951, 2311, 3001, 3301, 3361, 3529, 3583, 3919, 3931, 4339, 5419, 5479, 5851, 6361
Offset: 1

Views

Author

M. F. Hasler, Jan 06 2016

Keywords

Comments

Goldbach conjecture related. The values of A266948(n) range theoretically between 2 and n. The smaller they are, the easier it was to find a suitable twin prime, and thus the more likely the conjecture "A266948(n) > 0 for all n > 1" holds. These record values vs. the respective indices A266950 yield an estimate for the growth of an upper bound.

Crossrefs

Cf. A266948, A266950.

Extensions

a(28)-a(37) from Pontus von Brömssen, Jan 06 2024

A266952 Least prime p such that p-2 and 6n-p and 6n+2-p are also prime, or 0 if no such prime exists.

Original entry on oeis.org

0, 0, 7, 7, 7, 13, 7, 13, 7, 13, 19, 7, 13, 7, 13, 19, 0, 31, 7, 7, 13, 19, 31, 31, 7, 13, 7, 13, 19, 73, 31, 7, 13, 7, 7, 13, 19, 31, 31, 7, 13, 7, 13, 19, 73, 31, 7, 13, 7, 13, 19, 109, 31, 7, 13, 19, 109, 31, 109, 7, 13, 19, 61, 31, 73, 43, 199, 0, 61, 103, 73, 7, 13, 7, 13, 19, 109, 31, 7, 13, 19, 139, 31, 151, 43, 199, 0, 61, 7, 13, 19, 199, 31, 139, 43
Offset: 0

Views

Author

M. F. Hasler, Jan 06 2016

Keywords

Comments

If a(n) > 0, then the triple {6n-2, 6n, 6n+2} of consecutive even numbers allows a "simultaneous Goldbach decomposition" using two pairs of twin primes, 6n-2 = p-2 + 6n-p ; 6n = p + 6n-p ; 6n+2 = p + 6n+2-p.
Up to 10^5, the only indices for which a(n)=0 are {0, 1, 16, 67, 86, 131, 151, 186, 191, 211, 226, 541, 701}. I conjecture that this list is finite, and probably complete. Is it a coincidence that all odd numbers > 1 in this list are primes? (See also A144094.)
This seems equivalent to a conjecture Zwillinger made in 1978, see reference in LINKS.
See A266953 for another variant with a slightly relaxed condition (instead of 6n+2-p one can also have 6n+4-p prime, but this affects only n=2 and n=67), and A266948 for another variant with less restrictive conditions (only p-2 and 6n-p have to be prime).

Links

Crossrefs

Cf. A007534, A266948, A266953.

Programs

  • PARI
    A266952(n)=my(GP(n, p=2)=forprime(p=p, n+1, isprime(n*2-p)&&return(p))); for(p=1, 3*n, isprime(-2+p=GP(3*n, p))+!p&&(!p||isprime(6*n+2-p))&&return(p))

A266953 Least prime p such that p-2 and 6n-p and either 6n+2-p or 6n+4-p is also prime, or 0 if no such prime exists.

Original entry on oeis.org

0, 0, 5, 5, 5, 13, 7, 5, 5, 13, 19, 7, 5, 7, 5, 19, 0, 5, 5, 5, 13, 19, 5, 31, 7, 13, 7, 13, 5, 73, 31, 7, 13, 5, 7, 13, 19, 31, 5, 5, 13, 7, 13, 19, 73, 31, 7, 5, 7, 13, 19, 109, 5, 5, 13, 19, 109, 31, 109, 5, 13, 19, 61, 31, 5, 43, 199, 5, 61, 103, 73, 7, 13, 7, 5, 19, 109, 5, 5, 13, 19, 139, 5, 151, 5, 199, 0, 61, 7, 13, 19, 199, 31, 139, 43, 109, 7, 13, 19
Offset: 0

Views

Author

M. F. Hasler, Jan 06 2016

Keywords

Comments

If a(n) > 0, then the triple {6n-2, 6n, 6n+2} of consecutive even numbers allows a "simultaneous Goldbach decomposition" using only 4 different primes, 6n-2 = p-2 + 6n-p ; 6n = p + 6n-p ; 6n+2 = p + 6n+2-p = p-2 + 6n+4-p.
See A266952 for the version which does not allow the second decomposition of the last member. See A266948 for a variant which does not require 6n+2-p to be prime.
Up to 10^5, the only indices for which a(n)=0 are {0, 1, 16, 86, 131, 151, 186, 191, 211, 226, 541, 701}. (Only 2 and 67 require the alternative primality of 6n+4-p and have thus A266952(n)=0.) I conjecture that this list is finite, and probably complete. Is it a coincidence that all odd numbers in this list are primes?

Programs

  • PARI
    A266953(n)=my(GP(n, p=2)=forprime(p=p,n+1,isprime(n*2-p)&&return(p))); for(p=1,3*n,isprime(-2+p=GP(3*n, p))+!p&&(!p||isprime(6*n+2-p)||isprime(6*n+4-p))&&return(p))

A144094 Prime numbers r such that there is no pair of twin primes p,p+2 and q,q+2 such that p+q+2 = 6r.

Original entry on oeis.org

67, 131, 151, 191, 211, 541, 701
Offset: 1

Views

Author

Pierre CAMI, Sep 10 2008

Keywords

Comments

Probably the sequence is complete.
More generally, {1, 2, 16, 66, 67, 86, 116, 131, 151, 186, 191, 211, 226, 541, 701} seem to be the only numbers such that {6n-2, 6n, 6n+2} do not have a Goldbach type of decomposition (sum of two primes) using only two pairs of twin primes. See also A266952, A266953, and A266948 and A007534. - M. F. Hasler, Jan 07 2016

Examples

			6*3=18, 5,7 twin primes as 11,13, 5+13=7+11=18 so 3 not in the sequence 6*5=30, 11,13 twin primes as 17,19, 13+19=11+17=30 so 5 not in the sequence
Showing 1-5 of 5 results.

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