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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A253073 Lexicographically earliest sequence of distinct numbers such that neither a(n) nor a(n-1)+a(n) is prime.

Original entry on oeis.org

0, 1, 8, 4, 6, 9, 12, 10, 14, 16, 18, 15, 20, 22, 24, 21, 25, 26, 28, 27, 30, 32, 33, 35, 34, 36, 38, 39, 42, 40, 44, 46, 45, 48, 50, 49, 51, 54, 52, 56, 55, 57, 58, 60, 62, 63, 65, 64, 66, 68, 70, 72, 69, 74, 76, 77, 75, 78, 80, 81, 84, 82, 86, 85, 87, 88
Offset: 1

Views

Author

N. J. A. Sloane, Feb 01 2015, based on a suggestion from Patrick Devlin

Keywords

Comments

Conjecture: this is a permutation of the nonprimes. [Proof outline given below by Semeon Artamonov and Pat Devlin.]
Let x be a number that's missing.
Then eventually every term must be of the form PRIME - x. (Otherwise, x would appear as that next term.)
In particular, this means there are only finitely many multiples of x that appear in the sequence. To make this cleaner, let Y be a multiple of x larger than all multiples of x appearing in the sequence.
Let q be a prime not dividing Y. Then since none of the terms Y, 2Y, 3Y, ..., 2qY appear, it must be that, eventually, every term in the sequence is of the form PRIME - Y and also of the form PRIME - 2Y and also of the form PRIME - 3Y, ... and also of the form PRIME - 2qY.
That means we have a prime p and a number Y such that p, p+Y, p+2Y, p + 3Y, p+4Y, ..., p+2qY are all prime. But take this sequence mod q. Since q does not divide Y, the terms 0, Y, ..., 2qY cover every residue class mod q twice. Therefore, p + kY covers each residue class mod q twice. Consequently, there are two terms congruent to 0 mod q. One can be q, but the other must be a multiple of it (contradicting its primality).

Links

Crossrefs

Cf. A254337, A002808, A253074.
Cf. A018252, A010051.

Programs

  • Haskell
    a253073 n = a253073_list !! (n-1)
a253073_list = 0 : f 0 a018252_list where
   f u vs = g vs where
     g (w:ws) | a010051' (u + w) == 1 = g ws
              | otherwise = w : f w (delete w vs)
-- Reinhard Zumkeller, Feb 02 2015

A153137 Smallest sequence of noncomposite numbers such that no sum of at least two terms is prime.

Original entry on oeis.org

1, 3, 5, 113, 181, 661, 10891, 927149, 88070399, 15288362671
Offset: 1

Views

Author

Benoit Jubin, Dec 19 2008

Keywords

Crossrefs

Cf. A153136, A153138, A133660, A052349, A254337, A254341.

Programs

  • PARI
    print1(1); a=[1]; for(n=1, 10, forprime(p=vecmin(a)+1, , setsearch(a, p)&&next; for(i=1, 2^#a-1, isprime(normlp(vecextract(a, i), 1)+p)&&next(2)); a=concat(a, p); print1(","p))) \\ Very simplistic, should at least avoid an odd number of terms in the partial sum of earlier terms. \\ M. F. Hasler, Jan 29 2015

Extensions

a(8)-a(10) from Donovan Johnson, Dec 23 2008

A153138 Smallest sequence of odd primes such that no sum of at least two terms is prime.

Original entry on oeis.org

3, 5, 7, 83, 317, 383, 29567, 423509, 118661483, 52542428123
Offset: 1

Views

Author

Benoit Jubin, Dec 19 2008

Keywords

Crossrefs

Cf. A153136, A153137, A133660, A052349, A254337, A254341.

Programs

  • PARI
    a=[]; for(n=1, 10, forprime(p=if(a,a[#a]+2,3), , setsearch(a, p)&&next; for(i=1, 2^#a-1, isprime(normlp(vecextract(a, i), 1)+p)&&next(2)); a=concat(a, p); print1(p", "))) \\ Very simplistic, should at least avoid an odd number of terms in the partial sum of earlier terms. \\ M. F. Hasler, Jan 29 2015

Extensions

a(8)-a(10) from Donovan Johnson, Dec 23 2008
Previous Showing 11-13 of 13 results.

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

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