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 1-4 of 4 results.

A096787 Primes of form 4n+3 that are the sum of two consecutive composite numbers.

Original entry on oeis.org

19, 31, 43, 67, 71, 79, 103, 127, 131, 139, 151, 163, 191, 199, 211, 223, 239, 251, 271, 283, 307, 311, 331, 367, 379, 419, 431, 439, 443, 463, 487, 491, 499, 523, 547, 571, 599, 607, 619, 631, 643, 647, 659, 683, 691, 727, 739, 743, 751, 787, 811, 823, 827
Offset: 1

Views

Author

Lekraj Beedassy, Jul 09 2004

Keywords

Links

Crossrefs

Subsequence of A060254. See A096788 for values 2n+1. See A096676 for n values.
Cf. A060254, A096784, A096785, A096786, A096788, A096676, A096679.

Programs

  • Mathematica
    2Select[ Range[ 450], PrimeQ[ # ] == PrimeQ[ # + 1] == False && PrimeQ[2# + 1, GaussianIntegers -> True] == True &] + 1 (* Robert G. Wilson v, Jul 11 2004 *)
Select[Total/@Partition[Select[Range[500],CompositeQ],2,1],PrimeQ[#] && IntegerQ[ (#-3)/4]&] (* Harvey P. Dale, Mar 06 2019 *)
  • PARI
    nextcomposite(k)=if(k<3,4,if(isprime(k),k+1,k));
{m=440;n=4;while(nKlaus Brockhaus, Jul 10 2004

Formula

Equals 1+2*A096788.

Extensions

Corrected and extended by Klaus Brockhaus and Ray Chandler, Jul 10 2004

A096788 Numbers m such that both m and m+1 are composite numbers whose sum is a prime of the form 4k+3.

Original entry on oeis.org

9, 15, 21, 33, 35, 39, 51, 63, 65, 69, 75, 81, 95, 99, 105, 111, 119, 125, 135, 141, 153, 155, 165, 183, 189, 209, 215, 219, 221, 231, 243, 245, 249, 261, 273, 285, 299, 303, 309, 315, 321, 323, 329, 341, 345, 363, 369, 371, 375, 393, 405, 411, 413, 429, 441
Offset: 1

Views

Author

Lekraj Beedassy, Jul 09 2004

Keywords

Comments

Odd composite numbers c such that 2*c + 1 is prime. - Alexandre Herrera, Jul 07 2023

Crossrefs

Subsequence (odd numbers) of A096784. See A096787 for the associated primes.
Cf. A060254, A096784, A096785, A096786, A096787, A096676.

Programs

  • Mathematica
    Select[ Range[ 450], PrimeQ[ # ] == PrimeQ[ # + 1] == False && PrimeQ[2# + 1, GaussianIntegers -> True] == True &] (* Robert G. Wilson v, Jul 11 2004 *)
  • PARI
    nextcomposite(k)=if(k<3,4,if(isprime(k),k+1,k));
{m=455;n=4;while(nKlaus Brockhaus, Jul 10 2004

Formula

Equals (A096787 - 1)/2.

Extensions

Corrected and extended by Klaus Brockhaus and Ray Chandler, Jul 10 2000
Incorrect comment about Gaussian primes deleted by N. J. A. Sloane, Mar 02 2011

A144313 Prime numbers p such that p - 1 is the fourth a-figurate number, seventh b-figurate number and possibly tenth c-figurate number for some a, b and c and not a d-figurate number for any nontrivial d.

Original entry on oeis.org

29, 71, 113, 239, 281, 449, 491, 659, 701, 827, 911, 953, 1373, 1499, 1583, 1667, 1709, 1877, 2003, 2087, 2129, 2213, 2339, 2423, 2549, 2591, 2633, 2801, 2843, 2969, 3221, 3347, 3389, 3557, 3767, 3851, 4229, 4271, 4397, 4481, 4649, 4691, 4733, 5153, 5279
Offset: 1

Views

Author

Reikku Kulon, Sep 17 2008

Keywords

Comments

Appears to be necessarily a subset of A007528.
The 46th Mersenne prime exponent (Mpe, A000043) 43112609 is a member: 43112608 is the fourth 7185436-figurate number and the seventh 2052983-figurate number and is not a k-figurate number for any other k except 43112608 (trivially). Several other Mersenne prime exponents are members of this sequence.
It is conjectured:
- that this sequence is infinite;
- that there is a unique set {4, 7, 10, 16, ...} (A138694?) giving the possible orders in k-figurate numbers for the set S of all Mpe for which Mpe - 1 is a (4, 7) or (4, 10) k-figurate number;
- that the ratio of Mpe in this sequence to those not approaches a nonzero value;
- that a characteristic function f(n) exists which equals 1 iff n is in S.
Contribution from Reikku Kulon, Sep 18 2008: (Start)
Subset of the integers n such that n is congruent to 29 modulo 42. The case where p - 1 is a tenth c-figurate number occurs when p is also congruent to 281 modulo 630.
The first three primes where c is defined are 281, 911 and 2801, with c = 8, 22, 64; c is congruent to 8 modulo 14. All such primes are necessarily congruent to 1 modulo 10.
The first invalid values of c are 36 and 50, which correspond to the semiprimes 1541 = 23 * 67 and 2171 = 13 * 167. Both of these are members of A071331 and A098237. The next invalid value of c, 78, corresponds to 3431 = 47 * 73, once again a member of both sequences.
The first primes where a, b, c and d are all defined (which therefore excludes them from this sequence) are the consecutive 6581, 7211 and 7841, all members of A140856, A140732, A142076, A142317 and A142905. (End)

Crossrefs

Cf. A000040, A007528, A138694, A131500, A096676, A000043, A000668
Contribution from Reikku Kulon, Sep 18 2008: (Start)
Cf. A071331, A098237 (semiprimes)
Cf. A140856, A140732, A142076, A142317, A142905 (a, b, c and d all defined) (End)

A144327 Prime numbers p such that p - 1 is the fourth a-figurate number and nineteenth b-figurate number for some a and b.

Original entry on oeis.org

191, 1217, 1559, 1901, 2243, 2927, 4637, 6689, 8741, 9767, 12161, 12503, 13187, 14897, 15581, 15923, 16607, 17291, 19001, 20369, 21737, 22079, 23447, 23789, 24473, 25841, 26183, 27551, 27893, 30971, 33023, 35759, 37811, 38153, 39521
Offset: 1

Views

Author

Reikku Kulon, Sep 17 2008

Keywords

Comments

Sequence of primes sufficient to determine the conjecture of A144326. Presumed to include many terms that cannot be Mersenne prime exponents, noting A096676: either infinitely many, or all.
These primes are very rare. There are only 10 less than 10000 and only 85 less than 100000.

Crossrefs

Cf. A096676, A144326
Showing 1-4 of 4 results.

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