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.

A086220 Numbers n such that p=n^2+2, p+2, p+6 and p+8 are four consecutive primes.

Original entry on oeis.org

3, 57, 32397, 54813, 61827, 62493, 98487, 98853, 119937, 213237, 254577, 306123, 322263, 328803, 438453, 603603, 619263, 656733, 671013, 675807, 821247, 875277, 1051173, 1121133, 1163697, 1230783, 1234317, 1337763, 1382223, 1388103, 1455927, 1538517, 1581237
Offset: 1

Views

Author

Zak Seidov, Sep 08 2003

Keywords

Links

Crossrefs

Cf. A086380, A086381, A086509.

Programs

  • PARI
    is(n)={my(p=n^2+2); isprime(p) && isprime(p+2) && isprime(p+6) && isprime(p+8)} \\ Andrew Howroyd, Jan 05 2020

Extensions

Terms a(15) and beyond from Andrew Howroyd, Jan 05 2020

A086509 Numbers n such that p=n^2+2, p+2, p+6, p+8 and p+12 are five consecutive primes.

Original entry on oeis.org

3, 32397, 213237, 254577, 1587597, 2305167, 3440307, 5622903, 6067893, 6895953, 7424157, 8304927, 8917707, 8936367
Offset: 1

Views

Author

Zak Seidov, Sep 09 2003

Keywords

Links

Crossrefs

Cf. A086220, A086380, A086381.

Extensions

John F. Brennen gives first 19575 terms of this sequence, n <= 165294372813.

A177173 Numbers n such that n^2 + 3^k is prime for k = 1, 2, 3.

Original entry on oeis.org

2, 10, 38, 52, 350, 542, 1102, 1460, 1522, 1732, 2510, 2642, 2768, 3692, 4592, 4658, 4690, 7238, 8180, 8320, 8960, 11392, 13468, 14920, 15908, 16600, 16832, 17878, 18820, 19100, 21532, 22060, 23240, 23842, 23968, 24622, 26428, 26638, 27170
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 04 2010

Keywords

Comments

p = n^2 + 3, q = n^2 + 3^2 = p+6, r = n^2 + 3^3 = p+18 to be primes.
Trivially n is not a multiple of 3 and necessarily LSD of such n is e = 0, 2 or 8 as k^2+3^2 is a multiple of 5 for k = 4 or 6.
Note n^2 + m^k prime (k = 1, 2, 3) in case of m = 2 is (n^2+2,n^2+2^2,n^2+2^3) = (p,p+2,p+6): i.e., a "near square" prime triple of the first kind.
Case k=2: q is also a Pythagorean prime (A002144)
n = 350: first case where p = 122503 = prime(i), q and r are consecutive primes (i = 122503), sod(p) = sod(i) = 13, a so-called Honaker prime.
p = prime(i), q, r consecutive primes, (n,i): (350,11524) (542,25517) (1460,157987) (3692,887608) (4592,1335102) (4690,1389018).

Examples

			2^2 + 3 = 7 = prime(4), 2^2 + 3^2 = 13 = prime(6), 2^2 + 3^3 = 31 = prime(11), 2 is first term.
10^2 + 3 = 103 = prime(27), 10^2 + 3^2 = 109 = prime(29), 10^2 + 3^3 = 127 = prime(31), 10 is 2nd term.
Curiously k=0: 10^2 + 3^0 = 101 = prime(26), k=4: 10^2 + 3^4 = 181 = prime(42), necessarily LSD for such n is e = 0, k= 5: 10^2 + 3^5 = 7^3, k=6: 10^2 + 3^6 = 829 = prime(145), 10^2 + 3^7 = 2287 = prime(340), 10^2 + 3^8 = 6661 = prime(859)
n = 8180, primes for exponents k = 0, 1, 2, 3 and 4: p=66912403=prime(3946899), q=66912409=prime(3946900), r=66912427=prime(3946902), n^2+3^0=66912401=prime(3946898) and n^2+3^4=66912481=prime(3946905).
n = 8960, primes for exponents k = 1, 2, 3, 4, 5 and 6: p=80281603=prime(4684862), q=80281609=prime(4684863), r=80281627=prime(4684865), n^2+3^4=80281681=prime(4684868), n^2+3^5=80281843=prime(4684877), n^2+3^5=80282329=prime(4684904).

References

  • F. Padberg, Zahlentheorie und Arithmetik, Spektrum Akademie Verlag, Heidelberg-Berlin 1999.
  • M. du Sautoy, Die Musik der Primzahlen: Auf den Spuren des groessten Raetsels der Mathematik, Deutscher Taschenbuch Verlag, 2006.

Links

  • C. Hooley On nonary cubic forms, Journal für die reine und angewandte Mathematik 386, pp. 32-98, 1988.

Crossrefs

Cf. A002144, A049422, A055394, A086380, A114272.

Programs

Extensions

More terms from R. J. Mathar, Nov 01 2010

A178639 Numbers m such that all three values m^2 + 13^k, k = 1, 2, 3, are prime.

Original entry on oeis.org

10, 12, 200, 268, 340, 418, 488, 530, 838, 840, 1102, 1720, 1830, 2240, 2410, 2768, 3148, 3202, 3318, 3322, 3958, 4162, 4610, 5080, 5672, 5700, 5722, 5870, 6178, 6302, 6480, 7490, 8130, 8262, 8888, 9132, 9602, 9618, 10638
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 31 2010

Keywords

Comments

Subsequence of A176969.
The least-significant digit of all terms is one of 0, 2 or 8, because for odd digits m^2 + 13^k would be even (not prime), and for digits 4 and 6 the number m^2 + 13^2 is a multiple of 5.

Examples

			m=10 is in the sequence because 10^2 + 13 = 113 = prime(30), 10^2 + 13^2 = 269 = prime(57), 10^2 + 13^3 = 2297 = prime(342).
m=8888 is in the sequence because 8888^2 + 13 = 78996557 = prime(4614261), 8888^2 + 13^2 = 78996713 = prime(4614269), 8888^2 + 13^3 = 78998741 = prime(4614379).
m=6480 yields a prime 6480^2 + 13^k even for k=0.
m=7490 yields a prime 7490^2 + 13^k even for k=0 and k=4.

References

  • B. Bunch: The Kingdom of Infinite Number: A Field Guide, W. H. Freeman, 2001.
  • R. Courant, H. Robbins: What Is Mathematics? An Elementary Approach to Ideas and Methods, Oxford University Press, 1996.
  • G. H. Hardy, E. M. Wright, E. M., An Introduction to the Theory of Numbers (5th edition), Oxford University Press, 1980.

Crossrefs

Cf. A000040, A000290, A055394, A056899, A086380, A113536, A176371, A176969.

Extensions

keyword:base removed by R. J. Mathar, Jul 13 2010
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