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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A023303 Primes that remain prime through 4 iterations of function f(x) = 2x + 3.

Original entry on oeis.org

47, 67, 97, 137, 307, 1427, 2857, 6047, 6997, 9377, 12097, 16057, 24197, 32117, 35117, 36877, 44507, 46687, 54517, 55487, 64877, 71327, 71807, 76537, 89017, 92387, 94427, 100057, 132707, 142057, 153077, 160207, 184777, 186647, 194027, 200117, 205237
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Primes p such that 2*p+3, 4*p+9, 8*p+21 and 16*p+45 are also primes. - Vincenzo Librandi, Aug 04 2010

Links

Crossrefs

Subsequence of A023204.

Programs

  • Magma
    [ p: p in PrimesUpTo(300000) | IsPrime(p) and IsPrime(2*p+3) and IsPrime(4*p+9) and IsPrime(8*p+21) and IsPrime(16*p+45)] // Vincenzo Librandi, Aug 04 2010
  • Mathematica
    Select[Prime[Range[20000]],And@@PrimeQ[NestList[2#+3&,#,4]]&] (* Harvey P. Dale, Aug 12 2012 *)

A171517 Primes p such that 2*p+11 is prime.

Original entry on oeis.org

3, 13, 31, 43, 73, 109, 151, 163, 181, 193, 199, 211, 223, 283, 331, 349, 373, 379, 409, 421, 433, 463, 499, 541, 571, 601, 613, 619, 643, 709, 739, 769, 823, 829, 883, 991, 1009, 1021, 1039, 1051, 1063, 1129, 1213, 1231, 1291, 1303, 1423, 1453, 1471, 1549
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 10 2009

Keywords

Examples

			2*3+11=17, which is prime.

Links

Crossrefs

Cf. A000040, A005382, A005384, A023204 - A023207, A063908 - A063912.

Programs

  • Magma
    [p: p in PrimesUpTo(1600) | IsPrime(2*p+11)]; // Vincenzo Librandi, Apr 27 2014
  • Mathematica
    Select[Prime[Range[6! ]],PrimeQ[2*#+11]&]

A089527 p(k) such that 2*p(k)+3 and 2*p(k+1) + 3 are consecutive primes, where p(i) denotes the i-th prime.

Original entry on oeis.org

5, 17, 43, 137, 193, 197, 223, 227, 349, 379, 463, 1033, 1109, 1303, 1427, 1559, 1607, 1609, 3163, 3613, 3767, 3823, 4289, 4337, 4637, 4969, 5657, 5839, 6199, 6709, 6733, 7823, 8363, 8699, 8747, 8863, 9127, 9643, 9743, 9967, 10253, 10459, 10979, 11593
Offset: 1

Views

Author

Ray Chandler, Nov 07 2003

Keywords

Examples

			p(3)=5, 2*5 + 3 = 13 = p(6);
p(4)=7, 2*7 + 3 = 17 = p(7).

Crossrefs

Subsequence of A023204.
Cf. A089526, A089528, A089529.

Formula

a(n) = A000040(A089526(n)).

Extensions

Offset changed to 1 by Jinyuan Wang, Aug 04 2021

A089532 A089531 indexed by A000040.

Original entry on oeis.org

4, 6, 7, 10, 12, 13, 18, 24, 25, 29, 33, 35, 42, 45, 50, 55, 59, 60, 66, 68, 70, 77, 78, 79, 87, 88, 89, 100, 102, 104, 113, 123, 126, 127, 135, 136, 139, 142, 152, 158, 159, 165, 169, 172, 176, 184, 187, 189, 197, 199, 201, 203, 209, 211, 216, 234, 237, 244, 251
Offset: 1

Views

Author

Ray Chandler, Nov 07 2003

Keywords

Crossrefs

Cf. A023204, A089530, A089531.

Formula

a(n)=k such that A089531(n)=A000040(k).

A163769 Primes p such that 2*p+3 is not prime.

Original entry on oeis.org

3, 11, 23, 31, 37, 41, 59, 61, 71, 79, 83, 101, 103, 107, 109, 131, 149, 151, 163, 179, 181, 191, 211, 233, 239, 241, 251, 257, 263, 271, 281, 293, 311, 313, 317, 331, 347, 359, 367, 373, 389, 401, 419, 421, 431, 433, 443, 449, 457, 461, 479, 491, 499, 521
Offset: 1

Views

Author

Vincenzo Librandi, Aug 04 2009

Keywords

Comments

All those p appear in A144562. [Proof: since 2p+3 is odd and not prime, it can be written as a product of two odd numbers, 2p+3=(2k+1)*(2s+1), therefore p=2ks+k+s-1. - R. J. Mathar, Aug 06 2009]

Examples

			3 is in the sequence because 2*3+3=9 is composite; 23 is in the sequence because 2*23+3=49 is composite.

Links

Crossrefs

Cf. A144562.

Programs

  • Magma
    [p: p in PrimesUpTo(700) | not IsPrime(2*p+3)]; // Vincenzo Librandi, Apr 08 2013
  • Mathematica
    Select[Prime[Range[200]],!PrimeQ[2#+3]&] (* Harvey P. Dale, Feb 02 2012 *)

Formula

A153238 INTERSECT A000040. - R. J. Mathar, Aug 05 2009
A000040 \ A023204. - R. J. Mathar, Aug 05 2009

Extensions

Entries checked - R. J. Mathar, Aug 06 2009

A230117 Primes p such that 2*p+1 is prime and 2*p+3 is not prime.

Original entry on oeis.org

3, 11, 23, 41, 83, 131, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 593, 641, 653, 683, 719, 761, 911, 953, 1019, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1439, 1451, 1481, 1511, 1601, 1811, 1889, 1901, 1931, 1973, 2003, 2039, 2069, 2141
Offset: 1

Views

Author

Vincenzo Librandi, Oct 10 2013

Keywords

Comments

Intersection of A005384 and A163769. - Felix Fröhlich, Jan 14 2017

Examples

			23 is in the sequence because 2*23+1=47 (prime) and 2*23+3=49 (not prime).

Links

Crossrefs

Cf. A005384, A023204, A053176, A126107, A163769, A230039.

Programs

  • Magma
    [p: p in PrimesUpTo(2500)| IsPrime(2*p+1) and not IsPrime(2*p+3)];
  • Mathematica
    Select[Range[10^6],PrimeQ[#]&& PrimeQ[2#+1]&&!PrimeQ[2#+3]&]
  • PARI
    is(n) = ispseudoprime(n) && ispseudoprime(2*n+1) && !ispseudoprime(2*n+3) \\ Felix Fröhlich, Jan 14 2017

A106067 Primes p such that 3*p + 2 and 2*p + 3 are primes.

Original entry on oeis.org

5, 7, 13, 17, 19, 29, 43, 89, 97, 127, 139, 167, 173, 197, 199, 227, 269, 337, 349, 353, 383, 397, 409, 439, 503, 523, 607, 643, 659, 797, 859, 887, 929, 1013, 1039, 1063, 1069, 1109, 1153, 1193, 1259, 1277, 1303, 1307, 1427, 1429, 1483, 1559, 1567, 1583
Offset: 1

Views

Author

Zak Seidov, May 07 2005

Keywords

Links

Crossrefs

Intersection of A023204 and A023208.

Programs

  • Magma
    [p: p in PrimesUpTo(10000)| IsPrime(3*p+2) and IsPrime(2*p+3)]; // Vincenzo Librandi, Nov 13 2010
  • Mathematica
    Select[Prime[Range[20000]], PrimeQ[2#+3]&&PrimeQ[3#+2]&]
  • PARI
    is(p) = isprime(p) && isprime(3*p+2) && isprime(2*p+3); \\ Amiram Eldar, Nov 08 2024

A230039 Primes p such that 2*p+1 is not prime and 2*p+3 is prime.

Original entry on oeis.org

7, 13, 17, 19, 43, 47, 67, 73, 97, 127, 137, 139, 157, 167, 193, 197, 199, 223, 227, 229, 269, 277, 283, 307, 337, 349, 353, 379, 383, 397, 409, 439, 463, 467, 487, 503, 523, 547, 557, 563, 599, 607, 613, 617, 643, 647, 739, 773, 797, 827, 853, 859, 887, 929
Offset: 1

Views

Author

Vincenzo Librandi, Oct 10 2013

Keywords

Comments

Intersection of A023204 and A053176. - Felix Fröhlich, Jan 14 2017

Examples

			43 is in the sequence because 2*43+1=87 (not prime) and 2*43+3=89 (prime).

Links

Crossrefs

Cf. A005384, A023204, A053176, A126107, A163769, A230117.

Programs

  • Magma
    [p: p in PrimesUpTo(2500)|not IsPrime(2*p+1) and IsPrime(2*p+3)];
  • Mathematica
    Select[Range[10^5],PrimeQ[#]&& !PrimeQ[2#+1]&& PrimeQ[2#+3]&]
  • PARI
    is(n) = ispseudoprime(n) && !ispseudoprime(2*n+1) && ispseudoprime(2*n+3) \\ Felix Fröhlich, Jan 14 2017

A261810 n and (2*n^2 + 2*n - 1) are primes.

Original entry on oeis.org

2, 3, 5, 11, 23, 59, 71, 113, 131, 137, 149, 179, 227, 257, 263, 269, 293, 317, 347, 353, 401, 419, 443, 449, 467, 557, 653, 659, 677, 683, 743, 773, 809, 839, 857, 881, 911, 929, 947, 977, 1019, 1049, 1277, 1301, 1319, 1433, 1571, 1697, 1847, 1871, 1901, 1913
Offset: 1

Views

Author

Jaroslav Krizek, Sep 01 2015

Keywords

Comments

Primes p such that (number of divisors of p * sum of divisors of p * product of divisors of p - 1) is also a prime.
Primes p such that (A000005(p) * A000203(p) * A007955(p) - 1) is also a prime.
See similar sequences of type primes p such that x is also a prime for some x wherein tau(p) = A000005(p) = number of divisors of p, sigma(p) = A000203(p) = sum of divisors of p and pod(p) = A007955(p) = product of divisors of p:
A001359 (for x = tau(p) + sigma(p) - 1 and x = tau(p) + pod(p)),
A005382 (for x = tau(p) * pod(p) - 1),
A005384 (for x = sigma(p) + pod(p), x = tau(p) * sigma(p) - 1 and x = tau(p) * pod(p) + 1),
A023200 (for x = tau(p) + sigma(p) + 1),
A023204 (for x = tau(p) + sigma(p) + pod(p) and x = tau(p) * sigma(p) + 1),
A053182 (for x = sigma(p) * pod(p) + 1),
A053184 (for x = sigma(p) * pod(p) - 1),
A158526 (for x = tau(p) * sigma(p) * pod(p) + 1).
For n >= 3, a(n) == 5 mod 6. - Robert Israel, Sep 02 2015

Examples

			3 and 2*3^2 + 2*3 - 1 = 23 are primes.

Links

Crossrefs

Cf. A000005, A000203, A007955.
Cf. A001359, A005382, A005384, A023200, A023204, A053182, A053184, A158526.

Programs

  • Magma
    [n: n in[1..10000] | IsPrime(n) and IsPrime(2*n*n + 2*n - 1)];
  • Maple
    select(t -> isprime(t) and isprime(2*t^2 + 2*t-1), [2,3,seq(6*i-1,i=1..1000)]); # Robert Israel, Sep 02 2015
  • Mathematica
    Select[Prime[Range[300]], PrimeQ[2 #^2 + 2 # - 1] &] (* Vincenzo Librandi, Sep 02 2015 *)
  • PARI
    is(n)=isprime(n)&&isprime(2*n^2 + 2*n - 1) \\ Anders Hellström, Sep 01 2015

A015827 Numbers k such that phi(k + 9) | sigma(k).

Original entry on oeis.org

3, 5, 7, 10, 15, 21, 24, 30, 31, 33, 42, 47, 57, 69, 78, 79, 93, 102, 114, 127, 129, 135, 145, 161, 174, 177, 186, 210, 213, 216, 223, 231, 237, 238, 239, 249, 258, 264, 270, 282, 297, 309, 318, 355, 367, 371, 376, 393, 399, 402, 417, 418, 435, 438, 455, 456
Offset: 1

Views

Author

Robert G. Wilson v

Keywords

Comments

Includes 6*A023204. Thus Dickson's conjecture implies the sequence is infinite. - Robert Israel, Jan 10 2019

Links

Crossrefs

Cf. A015849, A023204.

Programs

  • Maple
    filter:= n -> (numtheory:-sigma(n)/numtheory:-phi(n+9))::integer:
select(filter, [$1..1000]); # Robert Israel, Jan 10 2019
  • Mathematica
    Select[Range[1000], Divisible[DivisorSigma[1, #], EulerPhi[9 + #]] &] (* David Nacin, Mar 01 2012 *)
  • PARI
    is(n)=sigma(n)%(eulerphi(n)+9)==0 \\ Charles R Greathouse IV, Sep 25 2012
Previous Showing 11-20 of 31 results. Next

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