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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A109132 Numbers n such that n! + 1 is a Chen prime. n! + 1 is then called a factorial Chen prime.

Original entry on oeis.org

0, 1, 2, 3, 11, 77
Offset: 1

Views

Author

Jason Earls, Aug 17 2005

Keywords

Comments

Conjecture: 77! + 1 is the largest factorial Chen prime.
For n>2, n is in the sequence iff both numbers n!/3+1 and n!+1 are primes. - Farideh Firoozbakht, Mar 24 2006

Crossrefs

Cf. A002981, A109611.

A109346 Chen primes p such that p + 2 is a brilliant number.

Original entry on oeis.org

2, 7, 13, 19, 23, 47, 167, 251, 317, 359, 389, 401, 449, 479, 491, 557, 587, 647, 701, 797, 839, 911, 941, 947, 977, 1187, 1217, 1259, 1361, 1367, 1409, 1499, 1511, 1709, 1847, 1889, 1907, 2207, 2417, 2447, 2477, 2579, 2621, 2699, 2879, 2909, 3137, 3719
Offset: 1

Views

Author

Jason Earls, Aug 21 2005

Keywords

Crossrefs

Cf. A109611, A078972.

A109364 Least positive k such that 2^n + k is a Chen prime and 2^n + k + 2 is a brilliant number.

Original entry on oeis.org

5, 3, 5, 3, 15, 103, 39, 61, 45, 163, 159, 301, 2415, 403, 339, 331, 195, 553, 933, 24493, 1359, 85, 3999, 253, 267, 7567, 12969, 2815, 4419, 2047, 3171, 8203, 14019, 5665, 951, 5365, 5949, 6565, 7653, 77785, 3759, 5833, 1311, 4771, 18807, 7303, 25431
Offset: 1

Views

Author

Jason Earls, Aug 23 2005

Keywords

Crossrefs

Cf. A109611, A078972.

A110060 Least positive k such that n^n + k is a Chen prime.

Original entry on oeis.org

1, 1, 2, 1, 12, 23, 4, 73, 100, 19, 450, 281, 114, 241, 56, 51, 582, 527, 412, 771, 230, 997, 440, 95, 1882, 1117, 224, 1213, 1968, 29, 168, 421, 104, 2383, 876, 1085, 5776, 253, 742, 4909, 402, 2045, 3414, 403, 2366, 705, 2124, 6455, 5662, 2329, 1568, 20101
Offset: 1

Views

Author

Jason Earls, Sep 04 2005

Keywords

Crossrefs

Cf. A109611.

Programs

  • Mathematica
    (* After loading the NumberTheory package *) SemiPrimeQ[n_] := (f = FactorIntegerECM[n]; PrimeQ[f] && PrimeQ[n/f]); ShenPrimeQ[n_] := (PrimeQ[n] && (PrimeQ[n + 2] || SemiPrimeQ[n + 2])); Do[k = 1; While[ !ShenPrimeQ[n^n + k], k++ ]; Print[k], {n, 100}] (* Ryan Propper, Sep 27 2006 *)

Extensions

More terms from Ryan Propper, Sep 27 2006

A110905 a(n) is the least number k such that k*prime(n)# - 1 is prime and (k*prime(n)# - 1)^2 - 2 is a Chen prime.

Original entry on oeis.org

2, 1, 1, 11, 1, 6, 41, 17, 8, 13, 14, 107, 84, 23, 4, 101, 13, 89, 211, 58, 83, 75, 260, 414, 35, 39, 871, 79, 27, 42, 915, 44, 349, 142, 249, 404, 140, 84, 1068, 693, 972, 236, 1571, 1200, 298, 423, 970, 183, 173, 659, 523, 645, 1596, 448, 40, 201, 195, 1859, 427, 1732
Offset: 1

Views

Author

Pierre CAMI, Sep 21 2005

Keywords

Examples

			2*2-1 = 3, (2*2-1)^2-2 = 7, 3 and 7 are primes, so a(1) = 2.
11*2*3*5*7-1 = 2309, (11*2*3*5*7-1)^2-2 = 5331479, 2309 and 5331479 are primes, so a(4) = 11.

Crossrefs

Cf. A002110, A109611.

Programs

  • Mathematica
    chenQ[n_] := PrimeQ[n] && PrimeOmega[n+2] <= 2; a[n_] := Module[{p = Product[Prime[i], {i, 1, n}], k = 0}, While[!PrimeQ[k*p - 1] || !chenQ[(k*p-1)^2-2], k++]; k]; Array[a, 60] (* Amiram Eldar, Sep 11 2021 *)
  • PARI
    isok(k, q) = if (isprime(k*q-1), my(c=(k*q-1)^2-2); (isprime(c) && (bigomega(c+2)<=2)));
a(n) = my(k=1, q=prod(i=1, n, prime(i))); while (!isok(k, q), k++); k; \\ Michel Marcus, Sep 11 2021

A115606 Partial sums of A102540 (primes that are not Chen primes).

Original entry on oeis.org

0, 43, 104, 177, 256, 353, 456, 607, 770, 943, 1136, 1359, 1588, 1829, 2100, 2377, 2660, 2973, 3304, 3653, 4020, 4393, 4776, 5173, 5594, 6027, 6466, 6923, 7386, 7909, 8456, 9049, 9650, 10257, 10870, 11489, 12132, 12793, 13466, 14157, 14866, 15593
Offset: 0

Views

Author

Jonathan Vos Post, Mar 09 2006

Keywords

Comments

See also A109611 (Chen primes: primes p such that p + 2 is either a prime or a semiprime), A102540 (primes that are not Chen primes). a(n) is prime for a(1) = 43, a(5) = 353 [Chen prime], a(7) = 607, a(15) = 2377, a(31) = 9049, a(35) = 11489 [Chen prime], a(49) = 22013. a(n) is semiprime for a(3) = 177, a(9) = 943, a(13) = 1829, a(17) = 2973, a(19) = 3653, a(21) = 4393, a(23) = 5173, a(24) = 5594, a(29) = 7909, a(37) = 12793, a(38) = 13466, a(40) = 14866, a(41) = 15593, a(43) = 17065, a(45) = 18595.

Examples

			a(5) = 43 + 61 + 73 + 79 + 97 = 353, which happens to be the Chen prime A109611(52).

Crossrefs

Cf. A000040, A001358, A007504, A102540, A109611.

Formula

a(n) = SUM[k=1..n] A102540(k).

A115719 Products of two primes that are not Chen primes.

Original entry on oeis.org

1849, 2623, 3139, 3397, 3721, 4171, 4429, 4453, 4819, 5329, 5767, 5917, 6241, 6283, 6493, 7009, 7081, 7439, 7519, 7663, 8137, 8299, 9211, 9409, 9589, 9847, 9943, 9991, 10363, 10553, 10609, 11023, 11653, 11773, 11899, 11911, 11929, 12169, 12629, 12877
Offset: 1

Views

Author

Jonathan Vos Post, Mar 09 2006

Keywords

Comments

Subset of semiprimes (A001358) such that neither prime factor is a Chen prime (A109611).

Examples

			a(1) = 1849 = A102540(1)*A102540(1) = 43*43.
a(2) = 2623 = A102540(1)*A102540(2) = 43*61.
a(36) = 14701 = A102540(2)*A102540(13) = 61 * 241.

Crossrefs

Cf. A001358, A109611, A102540.

A117243 Chen numbers which are also twin primes.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 59, 71, 101, 107, 109, 137, 139, 149, 179, 181, 191, 197, 199, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 571, 599, 617, 641, 659, 809, 811, 821, 827, 829, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229
Offset: 1

Views

Author

Jani Melik, Apr 22 2006

Keywords

Examples

			(3, 5) are twin numbers and also Chen numbers.

Crossrefs

Cf. A001097, A109611.

Programs

  • Maple
    ischenprime:=proc(n); if (isprime(n) = 'true') then if (isprime(n+2) = 'true' or numtheory[bigomega](n+2) = 2) then RETURN('true') else RETURN('false') fi fi end:
ts_chen_twin_primes:=proc(n) local i, ans; ans:=[ ]: for i from 1 to n do if (ischenprime(i) = 'true') and (isprime(i+2) = 'true' or isprime(i-2) = 'true') then ans:=[op(ans), i]: fi od; RETURN(ans) end:
ts_chen_twin_primes(1230);
  • Mathematica
    Select[Prime[Range[250]],PrimeOmega[#+2]<3&&AnyTrue[#+{2,-2},PrimeQ]&] (* Harvey P. Dale, Jul 28 2023 *)

Extensions

Corrected and extended by Jani Melik, May 04 2013

A117740 Prime Fibonacci numbers that are Chen primes.

Original entry on oeis.org

2, 3, 5, 13, 89, 233, 514229
Offset: 1

Views

Author

Jani Melik, Apr 28 2006; corrected Apr 28 2006

Keywords

Crossrefs

Cf. A005478, A109611.

Programs

  • Maple
    ischenprime:=proc(n); if (isprime(n) = 'true') then if (isprime(n+2) = 'true' or numtheory[bigomega](n+2) = 2) then RETURN('true') else RETURN('false') fi fi end: ts_prime_fibonacci_chen:=proc(n) local i, tren, ans; ans:= [ ]: for i from 0 to n do tren := combinat[fibonacci](i): if (isprime( tren ) = 'true' and ischenprime(tren) = 'true') then ans:=[op(ans), tren]: fi od; RETURN(ans) end: ts_prime_fibonacci_chen(300); # Jani Melik, May 05 2006

A117743 Chen primes that are not Fibonacci numbers.

Original entry on oeis.org

7, 11, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 67, 71, 83, 101, 107, 109, 113, 127, 131, 137, 139, 149, 157, 167, 179, 181, 191, 197, 199, 211, 227, 239, 251, 257, 263, 269, 281, 293, 307, 311, 317, 337, 347, 353, 359, 379, 389, 401, 409, 419, 431, 443, 449, 461, 467
Offset: 1

Views

Author

Jani Melik, Apr 28 2006; corrected Apr 28 2006

Keywords

Crossrefs

Cf. A005478, A109611.

Extensions

Missing term 83 inserted and more terms from Georg Fischer, Mar 26 2025
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