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 1-5 of 5 results.

A141426 a(n) = A140555(n) - 1.

Original entry on oeis.org

1, 2, 18, 28, 42, 58, 70, 78, 88, 108, 112, 126, 136, 138, 148, 162, 178, 180, 196, 198, 210, 228, 238, 240, 268, 280, 282, 292, 312, 316, 336, 348, 358, 378, 388, 396, 400, 408, 418, 420, 430, 438, 448, 462, 466, 478, 486, 490, 498, 508, 520, 522, 546, 568
Offset: 1

Views

Author

Giovanni Teofilatto, Aug 06 2008

Keywords

Comments

Previous name was: Count of numbers smaller than and coprime to the prime A140555(n).

Crossrefs

Cf. A140555.

Programs

  • Maple
    isA140555 := proc(n) RETURN( isprime(n) and not isprime(n+6) ) ; end: A140555 := proc(n) option remember ; if n = 1 then 2; else for a from procname(n-1)+1 do if isA140555(a) then RETURN(a) ; fi; od: fi; end: A141426 := proc(n) A140555(n)-1 ; end: for n from 1 to 100 do printf("%d,",A141426(n)) ; od: # R. J. Mathar, Aug 12 2008
  • Mathematica
    lim=255;a140555=Select[Prime[Range[lim]], !PrimeQ[#+6]&];a[n_]:=Total[Boole[CoprimeQ[Range[a141426[[n]]-1],a140555[[n]]]]];Array[a,PrimePi[lim]] (* James C. McMahon, Jul 22 2025 *)

Formula

a(n) = phi(A140555(n)) = A140555(n) - 1.

Extensions

Edited and extended by R. J. Mathar, Aug 12 2008
New name from Joerg Arndt, Jul 22 2025

A067775 Primes p such that p + 4 is composite.

Original entry on oeis.org

2, 5, 11, 17, 23, 29, 31, 41, 47, 53, 59, 61, 71, 73, 83, 89, 101, 107, 113, 131, 137, 139, 149, 151, 157, 167, 173, 179, 181, 191, 197, 199, 211, 227, 233, 239, 241, 251, 257, 263, 269, 271, 281, 283, 293, 311, 317, 331, 337, 347, 353, 359, 367, 373, 383, 389
Offset: 1

Views

Author

Benoit Cloitre, Feb 06 2002

Keywords

Comments

Primes n such that n!*B(n+3) is an integer where B(k) are the Bernoulli numbers B(1) = -1/2, B(2) = 1/6, B(4) = -1/30, ..., B(2m+1) = 0 for m > 1.
If n is prime n!*B(n-1) is always an integer. Note that if Goldbach's conjecture (2n = p1 + p2 for all n >= 2) is false and K is the smallest value of n for which it fails, then for 2(K-2) = p3 + p4, the primes p3 and p4 must be taken from this list. See similar comment for A140555. - Keith Backman, Apr 06 2012
Complement of A023200 (primes p such that p + 4 is also prime) with respect to A000040 (primes). For p > 2: primes p such that there is no prime of the form r^2 + p where r is prime, subsequence of A232010. Example: the prime 7 is not in the sequence because 2^2 + 7 = 11 (prime). A232009(a(n)) = 0 for n > 1 . - Jaroslav Krizek, Nov 22 2013

Links

Crossrefs

Cf. A049591, A140555, A232009, A232010, A232012, A023200.

Programs

  • Mathematica
    A067775 = {}; Do[p = Prime@ n; If[ IntegerQ[ p! BernoulliB[p + 3]], AppendTo[A067775, p]], {n, 77}]; A067775 (* Robert G. Wilson v, Aug 19 2008 *)
Select[Prime[Range[80]], Not[PrimeQ[# + 4]] &] (* Alonso del Arte, Apr 02 2014 *)
  • PARI
    lista(nn) = {forprime(p=1, nn, if (! isprime(p+4), print1(p, ", ")););} \\ Michel Marcus, Nov 22 2013

Formula

a(n) ~ n log n. - Charles R Greathouse IV, Nov 22 2013

Extensions

New name from Klaus Brockhaus at the suggestion of Michel Marcus, Nov 22 2013

A233314 Primes p such that p+6 and p+12 are composite.

Original entry on oeis.org

2, 3, 43, 79, 109, 113, 149, 163, 197, 241, 283, 293, 313, 317, 349, 359, 379, 401, 439, 463, 499, 521, 523, 547, 569, 577, 599, 617, 643, 659, 673, 683, 691, 701, 709, 719, 743, 769, 773, 787, 829, 839, 859, 863, 883, 887, 911, 919, 937, 953, 967, 983, 1031
Offset: 1

Views

Author

Zak Seidov, Dec 07 2013

Keywords

Comments

Subsequence of A140555: a(3) = 43 = A140555(5), a(1000) = 18503 = A140555(1468).

Links

Crossrefs

Cf. A140555.

Programs

  • Mathematica
    Select[Prime[Range[200]], ! PrimeQ[# + 6] && ! PrimeQ[# + 12] &] (* T. D. Noe, Dec 10 2013 *)

A233318 Primes p such that p+6, p+12 and p+18 are composite.

Original entry on oeis.org

2, 3, 197, 241, 283, 317, 359, 463, 499, 521, 547, 577, 617, 719, 773, 787, 829, 883, 887, 937, 967, 983, 1061, 1093, 1109, 1129, 1171, 1193, 1229, 1249, 1319, 1327, 1373, 1399, 1451, 1489, 1523, 1627, 1637, 1667, 1669, 1699, 1733, 1787, 1801, 1823, 1831, 1847
Offset: 1

Views

Author

Zak Seidov, Dec 07 2013

Keywords

Comments

Subsequence of A233314 and A140555: a(3) = 197 = A233314(7) = A140555(19).

Links

Crossrefs

Cf. A140555, A233314.

Programs

  • Mathematica
    Select[Prime[Range[400]], ! PrimeQ[# + 6] && ! PrimeQ[# + 12] && ! PrimeQ[# + 18] &] (* T. D. Noe, Dec 10 2013 *)
Select[Prime[Range[400]],AllTrue[#+{6,12,18},CompositeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 15 2019 *)

A297847 Sexiness of p = prime(n): number of iterations of the function f(x) = x + 6 that leave p prime.

Original entry on oeis.org

0, 0, 4, 2, 3, 1, 2, 0, 1, 0, 2, 1, 3, 0, 2, 1, 0, 3, 2, 0, 1, 0, 1, 0, 2, 2, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 1, 1, 0, 0, 0, 1, 2, 0, 1, 0, 0, 3, 2, 1, 0, 2, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 2, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0
Offset: 1

Views

Author

Felix Fröhlich, Jan 07 2018

Keywords

Comments

a(n) > 0 iff p is a term of A023201.
a(n) = 0 iff p is a term of A140555.
a(n) = 2 iff p is a term of A046118.
a(n) > 2 iff p is a term of A023271.
a(n) < 4 except for n = 3. Proof: The last digits of the numbers in the progression repeat 1, 7, 3, 9, 5, 1, 7, 3, 9, 5, ..., so a(n) is at most 4, which only happens for p = 5, since A007652(n) = 5 only for n = 3.

Examples

			For n = 13: prime(13) = 41 and 41 remains prime through exactly 3 iterations of f(x) = x + 6, since 47, 53 and 59 are prime, but 65 is composite, so a(13) = 3.

Links

Crossrefs

Cf. A023201, A023271, A046118, A140555.

Programs

  • Mathematica
    Array[-2 + Length@ NestWhileList[# + 6 &, Prime@ #, PrimeQ] &, 105] (* Michael De Vlieger, Jan 11 2018 *)
  • PARI
    a(n) = my(p=prime(n), x=p, i=0); while(1, x=x+6; if(!ispseudoprime(x), return(i), i++))
Showing 1-5 of 5 results.

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