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 1-9 of 9 results.

A212266 Primes p such that p - m! is composite, where m is the greatest number such that m! < p.

Original entry on oeis.org

59, 73, 79, 89, 101, 109, 197, 211, 239, 241, 263, 281, 307, 337, 367, 373, 379, 409, 419, 421, 439, 443, 449, 461, 463, 491, 523, 547, 557, 571, 593, 601, 613, 617, 631, 647, 653, 659, 673, 701, 709, 769, 797, 811, 839, 853, 863, 881, 907, 929, 937, 941, 967
Offset: 1

Views

Author

Balarka Sen, May 12 2012

Keywords

Comments

The first five terms 59, 73, 79, 89, 101 belong to A023209. The terms 409, 419, 421, 439, 443, 449 also belong to A127209.
It seems likely that a(n) ~ n log n, can this be proved? - Charles R Greathouse IV, Sep 20 2012

Examples

			29 is not a member because 29 - 4! = 5 is prime.
59 is a member because 59 - 4! = 35 is composite.

Links

Crossrefs

Cf. A212600, A212598, A136437, A023209, A127209.

Programs

  • Mathematica
    Select[Prime[Range[200]],Module[{m=9},CompositeQ[While[m!>=#,m--];#-m!]]&] (* The initial m constant (set at 9 in the program) needs to be increased if the prime Range constant (set at 200 in the program) is increased beyond 30969. *) (* Harvey P. Dale, Dec 01 2023 *)
  • PARI
    for(n=3,5,N=n!;forprime(p=N+3,N*(n+1),if(!isprime(p-N), print1(p", ")))) \\ Charles R Greathouse IV, May 12 2012
  • PARI
    is_A212266(p)=isprime(p) && for(n=1,p, n!
    1))  \\ M. F. Hasler, May 20 2012

A023278 Primes that remain prime through 3 iterations of function f(x) = 3x + 4.

Original entry on oeis.org

23, 683, 1663, 2753, 3203, 6073, 6323, 7523, 8243, 9293, 9613, 15173, 19913, 21023, 21683, 25183, 26633, 29663, 34613, 34703, 39293, 41953, 43283, 45533, 52813, 59393, 62473, 65053, 66763, 71713, 71993, 81533, 81953, 84523, 87833, 103843, 104183
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Primes p such that 3*p+4, 9*p+16 and 27*p+52 are also primes. - Vincenzo Librandi, Aug 04 2010

Links

Crossrefs

Subsequence of A023209, A023247, and of A034936.

Programs

  • Magma
    [n: n in [1..150000] | IsPrime(n) and IsPrime(3*n+4) and IsPrime(9*n+16) and IsPrime(27*n+52)] // Vincenzo Librandi, Aug 04 2010
  • Mathematica
    Select[Prime@ Range[10^4], Times @@ Boole@ PrimeQ@ Rest@ NestList[3 # + 4 &, #, 3] > 0 &] (* Michael De Vlieger, Sep 19 2016 *)
  • PARI
    is(n)=isprime(n) && isprime(3*n+4) && isprime(9*n+16) && isprime(27*n+52) \\ Charles R Greathouse IV, Sep 20 2016

Formula

a(n) == 3 (mod 10). - John Cerkan, Sep 16 2016

A023308 Primes that remain prime through 4 iterations of the function f(x) = 3x + 4.

Original entry on oeis.org

3203, 21683, 34613, 52813, 103843, 116933, 117443, 165443, 172933, 193603, 195053, 213973, 226783, 321053, 322193, 357613, 360323, 362233, 363403, 368743, 472393, 474143, 496333, 518543, 528673, 569083, 571303, 631853, 654623, 714893, 758503
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Primes p such that 3*p+4, 9*p+16, 27*p+52 and 81*p+160 are also primes. - Vincenzo Librandi, Aug 04 2010
All a(n) == 33 or 53 (mod 70). - John Cerkan, Oct 04 2016

Links

Crossrefs

Subsequence of A023209, A023247, A023278, and A034936.

Programs

  • GAP
    Filtered([1..760000],n->IsPrime(n) and IsPrime(3*n+4) and IsPrime(9*n+16) and IsPrime(27*n+52) and IsPrime(81*n+160)); # Muniru A Asiru, Dec 07 2018
  • Magma
    [n: n in [1..1000000] | IsPrime(n) and IsPrime(3*n+4) and IsPrime(9*n+16) and IsPrime(27*n+52) and IsPrime(81*n+160)] // Vincenzo Librandi, Aug 04 2010
  • Maple
    select(n->isprime(n) and isprime(3*n+4) and isprime(9*n+16) and isprime(27*n+52) and isprime(81*n+160),[$1..760000]); # Muniru A Asiru, Dec 07 2018
  • Mathematica
    Select[Prime[Range[10000]], Union[PrimeQ[NestList[(3# + 4 &), #, 4]]] == {True} &] (* Alonso del Arte, Nov 30 2018 *)
  • PARI
    is(n) = my(x=3*n+4, i=0); while(1, if(!ispseudoprime(x), return(0), i++); if(i==4, return(1)); x=3*x+4)
forprime(p=1, 760000, if(is(p), print1(p, ", "))) \\ Felix Fröhlich, Dec 07 2018

A258261 Primes p such that 3p - 4 is also prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 19, 29, 31, 37, 47, 59, 61, 67, 79, 89, 107, 131, 149, 151, 157, 191, 197, 199, 227, 229, 241, 271, 277, 281, 311, 317, 367, 389, 397, 409, 421, 431, 457, 479, 499, 509, 521, 541, 547, 557, 571, 617, 631, 659, 661, 677, 691, 701, 719
Offset: 1

Views

Author

Zak Seidov, May 24 2015

Keywords

Comments

This sequence is interesting because of the comments in A258233: for n > 1, if 3 * prime(n) - 4 is prime then A258233(n) = 1 + A071704(n), otherwise A258233 (n) = A071704(n). - Zak Seidov, Jun 04 2015
Subsequence of primes of A228121. - Michel Marcus, May 30 2015

Examples

			3 * 2 - 4 = 2, 3 * 3 - 4 = 5, 3 * 5 - 4 = 11, 3 * 7 - 4 = 17, 3 * 11 - 4 = 29 are all prime, so 2, 3, 5, 7, 11 are all in the sequence.
3 * 13 - 4 = 35 = 5 * 7, so 13 is not in the sequence.

Crossrefs

Cf. A023209 (3p + 4), A071704, A258233, A228353, A228121.

Programs

  • Magma
    [p: p in PrimesUpTo(1000) | IsPrime(3*p-4)]; // Vincenzo Librandi, May 25 2015
  • Mathematica
    Select[Prime[Range[200]], PrimeQ[3# - 4] &]
  • PARI
    forprime(p=1,10^3,if(isprime(3*p-4),print1(p,", "))) \\ Derek Orr, May 27 2015

A337185 Composite numbers k such that A337183(k) is prime.

Original entry on oeis.org

6, 10, 15, 21, 22, 26, 28, 33, 35, 38, 39, 42, 44, 45, 46, 50, 54, 62, 65, 66, 68, 69, 75, 76, 77, 82, 85, 95, 98, 99, 102, 105, 106, 111, 115, 116, 117, 118, 120, 123, 124, 126, 129, 132, 135, 141, 143, 145, 146, 152, 155, 158, 161, 168, 170, 176, 178, 186, 188, 198, 200, 201, 202, 203, 205, 206
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Jan 29 2021

Keywords

Examples

			15 is a term because it is not prime and A337183(15) = 29 is prime.

Links

Crossrefs

Cf. A337183. Disjoint from A001597. Includes 2*k for k in A023209.

Programs

  • Maple
    f:= proc(n) local F, b, i;
  F:= sort(map(t -> t[1]$t[2], ifactors(n)[2]), `>`);
  b:= convert(F, `+`);
  (add(F[i]*b^(i-1), i=1..nops(F)));
end proc:
select(t -> not isprime(t) and isprime(f(t)), [$2..300]);

A023336 Primes that remain prime through 5 iterations of function f(x) = 3x + 4.

Original entry on oeis.org

34613, 165443, 321053, 363403, 474143, 496333, 528673, 631853, 834503, 957563, 1199623, 1274803, 1817093, 1918733, 2063423, 2611663, 2889703, 3224233, 3652703, 3697433, 3824413, 3852973, 4655873, 4708793, 5089943, 5508263, 5937853, 6067073
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Primes p such that 3*p+4, 9*p+16, 27*p+52, 81*p+160 and 243*p+484 are also primes. - Vincenzo Librandi, Aug 05 2010

Links

Crossrefs

Subsequence of A023209, A023247, A023278, A023308, and A034936.

Programs

  • Magma
    [n: n in [1..25000000] | IsPrime(n) and IsPrime(3*n+4) and IsPrime(9*n+16) and IsPrime(27*n+52) and IsPrime(81*n+160) and IsPrime(243*n+484)] // Vincenzo Librandi, Aug 05 2010

Formula

a(n) == 33 (mod 70). - John Cerkan, Oct 10 2016

A322923 Primes of the form 3*p + 4, where p is a prime.

Original entry on oeis.org

13, 19, 37, 43, 61, 73, 97, 127, 163, 181, 223, 241, 271, 307, 313, 331, 397, 421, 457, 523, 541, 547, 577, 601, 673, 691, 727, 757, 811, 853, 883, 937, 997, 1051, 1063, 1123, 1153, 1171, 1231, 1297, 1303, 1321, 1531, 1567, 1627, 1693, 1783, 1801
Offset: 1

Views

Author

Vincenzo Librandi, Mar 12 2019

Keywords

Links

Crossrefs

Cf. A023209, A100202, A102851, A142120, A142262, A142817.

Programs

  • GAP
    P:=Filtered([1..1000],IsPrime);;
a:=Filtered(List(P,i->3*i+4),k->IsPrime(k)); # Muniru A Asiru, Mar 23 2019
  • Magma
    [a: p in PrimesUpTo(600) | IsPrime(a) where a is 3*p+4];
  • Maple
    select(isprime,[3*ithprime(p)+4$p=1..120]); # Muniru A Asiru, Mar 23 2019
  • Mathematica
    Select[Table[p=Prime[n];3p+4,{n,85}],PrimeQ]
  • PARI
    terms(n) = my(x=0, i=0); forprime(p=1, , if(i >= n, break); x=3*p+4; if(ispseudoprime(x), print1(x, ", "); i++))
/* Print initial 50 terms as follows: */
terms(50) \\ Felix Fröhlich, Mar 23 2019

A342035 Numbers k such that both bigomega(k)+sopfr(k) and bigomega(k)+sopfr(k)+k are prime.

Original entry on oeis.org

2, 6, 18, 24, 26, 30, 38, 56, 72, 90, 104, 120, 152, 158, 162, 174, 206, 218, 288, 294, 318, 342, 344, 350, 354, 360, 378, 408, 446, 458, 486, 510, 522, 534, 558, 690, 696, 698, 726, 776, 792, 824, 878, 894, 910, 936, 990, 992, 1016, 1056, 1078, 1098, 1152, 1170, 1184, 1256, 1278, 1286, 1330
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Feb 25 2021

Keywords

Comments

Numbers k such that A001222(k)+A001414(k) and A001222(k)+A001414(k)+k are prime.
All terms are even.
Semiprimes in the sequence are 2*p where p is in the intersection of A023200 and A023209.

Examples

			a(4) = 24 = 2^3*3 is in the sequence because A001222(24) = 3+1 = 4, A001414(24) = 3*2+3 = 9, and 4+9 = 13 and 4+9+24 = 37 are prime.

Links

Crossrefs

Cf. A001222, A001414, A023200, A023209.

Programs

  • Maple
    filter:= proc(n) local k,v;
  v:= add(k[2]*(1+k[1]), k = ifactors(n)[2]);
  isprime(v) and isprime(n+v)
end proc:
select(filter, [seq(i,i=2..2000,2)]);

A352170 Primes p such that p+4, 3*p+4 and 3*p+8 are also prime.

Original entry on oeis.org

3, 13, 103, 223, 823, 2953, 7873, 11113, 11863, 13033, 13963, 16063, 22153, 23743, 24763, 27733, 30133, 31513, 34213, 35593, 39883, 41893, 43063, 50383, 51043, 54493, 62983, 65323, 66343, 68473, 71593, 72643, 87793, 88423, 98893, 101203, 106363, 110563, 127873, 134593, 136603, 158563, 164623, 165703
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Mar 07 2022

Keywords

Comments

Members p of A023200 such that 3*p+4 is also in A023200.
Except for 3, all terms == 13 (mod 30).

Examples

			a(4) = 223 is a term because 223, 223+4 = 227, 3*223+4 = 673 and 3*223+8 = 677 are all prime.

Links

Crossrefs

Intersection of A023200, A023209 and A023210.

Programs

  • Maple
    select(p -> isprime(p) and isprime(p+4) and isprime(3*p+4) and isprime(3*p+8), [3,seq(i,i=13..10^6,30)]);
  • Mathematica
    Select[Range[200000], AllTrue[{#, # + 4, 3*# + 4, 3*# + 8}, PrimeQ] &] (* Amiram Eldar, Mar 07 2022 *)
  • Python
    from sympy import sieve, isprime
for p in sieve.primerange(0, 10**6):
    if(all(isprime(q) for q in [p+4, 3*p+4, 3*p+8])):
        print (p, end=", ") # Martin Ehrenstein, Mar 09 2022
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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