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 1-10 of 10 results.

A088423 a(n) is the number of primes in arithmetic progression starting with 11 and with d = 2n.

Original entry on oeis.org

2, 1, 4, 2, 1, 2, 1, 1, 3, 2, 1, 1, 2, 1, 5, 2, 1, 3, 1, 1, 2, 1, 1, 3, 2, 1, 1, 2, 1, 6, 2, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 4, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 2, 2, 1, 4, 2, 1, 1, 1, 1, 1, 2
Offset: 1

Views

Author

Zak Seidov, Sep 29 2003

Keywords

Comments

Arithmetic progression is stopped when next term is not prime. E.g., for n=3, a=4, that is 11,17,23,29 are prime, while next term, 35, is not prime.

Links

Crossrefs

Cf. A088420, A088421, A088422, A088424, A088425, A088426, A088427, A088428, A088429.

Programs

  • Mathematica
    bb={}; Do[s=1; Do[If[PrimeQ[11+k*d], s=s+1, bb={bb, s}; Break[]], {k, 10}], {d, 2, 200, 2}]; Flatten[bb]
  • PARI
    a(n) = my(p=11, x=p+2*n, i=1); while(1, if(ispseudoprime(x), i++; x=x+2*n, return(i))) \\ Felix Fröhlich, May 28 2021

A088426 Number of primes in arithmetic progression starting with 19 and with d=2n.

Original entry on oeis.org

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

Views

Author

Zak Seidov, Sep 29 2003

Keywords

Comments

Arithmetic progression is stopped when next term is not prime. E.g. for n=6 (d=12), a=3, that is 19,31,43 are prime, while next term, 55, is not prime.
From Robert Israel, Jul 27 2020: (Start)
a(n) = 1 if n == 1 (mod 3), a(n) <= 2 if n == 2 (mod 3).
If a(n) >= p where p is 3, 5, 7, 11, 13 or 17, then n is divisible by p.
All a(n) < 19.
Records:
a(1)=1
a(2)=2
a(6)=3
a(27)=4
a(210)=5
a(825)=6
a(16380)=7
a(273420)=9
a(17853675)=10 (End)
From David A. Corneth, Jul 29 2020: (Start)
Other first occurrences are:
a(779520) = 8
a(4918073160) = 11
a(3187366788375) = 12
a(6125952702870) = 13
If a(k) = 14 then k > 4.8*10^15.
If a(k) = 15 then k > 1.77 * 10^16. (End)

Links

Crossrefs

Cf. A005115, A088420, A088421, A088422, A088423, A088424, A088425, A088427, A088428, A088429.

Programs

  • Maple
    f:= proc(n) local d,k;
  d:= 2*n;
  for k from 1 while isprime(19+d*k) do od:
  k
end proc:
map(f, [$1..200]); # Robert Israel, Jul 27 2020
  • Mathematica
    bb={}; Do[s=1; Do[If[PrimeQ[19+k*d], s=s+1, bb={bb, s}; Break[]], {k, 10}], {d, 2, 200, 2}]; Flatten[bb]

A088421 Number of primes in arithmetic progression starting with 5 and with d=2n.

Original entry on oeis.org

2, 1, 5, 2, 1, 5, 2, 1, 4, 1, 1, 3, 2, 1, 1, 2, 1, 2, 2, 1, 5, 1, 1, 5, 1, 1, 4, 2, 1, 1, 2, 1, 3, 2, 1, 1, 2, 1, 2, 1, 1, 4, 1, 1, 1, 2, 1, 5, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 5, 1, 1, 4, 2, 1, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 2, 2, 1, 3, 2, 1, 1, 1
Offset: 1

Views

Author

Zak Seidov, Sep 29 2003

Keywords

Comments

Arithmetic progression is stopped when next term is not prime. E.g. for n=3, a=5, that is 5,11,17,23,29 are prime, while next term, 35, is not prime.

Crossrefs

Cf. A088420, A088422, A088423, A088424, A088425, A088426, A088427, A088428, A088429.

Programs

  • Mathematica
    bb={}; Do[s=1; Do[If[PrimeQ[5+k*d], s=s+1, bb={bb, s}; Break[]], {k, 10}], {d, 2, 200, 2}]; Flatten[bb]

A088422 Number of primes in arithmetic progression starting with 7 and with d=2n.

Original entry on oeis.org

1, 2, 3, 1, 2, 4, 1, 2, 1, 1, 2, 2, 1, 1, 6, 1, 2, 3, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 1, 2, 3, 1, 1, 4, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 3, 1, 2, 4, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 2, 2, 1, 1, 7, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 3, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1
Offset: 1

Views

Author

Zak Seidov, Sep 29 2003

Keywords

Comments

Arithmetic progression is stopped when next term is not prime. E.g. for n=3, a=3, that is 7,13,19 are prime, while next term, 25, is not prime.

Crossrefs

Cf. A088420, A088421, A088423, A088424, A088425, A088426, A088427, A088428, A088429.

Programs

  • Mathematica
    bb={}; Do[s=1; Do[If[PrimeQ[7+k*d], s=s+1, bb={bb, s}; Break[]], {k, 10}], {d, 2, 200, 2}]; Flatten[bb]

A088424 Number of primes in arithmetic progression starting with 13 and with d=2n.

Original entry on oeis.org

1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 3, 1, 2, 4, 1, 2, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 6, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 5, 1, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 2, 3, 1, 2, 2, 1, 1, 1, 1, 1, 4, 1
Offset: 1

Views

Author

Zak Seidov, Sep 29 2003

Keywords

Comments

Arithmetic progression is stopped when next term is not prime. E.g. for n=15, a=4, that is 13,43,73,103 are prime, while next term, 133, is not prime.

Crossrefs

Cf. A088420, A088421, A088422, A088423, A088425, A088426, A088427, A088428, A088429.

Programs

  • Mathematica
    bb={}; Do[s=1; Do[If[PrimeQ[13+k*d], s=s+1, bb={bb, s}; Break[]], {k, 10}], {d, 2, 200, 2}]; Flatten[bb]

A088425 Number of primes in arithmetic progression starting with 17 and with d=2n.

Original entry on oeis.org

2, 1, 3, 1, 1, 4, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 3, 1, 1, 3, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 3, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 2, 1, 5, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1
Offset: 1

Views

Author

Zak Seidov, Sep 29 2003

Keywords

Comments

Arithmetic progression is stopped when next term is not prime. E.g. for n=6 (d=12), a=4, that is 17,29,41,53 are prime, while next term, 65, is not prime.

Crossrefs

Cf. A088420, A088421, A088422, A088423, A088424, A088426, A088427, A088428, A088429.

Programs

  • Mathematica
    bb={}; Do[s=1; Do[If[PrimeQ[17+k*d], s=s+1, bb={bb, s}; Break[]], {k, 10}], {d, 2, 200, 2}]; Flatten[bb]

A088427 Number of primes in arithmetic progression starting with 23 and with d=2n.

Original entry on oeis.org

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

Views

Author

Zak Seidov, Sep 29 2003

Keywords

Comments

Arithmetic progression is stopped when next term is not prime. E.g. for n=15 (d=30), a=3, that is 23,53,83,113 are prime, while next term, 143, is not prime.

Crossrefs

Cf. A088420, A088421, A088422, A088423, A088424, A088425, A088426, A088428, A088429.

Programs

  • Mathematica
    bb={}; Do[s=1; Do[If[PrimeQ[23+k*d], s=s+1, bb={bb, s}; Break[]], {k, 10}], {d, 2, 200, 2}]; Flatten[bb]

A088428 Number of primes in arithmetic progression starting with 29 and with d=2n.

Original entry on oeis.org

2, 1, 1, 2, 1, 3, 2, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 1, 2, 1, 3, 2, 1, 1, 2, 1, 4, 1, 1, 3, 1, 1, 1, 2, 1, 3, 2, 1, 2, 2, 1, 4, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 5, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2
Offset: 1

Views

Author

Zak Seidov, Sep 29 2003

Keywords

Comments

Arithmetic progression is stopped when next term is not prime. E.g. for n=6 (d=12), a=3, that is 29,41,53 are prime, while next term, 65, is not prime.

Crossrefs

Cf. A088420, A088421, A088422, A088423, A088424, A088425, A088426, A088427, A088429.

Programs

  • Mathematica
    bb={}; Do[s=1; Do[If[PrimeQ[29+k*d], s=s+1, bb={bb, s}; Break[]], {k, 10}], {d, 2, 200, 2}]; Flatten[bb]

A088429 Number of primes in arithmetic progression starting with 31 and with d=2n.

Original entry on oeis.org

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

Views

Author

Zak Seidov, Sep 29 2003

Keywords

Comments

Arithmetic progression is stopped when next term is not prime. E.g. for n=3 (d=6), a=3, that is 31,37,43 are prime, while next term, 49, is not prime.

Crossrefs

Cf. A088420, A088421, A088422, A088423, A088424, A088425, A088426, A088427, A088428.

Programs

  • Mathematica
    bb={}; Do[s=1; Do[If[PrimeQ[31+k*d], s=s+1, bb={bb, s}; Break[]], {k, 10}], {d, 2, 200, 2}]; Flatten[bb]

A160394 Numbers n = p*q*r (p, q, r prime) congruent to 0 mod p+q+r.

Original entry on oeis.org

27, 30, 70, 105, 231, 286, 627, 646, 805, 897, 1581, 1798, 2967, 3055, 3526, 4543, 5487, 6461, 6745, 7198, 7881, 9717, 10366, 10707, 14231, 16377, 20806, 21091, 23326, 26331, 29607, 33901, 35905, 37411, 38086, 38843, 40587, 42211, 44998, 55581
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 12 2009

Keywords

Comments

Also numbers n = p*q*r where r = p*q-(p+q) and p, q, r are prime.
For each twin prime pair (q, q+2) the number n = 2*p*(p+2) is in the sequence, since 2+p+(p+2) divides n.
In some cases the factors of n are in arithmetic progression; occurring common differences are 2, 4, 8, 10, 14, 20, 28, 34, 38, 40, 50, 68, 80, 94, 98, ...
All those arithmetic progressions have first term 3, their common differences are the numbers d such that A088420(d/2) = 3. - Klaus Brockhaus, May 17 2009

Examples

			27 = 3*3*3 = (3+3+3)*3, hence 27 is in the sequence; r = 3*3-(3+3).
30 = 2*5*3 = (2+5+3)*3, hence 30 is in the sequence; r = 2*5-(2+5).
165 = 3*5*11 is not a multiple of 3+5+11 = 19, hence 165 is not in the sequence.
627 = 3*11*19 = (3+11+19)*19, hence 627 is in the sequence; r = 3*11-(3+11). The factors 3, 11, 19 are in arithmetic progression (d=8).
40587 = 3*83*163 = (3+83+163)*163, hence 40587 is in the sequence; r = 3*83-(3+83). The factors 3, 83, 163 are in arithmetic progression (d=80).

Links

Crossrefs

Cf. A014612 (3-almost primes, numbers that are divisible by exactly 3 primes (counted with multiplicity)).
Cf. A001359 (lesser of twin primes), A115334 (numbers n such that 3+2n and 3+4n are prime), A088420 (number of primes in arithmetic progression starting with 3 and with d=2n). [From Klaus Brockhaus, May 17 2009]

Programs

  • Magma
    [ n: n in [2..56000] | &+[ d[2]: d in f ] eq 3 and n mod &+[ d[1]*d[2]: d in f ] eq 0 where f is Factorization(n) ]; // Klaus Brockhaus, May 17 2009
  • PARI
    list(lim)=my(v=List()); forprime(p=2,lim\4, forprime(q=2,lim\(2*p), my(pq=p*q, r=pq-p-q); if(isprime(r), listput(v, pq*r)))); Set(v) \\ Charles R Greathouse IV, Jan 31 2017

Extensions

Partially edited by N. J. A. Sloane, May 14 2009
Missed entry 27 contributed by Zak Seidov, May 14 2009
Further edited by Klaus Brockhaus, May 17 2009
Showing 1-10 of 10 results.

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