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 1-3 of 3 results.

A329916 Smallest k such that 6*k*A057130(n)-1 and 6*k*A057130(n)+1 are twin primes.

Original entry on oeis.org

1, 2, 3, 23, 11, 18, 77, 46, 84, 76, 22, 30, 3, 107, 26, 198, 136, 23, 236, 284, 167, 269, 381, 405, 379, 374, 620, 481, 606, 505, 163, 1414, 348, 639, 1696, 1429, 850, 2050, 740, 117, 362, 35, 3961, 72, 1307, 1816, 9410, 5705, 972, 368, 5083, 4387, 3296, 6039
Offset: 1

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Author

Pierre CAMI, Nov 24 2019

Keywords

Comments

A057130 gives the product of prime numbers (-1 mod 6) in the order of occurrence.

Examples

			A057130(1)=5, 6*1*5-1=29, and 29 and 31 are twin primes, so a(1)=1.
A057130(2)=55, 6*2*55-1=659, and 659 and 661 are twin primes, so a(2)=2.

Crossrefs

Cf. A001359, A006512, A007528, A057130, A060256, A173937, A329736.

Programs

  • PARI
    lista(nn) = {my(pp = 1); forprime (p = 1, nn, if (Mod(p, 6) == -1, pp *= p; my(k=1); while (!isprime(6*k*pp-1) || !isprime(6*k*pp+1), k++); print1(k, ", ");););} \\ Michel Marcus, Nov 25 2019

A057131 One less than six times product of first n primes of form 6k-1.

Original entry on oeis.org

29, 329, 5609, 129029, 3741869, 153416669, 7210583489, 382160924969, 22547494573229, 1600872114699329, 132872385520044389, 11825642311283950709, 1194389873439679021709, 127799716458045655322969, 14441367959759159051495609, 1891819202728449835745924909
Offset: 1

Views

Author

Henry Bottomley, Aug 11 2000

Keywords

Comments

a(n)=5 mod 6, so a(n) has at least one prime factor of form 6k-1 and this is not one of those included in the calculation of a(n); for example 5609 has 71 as a prime factor. Therefore there are an infinite number of prime numbers of form 6k-1 (and also of form 3k-1).

Examples

			a(3) = 6*(5*11*17)-1 = 5609.

Crossrefs

Cf. A007528, A057130.

Programs

  • PARI
    lista(nn) = {pp = 6; for (n = 1, nn, p = prime(n); if (Mod(p, 6) == -1, pp *= p; print1(pp-1, ", ")););} \\ Michel Marcus, Sep 08 2013

Formula

a(n) = (a(n-1)+1)*A007528(n)-1 = 6*A057130(n)-1.

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Oct 06 2000
More terms from Michel Marcus, Sep 08 2013

A307162 a(n) is the smallest k such that A319100(k) = A025610(n).

Original entry on oeis.org

1, 3, 8, 7, 24, 21, 120, 56, 1320, 63, 168, 22440, 252, 840, 516120, 504, 9240, 819, 14967480, 2184, 157080, 3276, 613666680, 10920, 3612840, 6552, 28842333960, 120120, 15561, 104772360, 32760, 1528643699880, 2042040, 62244, 4295666760, 207480, 90189978292920, 46966920, 124488
Offset: 1

Views

Author

Jianing Song, Mar 27 2019

Keywords

Comments

A025610 is the range of A319100.
Let b = A319100. Note that:
- if k is an odd number, then b(2*k) = b(k), b(4*k) = 2*b(k), b(2^e*k) = 4*b(k) for e >= 3;
- if k is not divisible by 3, then b(3*k) = 2*b(k), b(3^e*k) = 6*b(k) for e >= 2;
- for all primes p > 3, if k is not divisible by p, then b(p^e*k) = b(p*k).
As a result, it is easy to see that for every n, a(n) is not congruent to 2 modulo 4 and is not divisible by 16 or 27 or p^2 for any prime p > 3.

Links

Crossrefs

Cf. A025610, A319100, A002476, A007528, A121940, A057130.

Programs

  • PARI
    isA025610(n) = omega(6*n)==2&&valuation(n,2)>=valuation(n,3)
b(n) = if(isA025610(n), i=1; while(A319100(i)!=n, i++); i)
for(n=1, 216, if(isA025610(n), print1(b(n), ", "))) \\ See A319100 for its program
  • PARI
    p(j) = my(t=0,v=vector(j)); for(k=1, oo, if(prime(k)%6==1, t++; v[t]=prime(k)); if(t==j, return(v)))
q(i) = my(t=0,v=vector(i)); for(k=1, oo, if(prime(k)%6==5, t++; v[t]=prime(k)); if(t==i, return(v)))
b(i,j) = {
if(j<=1 && i<=2, my(M=[1,3,8;7,21,56]); return(M[j+1,i+1]));
if(j==0 && i>=3, my(Q=q(i-3)); return(24*prod(k=1, i-3, Q[k])));
if(j>=2 && i<=2, my(P=p(j-1), w=[9,36,72]); return(w[i+1]*prod(k=1, j-1, P[k])));
if(j>=1 && i>=3, my(P=p(j), Q=q(i-2)); return(prod(k=1, j-1, P[k])*8*prod(k=1, i-3, Q[k])*min(9*Q[i-2], 3*P[j])));
}
list(lim) = my(v=A025610(lim), u=vector(#v)); for(k=1, #v, my(i=valuation(v[k],2)-valuation(v[k],3), j=valuation(v[k],3)); u[k]=b(i,j)); u \\ Jianing Song, Jun 04 2019, See A025610 for its program

Formula

Let p(j) = A002476(j), q(i) = A007528(i), P(j) = Product_{k=1..j} p(k) = A121940(j) if j > 0, Q(i) = Product_{k=1..i} q(k) = A057130(i) if i > 0. If A025610(n) = 2^i*6^j, then:
(a) if i = 0, then a(n) = 1 if j = 0, 7 if j = 1 and 9*P(j-1) if j >= 2;
(b) if i = 1, then a(n) = 3 if j = 0, 21 if j = 1 and 36*P(j-1) if j >= 2;
(c) if i = 2, then a(n) = 8 if j = 0, 56 if j = 1 and 72*P(j-1) if j >= 2;
(d) if i >= 3, then a(n) = 24*Q(i-3) if j = 0 and P(j-1)*8*Q(i-3)*min{9*q(i-2), 3*p(j)} if j >= 1. [Rewritten by Jianing Song, Jun 04 2019]
Showing 1-3 of 3 results.

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