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 1-4 of 4 results.

A066735 Primes p dividing 1 + the product of the primes less than p.

Original entry on oeis.org

2, 3, 19, 1471, 3001
Offset: 1

Views

Author

Joseph L. Pe, Jan 15 2002

Keywords

Comments

No further terms up to prime(216816) = 2999999. Is the sequence finite? - Klaus Brockhaus, Jan 17 2002
From Lévai Gábor (gablevai(AT)vipmail.hu), Nov 23 2004: (Start)
Let p(1)=2, p(2)=3, p(3)=5, ... denote the primes and let E(n) = 1 + p(1) * p(2) * ... * p(n). For k >= 1, list the primes p such that p(n+k) | E(n). For k=1 we get this sequence, for k=2 we get A100465.
For k >= 3 the known results are as follows: if k = 3: no solutions for p < 80000000; if k = 4: 463, 2908123 and no others for p < 80000000; if k = 5: 61, 73 and no others for p < 80000000; if k = 6: 21687203 and no others for p < 80000000; if k = 7: 149, 43951591 and no others for p < 80000000; if k = 8: 31, 131 and no others for p < 80000000; if k = 9: 58691999 and no others for p < 80000000. (End)
No further terms up to 80000000. - Lévai Gábor (gablevai(AT)vipmail.hu), Nov 23 2004
a(6) > 179424673 = prime(10^7). - Giovanni Resta, Apr 13 2017
a(6) > 914799232 > prime(46727379). - Max Z. Scialabba, Feb 26 2024

Examples

			1 + Product of the primes < 19 = 1 + 2*3*5*7*11*13*17 = 510511 = 19*26869; so 19 is a term of the sequence.

Links

Crossrefs

Cf. A002110, A002585, A100465, A081618.

Programs

  • Mathematica
    p = 2; Do[q = Prime[n]; If[ IntegerQ[(p + 1)/q], Print[q]]; p = p*q, {n, 2, 86120} ]
  • PARI
    a066735(m) =local(k,p); k=1; forprime(p=2,m, if((k+1)%p==0,print1(p,",")); k=k*p)

A341805 Numbers k such that (product of first k primes)-1 is divisible by the (k+1)-th prime.

Original entry on oeis.org

0, 2, 4, 9823712
Offset: 1

Views

Author

Jeppe Stig Nielsen, Feb 20 2021

Keywords

Examples

			4 is a member because 2*3*5*7-1 (product of first 4 primes, minus one) is divisible by the 5th prime, 11.
9823712 is a member because 2*3*5*...*176078267-1 is divisible by 176078293, where 176078267 is the 9823712th prime.

Crossrefs

Cf. A079276, A081618, A338543, A341804, A341812.

Programs

  • PARI
    isok(k) = ((vecprod(primes(k)) - 1) % prime(k+1)) == 0; \\ Michel Marcus, Mar 03 2021

Formula

a(n) = A000720(A341804(n)) - 1.

A338543 Numbers k such that 1 + Product_{i

Original entry on oeis.org

1, 2, 8, 233, 431
Offset: 1

Views

Author

Robert Israel, Nov 01 2020

Keywords

Comments

Numbers k such that A062347(k-1) == -1 (mod prime(k)).

Examples

			a(3)=8 is a term because 1 + 2*3*5*7*11*13*17 = 510511 is divisible by prime(8)=19.

Links

Crossrefs

Cf. A002110, A006862, A081618, A062347, A341805.

Programs

  • Maple
    select(t -> 1+mul(ithprime(i),i=1..t-1) mod ithprime(t)=0, [$1..1000]);
  • PARI
    isok(n) = ((1+vecprod(primes(n-1))) % prime(n)) == 0; \\ Michel Marcus, Nov 03 2020

Formula

a(n) = A081618(n)+1 for n >= 2.

A081617 Smallest k such that (product of first n primes)*k+1 is divisible by the (n+1)-th prime. Also (A075306(n)-1)/A002110(n).

Original entry on oeis.org

1, 4, 3, 10, 10, 2, 1, 3, 17, 13, 10, 34, 38, 4, 51, 55, 51, 29, 68, 13, 59, 30, 27, 45, 18, 92, 77, 82, 64, 14, 68, 58, 114, 68, 8, 77, 42, 114, 31, 98, 129, 110, 43, 61, 159, 35, 109, 60, 91, 149, 193, 2, 38, 120, 259, 147, 135, 22, 140, 10, 263, 285, 286, 134, 308
Offset: 1

Views

Author

Ralf Stephan, Mar 24 2003

Keywords

Examples

			The 8th prime, 19, divides 2*3*5*7*11*13*17+1=510511, thus a(7)=1.

Crossrefs

Cf. A081618 (positions of unity).

Programs

  • PARI
    for(n=1, 100, p=1; forprime(k=2, prime(n), p=p*k); pn=prime(n+1); s=0; while((s+1)%pn>0, s=s+p); print1(s/p", "))
Showing 1-4 of 4 results.

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