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.

A215658 Primes p such that the smallest positive integer k for which p# + k is square satisfies p# + k = k^2, where p# = 2*3*5*7*11*...*p is a primorial.

Original entry on oeis.org

2, 3, 5, 7, 17
Offset: 1

Views

Author

Jonathan Sondow, Sep 02 2012

Keywords

Comments

The corresponding values of k are 2, 3, 6, 15, 715 = A215659.
The equation p# + k = k^2 has an integer solution k if and only if 1 + 4*p# is a square.
Conjecture: Not the same sequence as A192579, which is finite.
When p is in this sequence, p# = k(k-1) is in A161620, the intersection of A002110 and A002378. - Jeppe Stig Nielsen, Mar 27 2018

Examples

			The smallest square > 17# = 510510 is 715^2 = 17# + 715, so 17 is a member.

Links

Crossrefs

Cf. A002110, A060797, A145781, A216144, A215659, A161620.

Programs

  • Mathematica
    t = {}; pm = 1; Do[pm = pm*p; s = Floor[Sqrt[pm]]; If[pm == s*(s+1), AppendTo[t, p]], {p, Prime[Range[100]]}]; t (* T. D. Noe, Sep 05 2012 *)
  • PARI
    for (n=1, 10, if (ceil(sqrt(prod(i=1, n, prime(i))))^2 - prod(i=1, n, prime(i)) - ceil(sqrt(prod(i=1, n, prime(i)))) == 0, print(prime(n)));); \\ Michel Marcus, Sep 05 2012
  • Python
    from sympy import primorial, integer_nthroot, prime
A215658_list = [prime(i) for i in range(1,10**2) if integer_nthroot(4*primorial(i)+1,2)[1]] # Chai Wah Wu, Apr 01 2021

Formula

A145781(n) = A216144(n) if and only if prime(n) is a member.
a(n)# = A215659(n)*(A215659(n)-1).

A215659 Values of k such that k*(k - 1) is a primorial number.

Original entry on oeis.org

2, 3, 6, 15, 715
Offset: 1

Views

Author

Jonathan Sondow, Sep 07 2012

Keywords

Comments

Values of k in A215658.
See A161620 for the primorial values. - Jeppe Stig Nielsen, Mar 27 2018

Links

  • Carol Nelson, David E. Penney and Carl Pomerance, 714 and 715, J. Recreational Math. 7:2 (1994), pp. 87-89.

Crossrefs

Cf. A215658, A161620.

Programs

  • Mathematica
    Select[Range[10^5], Product[Prime@ i, {i, PrimeNu@ #}] == # &[# (# - 1)] &] (* Michael De Vlieger, Apr 10 2018 *)
  • Python
    from sympy import primorial, integer_nthroot
A215659_list = []
for i in range(1,10**2):
    a, b = integer_nthroot(4*primorial(i)+1,2)
    if b:
        A215659_list.append((a+1)//2) # Chai Wah Wu, Apr 01 2021

Formula

a(n) * (a(n) - 1) = A215658(n)#, where p# = 2 * 3 * 5 * 7 * 11 * ... * p is a primorial, the product of the primes from 2 to p.

Extensions

Name improved by Jeppe Stig Nielsen, Mar 27 2018

A374609 Numbers k such that k*(k+1) is primorial (a term in A002110).

Original entry on oeis.org

1, 2, 5, 14, 714
Offset: 1

Views

Author

David James Sycamore and Michael De Vlieger, Jul 13 2024

Keywords

Comments

All terms are squarefree numbers (A005117).
a(n) >> exp(n/2); presumably finite. If a(6) exists then a(6) > 10^200000. - Charles R Greathouse IV, Jul 13 2024

Examples

			714 is a term because 714*715 = 510510 = A002110(7) = 2*3*5*7*11*13*17.

Crossrefs

Subsequence of A141399 and A007674.
Cf. A000194, A002110, A002378, A005117, A161620 (resulting primorials).

Programs

  • Mathematica
    nn = 1000; MapIndexed[Set[P[First[#2]], #1] &, FoldList[Times, Prime@ Range[nn]]]; Monitor[Reap[Do[(d = Floor@ Sqrt[#]; If[Divisible[#, d], Sow[d]]) &[P[i]], {i, nn}]][[-1, 1]], i] (* Michael De Vlieger, Jul 14 2024 *)

Formula

From Alois P. Heinz, Jul 13 2024: (Start)
a(n) = A000194(A161620(n)).
A002378(a(n)) = A161620(n). (End)
Showing 1-3 of 3 results.

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