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.

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).

A145781 Least m >= 0 which when added to primorial(n) yields a square.

Original entry on oeis.org

2, 3, 6, 15, 91, 246, 715, 3535, 21099, 95995, 175470, 4468006, 31516774, 192339970, 212951314, 5138843466, 76699112491, 103728576730, 3051100701874, 14417674958635, 245230361204214, 2296196521511806, 10940476546738414
Offset: 1

Views

Author

Alexander R. Povolotsky, Oct 18 2008

Keywords

Examples

			a(1)= 4 - 2 = 2, a(2)= 9 - 2*3 = 3, a(3)= 36 - 2*3*5 = 6, a(4)= 225 - 2*3*5*7 = 15, and a(5)= 2401 - 2*3*5*7*11 = 91.

Links

Crossrefs

Cf. A002110, A216144, A215658, A215659.

Programs

  • Mathematica
    (IntegerPart[Sqrt[#]]+1)^2-#&/@Rest[FoldList[Times,1,Prime[Range[30]]]]  (* Harvey P. Dale, Jan 14 2011 *)
  • PARI
    j=[];for (n=1,30,j=concat(j,(ceil(sqrt(prod(i=1,n,prime(i))))^2 -prod(i=1,n,prime(i)))));j

Formula

a(n) = A216144(n)^2 - A002110(n). - Jonathan Sondow, Sep 02 2012

Extensions

Definition shortened and clarified by Jonathan Sondow, Sep 02 2012

A161620 Primorial numbers of the form n^2 + n for some integer n.

Original entry on oeis.org

2, 6, 30, 210, 510510
Offset: 1

Views

Author

Daniel Tisdale, Jun 14 2009

Keywords

Comments

Primorial numbers m such that 4m+1 is a square.
Intersection of the sequences A002110 and A002378.
If it exists, a(6) > A034386(10^11). - Max Alekseyev, Oct 23 2011
The form is n^2 + n = n(n + 1), and the values n + 1 constitute A215659. - Jeppe Stig Nielsen, Mar 27 2018

Examples

			2 = 1*2 = 2
2*3 = 2*3 = 6
2*3*5 = 5*6 = 30
2*3*5*7 = 14*15 = 210
2*3*5*7*11*13*17 = 714*715 = 510510

Links

  • C. Nelson, D. E. Penney, and C. Pomerance, 714 and 715, J. Recreational Mathematics (1974) 7(2), 87-89.

Crossrefs

Cf. A002110, A002378, A215658, A215659.

Programs

  • Mathematica
    p=1; Do[p=p*Prime[c]; f=Floor[Sqrt[p]]; If[p==f*(f+1), Print[p]],{c,1000}]
  • PARI
    N=10^8;si=30;q=vector(si,i,nextprime(i*N));a=vector(si,i,1);forprime(p=2,N,for(i=1,si,a[i]=(a[i]*p)%q[i]);v=1;for(i=1,si,if(kronecker(4*a[i]+1,q[i])==-1,v=0;break));if(v,T=1;forprime(r=2,p,T*=r);print1(T",")))
  • PARI
    pr=1;forprime(p=2,,pr*=p;s=sqrtint(pr);s*(s+1)==pr&&print1(pr,", ")) \\ Jeppe Stig Nielsen, Mar 27 2018

Formula

a(n) = A034386(A215658(n)). - Jeppe Stig Nielsen, Mar 27 2018

Extensions

Edited by Hans Havermann, Dec 02 2010
Edited by Max Alekseyev, Dec 03 2010
Edited by Robert Gerbicz, Dec 04 2010

A216144 Square root of smallest square greater than the product of first n primes.

Original entry on oeis.org

2, 3, 6, 15, 49, 174, 715, 3115, 14937, 80435, 447840, 2724104, 17442772, 114379900, 784149082, 5708691486, 43849291331, 342473913400, 2803269796342, 23620771158595, 201815957246322, 1793779464521956, 16342108667160302, 154171144824008980, 1518409682511777987
Offset: 1

Views

Author

Michel Marcus, Sep 02 2012

Keywords

Comments

Known values such that a(n)=A145781(n) are a(n)=2,3,6,15 and 715, i.e. for primes p=2,3,5,7 and 17.
(The relation a(n)=A145781(n) means that a(n)(a(n)-1) is a primorial number.) - M. F. Hasler, Sep 02 2012, - corrected by Jonathan Sondow, Sep 02 2012

Examples

			a(2) = sqrt(2*3 + A145781(2))= sqrt(2*3 + 3) = sqrt(9) = 3.

Links

Crossrefs

Cf. A002110, A060797, A145781, A215658, A215659.

Programs

  • PARI
    j=[];for (n=1, 30, p = prod(i=1, n, prime(i)); j=concat(j, floor(sqrt((ceil(sqrt(p))^2))));); j
  • PARI
    A216144(n)=sqrtint(prod(k=1,n,prime(k)))+1 \\ - M. F. Hasler, Sep 02 2012

Formula

a(n)=sqrt(A002110(n) + A145781(n)).
a(n)=A060797(n)+1. - M. F. Hasler, Sep 02 2012
Showing 1-4 of 4 results.

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