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

2, 6, 30, 210, 510510

Offset: 1

Author

Daniel Tisdale, Jun 14 2009

Keywords

• nonn
• hard
• more

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