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

2, 6, 30, 210, 510510

Offset

1,1

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

Links

Table of n, a(n) for n=1..5.

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

Formula

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

Example

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

Mathematica

p=1; Do[p=p*Prime[c]; f=Floor[Sqrt[p]]; If[p==f*(f+1), Print[p]], {c, 1000}]
Intersection[Table[n^2+n, {n, 750}], FoldList[Times, Prime[Range[10]]]] (* Harvey P. Dale, Nov 25 2025 *)

Prog

(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

Crossrefs

Cf. A002110, A002378, A215658, A215659.

Sequence in context: A091456 A354411 A293756 * A333508 A205569 A108204

Adjacent sequences: A161617 A161618 A161619 * A161621 A161622 A161623

Keyword

nonn,hard,more

Author

Daniel Tisdale, Jun 14 2009

Extensions

Edited by Hans Havermann, Dec 02 2010

Edited by Max Alekseyev, Dec 03 2010

Edited by Robert Gerbicz, Dec 04 2010

Status

approved