A215658 -id:A215658 - cp's OEIS frontend

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

Jonathan Sondow, Sep 07 2012

Values of k in A215658.

See A161620 for the primorial values. - Jeppe Stig Nielsen, Mar 27 2018

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

Cf. A215658, A161620.

Select[Range[10^5], Product[Prime@ i, {i, PrimeNu@ #}] == # &[# (# - 1)] &] (* Michael De Vlieger, Apr 10 2018 *)

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

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.

Name improved by Jeppe Stig Nielsen, Mar 27 2018

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

Alexander R. Povolotsky, Oct 18 2008

nonn

			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.

G. C. Greubel, Table of n, a(n) for n = 1..630
C. Aebi and G. Cairns, Partitions of primes, Parabola 45, Issue 1 (2009); see the table on p. 5. - _Jonathan Sondow_, Jun 21 2012

Cf. A002110, A216144, A215658, A215659.

(IntegerPart[Sqrt[#]]+1)^2-#&/@Rest[FoldList[Times,1,Prime[Range[30]]]]  (* Harvey P. Dale, Jan 14 2011 *)

j=[];for (n=1,30,j=concat(j,(ceil(sqrt(prod(i=1,n,prime(i))))^2 -prod(i=1,n,prime(i)))));j

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

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

Daniel Tisdale, Jun 14 2009

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

			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

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

Cf. A002110, A002378, A215658, A215659.

p=1; Do[p=p*Prime[c]; f=Floor[Sqrt[p]]; If[p==f*(f+1), Print[p]],{c,1000}]

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",")))

pr=1;forprime(p=2,,pr*=p;s=sqrtint(pr);s*(s+1)==pr&&print1(pr,", ")) \\ Jeppe Stig Nielsen, Mar 27 2018

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

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

Michel Marcus, Sep 02 2012

nonn

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

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

C. Aebi and G. Cairns, Partitions of primes, Parabola 45, Issue 1 (2009); see the table on p. 5.

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

j=[];for (n=1, 30, p = prod(i=1, n, prime(i)); j=concat(j, floor(sqrt((ceil(sqrt(p))^2))));); j

A216144(n)=sqrtint(prod(k=1,n,prime(k)))+1 \\ - M. F. Hasler, Sep 02 2012

a(n)=sqrt(A002110(n) + A145781(n)).

a(n)=A060797(n)+1. - M. F. Hasler, Sep 02 2012

A295266 Positive integers whose squares can be represented as the sum or difference of 3-smooth numbers.

Original entry on oeis.org

1, 2, 3, 5, 7, 17

Offset: 1

Tomohiro Yamada, Nov 19 2017

In Chapter 7 of de Weger's tract, it is shown that there are no other terms.

More generally, de Weger exposited how one can determine all squares which can be represented as the sum or difference of k-smooth numbers for any given k and determined all integers whose squares can be represented as the sum or difference of 7-smooth numbers, among which the largest one is 14117^2 = 199289869 = 3^13 * 5^3 - 2 * 7^3.

			a(6) = 17 ; 17^2 = 288 + 1 = 2^5 * 3^2 + 1.

B. M. M. de Weger, Algorithms for Diophantine Equations, Centrum voor Wiskunde en Informatica, Amsterdam, 1989.

Cf. A003586 (3-smooth numbers).

Coincides with A192579 and A215658 except the term 1.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A295266 Positive integers whose squares can be represented as the sum or difference of 3-smooth numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs