A078456 Number of numbers less than prime(1)*...*prime(n) having exactly one prime factor among (prime(1),...,prime(n)) where prime(n) is the n-th prime.

1, 3, 14, 92, 968, 12096, 199296, 3679488, 82607616, 2349508608, 71507128320, 2604912721920, 105300128563200, 4466750187110400, 207324589680230400, 10866166392736972800, 634672612705724006400, 38337584554108256256000

Offset: 1

Author

Benoit Cloitre, Dec 31 2002

Keywords

• nonn

Comments

For n>1 a(n) is the determinant of the (n-1) X (n-1) matrix with elements M[i,j] = Prime[i+1] if i=j and 1 otherwise. (See example lines.) - Alexander Adamchuk, Jun 02 2006

Second column of A096294. - Eric Desbiaux, Jun 20 2013

Examples

			a(2)=3 since 2*3=6 and 2,3,4 have 1 prime factor among (2,3)
3 1 1 1 1 ...
1 5 1 1 1 ...
1 1 7 1 1 ...
1 1 1 11 1 ...
1 1 1 1 13 ...
and so a(2) = 3, a(3) = 3*5 - 1*1 = 14, a(4) = 3*5*7 + 1*1*1 + 1*1*1 - 7*1*1 - 5*1*1 - 3*1*1 = 92, etc.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..350

Crossrefs

Cf. A135212, A120271.

Cf. A117494, A002110.

Programs

Mathematica

Table[ Det[ DiagonalMatrix[ Table[ Prime[i+1]-1, {i, 1, n-1} ] ] + 1 ], {n, 1, 20} ] (* Alexander Adamchuk, Jun 02 2006 *)

PARI

a(n)=sum(k=1,prod(i=1,n, prime(i)),if(isprime(gcd(k,prod(i=1,n, prime(i)))),1,0))

PARI

a(n) = matdet(matrix(n-1, n-1, j, k, if (j==k, prime(j+1), 1))); \\ after Mathematica; Michel Marcus, Oct 02 2016

Formula

a(n) = (prime(n)-1)*a(n-1) + A005867(n). - Matthew Vandermast, Jun 06 2004

a(n) = A120071(n) * A135212(n). - Alexander Adamchuk, Nov 23 2007

a(n) = A117494(A002110(n)). - Ridouane Oudra, Sep 18 2022

Extensions

a(7) from Ralf Stephan, Mar 25 2003

a(8)-a(12) from Matthew Vandermast, Jun 06 2004

More terms from Alexander Adamchuk, Jun 02 2006