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A078456 Number of numbers less than p(1)*p(2)*...*p(n) having exactly one prime factor among (p(1),p(2)....,p(n)) where p(n) is the n-th prime. 5
1, 3, 14, 92, 968, 12096, 199296, 3679488, 82607616, 2349508608, 71507128320, 2604912721920, 105300128563200, 4466750187110400, 207324589680230400, 10866166392736972800, 634672612705724006400, 38337584554108256256000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

LINKS

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

FORMULA

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

a(n) = Det[ DiagonalMatrix[ Table[ Prime[i+1]-1, {i, 1, n-1} ] ] + 1 ] for n>1. - Alexander Adamchuk, Jun 02 2006

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

EXAMPLE

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.

MATHEMATICA

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

PROG

(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

CROSSREFS

Cf. A135212, A120271.

Sequence in context: A183611 A259903 A101220 * A195134 A089462 A088342

Adjacent sequences:  A078453 A078454 A078455 * A078457 A078458 A078459

KEYWORD

nonn

AUTHOR

Benoit Cloitre, Dec 31 2002

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

STATUS

approved

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Last modified October 23 05:43 EDT 2018. Contains 316519 sequences. (Running on oeis4.)