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A078456
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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.
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6
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1, 3, 14, 92, 968, 12096, 199296, 3679488, 82607616, 2349508608, 71507128320, 2604912721920, 105300128563200, 4466750187110400, 207324589680230400, 10866166392736972800, 634672612705724006400, 38337584554108256256000
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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EXAMPLE
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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.
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MATHEMATICA
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Table[ Det[ DiagonalMatrix[ Table[ Prime[i+1]-1, {i, 1, n-1} ] ] + 1 ], {n, 1, 20} ] (* Alexander Adamchuk, Jun 02 2006 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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