cp's OEIS Frontend

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.

%I A078456 #36 Sep 19 2022 06:41:22
%S A078456 1,3,14,92,968,12096,199296,3679488,82607616,2349508608,71507128320,
%T A078456 2604912721920,105300128563200,4466750187110400,207324589680230400,
%U A078456 10866166392736972800,634672612705724006400,38337584554108256256000
%N 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.
%C A078456 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
%C A078456 Second column of A096294. - _Eric Desbiaux_, Jun 20 2013
%H A078456 Charles R Greathouse IV, <a href="/A078456/b078456.txt">Table of n, a(n) for n = 1..350</a>
%F A078456 a(n) = (prime(n)-1)*a(n-1) + A005867(n). - _Matthew Vandermast_, Jun 06 2004
%F A078456 a(n) = A120071(n) * A135212(n). - _Alexander Adamchuk_, Nov 23 2007
%F A078456 a(n) = A117494(A002110(n)). - _Ridouane Oudra_, Sep 18 2022
%e A078456 a(2)=3 since 2*3=6 and 2,3,4 have 1 prime factor among (2,3)
%e A078456 3 1 1 1 1 ...
%e A078456 1 5 1 1 1 ...
%e A078456 1 1 7 1 1 ...
%e A078456 1 1 1 11 1 ...
%e A078456 1 1 1 1 13 ...
%e A078456 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.
%t A078456 Table[ Det[ DiagonalMatrix[ Table[ Prime[i+1]-1, {i, 1, n-1} ] ] + 1 ], {n, 1, 20} ] (* _Alexander Adamchuk_, Jun 02 2006 *)
%o A078456 (PARI) a(n)=sum(k=1,prod(i=1,n, prime(i)),if(isprime(gcd(k,prod(i=1,n, prime(i)))),1,0))
%o A078456 (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
%Y A078456 Cf. A135212, A120271.
%Y A078456 Cf. A117494, A002110.
%K A078456 nonn
%O A078456 1,2
%A A078456 _Benoit Cloitre_, Dec 31 2002
%E A078456 a(7) from _Ralf Stephan_, Mar 25 2003
%E A078456 a(8)-a(12) from _Matthew Vandermast_, Jun 06 2004
%E A078456 More terms from _Alexander Adamchuk_, Jun 02 2006