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%I #37 Aug 30 2022 20:39:48
%S 1,61,127,113,199,191,701,233,457,241,3701,557,3673,421,499,947,2437,
%T 4349,8513,4951,3229,937,4813,881,6863,1499,2803,12497,2029,88493,
%U 5857,10853,28627,9551,43691,85049,15973,75209,4933,5009,22613,14731,74489,16993,90887,307,3581,15083,12893,71317,3583,1907
%N Difference between A002110(n) and the largest semiprime b*c < A002110(n) where b is prime(n+1).
%C Only these 6 values are not prime numbers up to n=499: 1, 590221, 2807627, 5862793, 39109337, 13116283.
%C All a(n) are totatives of A002110(n); thus if a(n) < b^2 in the semiprime b*c then a(n) is prime, otherwise a(n) is either prime or semiprime.
%C The number c is prevprime(p_n# / p_(n+1)), where p_n# = A002110(n). Thus semiprime b*c = A000040(n+1)*prevprime(A002110(n) / A000040(n+1)), and a(n) = A002110(n) - A000040(n+1)*prevprime(A002110(n)/A000040(n+1)). - _Michael De Vlieger_, May 15 2017
%F a(n) = A002110(n) - A000040(n+1)*prevprime(A002110(n)/A000040(n+1)) for n >= 4. - _Michael De Vlieger_, May 15 2017
%e Sequence starts at n=4.
%e For n=5, a(n)=61.
%e Pn(5): a=2310, b=13, c=173, d=61.
%e I.e., d = a - (b*c) = 2310 - (13*173) = 2310 - 2249 = 61.
%e Pn(4): a=210, b=11, c=19, d=1,
%e Pn(5): a=2310, b=13, c=173, d=61,
%e Pn(6): a=30030, b=17, c=1759, d=127,
%e Pn(7): a=510510, b=19, c=26863, d=113,
%e Pn(8): a=9699690, b=23, c=421717, d=199,
%e Pn(9): a=223092870, b=29, c=7692851, d=191.
%e a(n) = a - (b*c) where a(n) has a high probability of being prime, and b*c is the largest semiprime below A002110(n) where b is prime (n+1).
%t Table[Function[{P, q}, P - NextPrime[P/q, -1] q] @@ {Product[Prime@ i, {i, n}], Prime[n + 1]}, {n, 4, 55}] (* _Michael De Vlieger_, May 15 2017 *)
%Y Cf. A000040, A001358, A002110, A285784, A285905.
%K nonn
%O 4,2
%A _Jamie Morken_, May 14 2017
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