A283425 Difference between A002110(n) and the largest semiprime b*c < A002110(n) where b is prime(n+1).

1, 61, 127, 113, 199, 191, 701, 233, 457, 241, 3701, 557, 3673, 421, 499, 947, 2437, 4349, 8513, 4951, 3229, 937, 4813, 881, 6863, 1499, 2803, 12497, 2029, 88493, 5857, 10853, 28627, 9551, 43691, 85049, 15973, 75209, 4933, 5009, 22613, 14731, 74489, 16993, 90887, 307, 3581, 15083, 12893, 71317, 3583, 1907

Offset: 4

Author

Jamie Morken, May 14 2017

Keywords

• nonn

Comments

Only these 6 values are not prime numbers up to n=499: 1, 590221, 2807627, 5862793, 39109337, 13116283.

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.

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

Examples

			Sequence starts at n=4.
For n=5, a(n)=61.
Pn(5): a=2310, b=13, c=173, d=61.
I.e., d = a - (b*c) = 2310 - (13*173) = 2310 - 2249 = 61.
Pn(4): a=210, b=11, c=19, d=1,
Pn(5): a=2310, b=13, c=173, d=61,
Pn(6): a=30030, b=17, c=1759, d=127,
Pn(7): a=510510, b=19, c=26863, d=113,
Pn(8): a=9699690, b=23, c=421717, d=199,
Pn(9): a=223092870, b=29, c=7692851, d=191.
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).
		

Crossrefs

Cf. A000040, A001358, A002110, A285784, A285905.

Programs

Mathematica

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 *)

Formula

a(n) = A002110(n) - A000040(n+1)*prevprime(A002110(n)/A000040(n+1)) for n >= 4. - Michael De Vlieger, May 15 2017