1,1

This is a "Proof of existence of infinite primes" sequence. Proof. Let N = (Product_{0<=e_i<=1} (Product_{1<=i<=n} p_i^e_i + Product_{1<=i<=n} p_i^(1-e_i)))^(1/2) . Suppose there are only a finite number of primes p_i, 1<=i<=n. If N is prime, then for all i, not (N=p_i). Because, for all i, p_i<N. If N is composite, then it must have a prime divisor p which is different from primes p_i. Because, for all i, not(N=0 mod p_i).

The numbers of decimal digits of a(n) are 1, 2, 5, 15, 39, 100, 246, 590, 1387, 3215, 7321, 16507, 36823, 81305, 178212, 388495, 842638, 1816984, ..., . - Robert G. Wilson v.

The numbers of prime factors of a(n) are 1, 2, 4, 8, 16, 33, 69, 136, 280, 566, 1107, ..., . - Robert G. Wilson v.

a(3)= ((1+p_1*p_2*p_3)*(p_3+p_1*p_2)*(p_2+p_1*p_3)*(p_2*p_3+p_1)*(p_1+p_2*p_3)*(p_1*p_3+p_2)*(p_1*p_2+p_3)*(p_1*p_2*p_3+1))^(1/2)

= (1+p_1*p_2*p_3)*(p_3+p_1*p_2)*(p_2+p_1*p_3)*(p_2*p_3+p_1) = 31*11*13*17.

f[n_] := Block[{a = 1, p = Prime@Range@n, k = 0, lmt = 2^(n - 1)}, While[k < lmt, e = IntegerDigits[k, 2, n]; a = a*(Times @@ (p^e) + Times @@ (p^(1 - e))); k++ ]; a]; Array[f, 7] (* Robert G. Wilson v *)

Cf. A111392.

Adjacent sequences: A112401 A112402 A112403 this_sequence A112405 A112406 A112407

Sequence in context: A132513 A034174 A119526 this_sequence A105758 A113799 A072682

nonn

Yasutoshi Kohmoto (zbi74583(AT)boat.zero.ad.jp)

Edited by Robert G. Wilson v (rgwv(at)rgwv.com), Dec 10 2005