A246373 - OEIS

%I #14 Aug 26 2014 01:28:18

%S 2,3,7,19,29,31,37,47,67,71,79,89,97,101,103,107,109,127,139,151,157,

%T 181,191,197,199,211,223,227,229,241,251,269,271,277,283,307,317,331,

%U 337,349,359,367,373,379,397,409,421,433,439,457,461,467,487,499,521,541,547,569,571,577,601

%N Primes p such that if 2p-1 = product_{k >= 1} A000040(k)^(c_k), then p <= product_{k >= 1} A000040(k-1)^(c_k).

%C Primes p such that A064216(p) >= p, or equally, A064989(2p-1) >= p.

%C All primes of A005382 are present here, because if 2p-1 is prime q, Bertrand's postulate guarantees (after cases 2 and 3 which are in A048674) that there exists at least one prime r larger than p and less than q = 2p-1, for which A064989(q) = r.

%H Antti Karttunen, <a href="/A246373/b246373.txt">Table of n, a(n) for n = 1..10000</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bertrand%27s_postulate">Bertrand's postulate</a>

%e 2 is present, as 2*2 - 1 = 3 = p_2, and p_{2-1} = p_1 = 2 >= 2.

%e 3 is present, as 2*3 - 1 = 5 = p_3, and p_{3-1} = p_2 = 3 >= 3.

%e 5 is not present, as 2*5 - 1 = 9 = p_2 * p_2, and p_1 * p_1 = 4, with 4 < 5.

%e 7 is present, as 2*7 - 1 = 13 = p_6, and p_5 = 11 >= 7.

%o (PARI)

%o A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};

%o n = 0; forprime(p=2,2^31, if((A064989((2*p)-1) >= p), n++; write("b246373.txt", n, " ", p); if(n > 9999, break)));

%o (Scheme, with _Antti Karttunen_'s IntSeq-library)

%o (define A246373 (MATCHING-POS 1 1 (lambda (n) (and (prime? n) (>= (A064216 n) n)))))

%Y Intersection of A000040 and A246372.

%Y Subsequence: A005382.

%Y A246374 gives the primes not here.

%Y Cf. A064216, A064989, A246281.

%K nonn

%O 1,1

%A _Antti Karttunen_, Aug 25 2014