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A336520 Primes in Pi: a(n) is the smallest prime factor of A090897(n) that does not appear in earlier terms of A090897, or 1, if no such factor exists.

%I A336520 #26 Aug 23 2020 14:29:02
%S A336520 3,2,53,379,58979,161923,2643383,1746893,6971,5,17,1499,11,
%T A336520 1555077581737,297707,4733,37,126541,2130276389911155737,1429,71971,
%U A336520 383,61,1559,29,193,12073,698543,157,20289606809,23687,1249,59,2393,251,101,15827173,82351,661
%N A336520 Primes in Pi: a(n) is the smallest prime factor of A090897(n) that does not appear in earlier terms of A090897, or 1, if no such factor exists.
%C A336520 Inspired by a comment of _Mario Cortés_ in A090897, who suggests that 1 might not appear in this sequence.
%C A336520 Differs from A336519 for n = 4, 16, 73, 83, 90, ....
%C A336520 a(n) is not 1 for the first 2000 terms. We can prove that a(n) has a prime factor p that does not divide LCM(A090897(1), ..., A090897(n-1)) without using prime number factorization. The method is explained in the link below. - _David A. Corneth_, Aug 22 2020
%H A336520 Peter Luschny, <a href="/A336520/a336520.txt">Prime factorization for n = 1..100</a>.
%H A336520 David A. Corneth, <a href="/A336520/a336520_1.gp.txt">Explanation of a method to determine the primality of a(n). Comes with example and PARI program. </a>
%e A336520 [ 1] 3,          {3}                  -> 3;
%e A336520 [ 2] 14,         {2, 7}               -> 2;
%e A336520 [ 3] 159,        {3, 53}              -> 53;
%e A336520 [ 4] 2653,       {7, 379}             -> 379;
%e A336520 [ 5] 58979,      {58979}              -> 58979;
%e A336520 [ 6] 323846,     {2, 161923}          -> 161923;
%e A336520 [ 7] 2643383,    {2643383}            -> 2643383;
%e A336520 [ 8] 27950288,   {2, 1746893}         -> 1746893;
%e A336520 [ 9] 419716939,  {6971, 60209}        -> 6971;
%e A336520 [10] 9375105820, {2, 5, 1163, 403057} -> 5.
%o A336520 (SageMath)
%o A336520 def Select(item, Selected):
%o A336520     return next((x for x in item if not (x in Selected)), 1)
%o A336520 def PiPart(n):
%o A336520     return floor(pi * 10^(n * (n + 1) // 2 - 1)) % 10^n
%o A336520 def A336520List(len):
%o A336520     prev = []; ret = []
%o A336520     for n in range(1, len + 1):
%o A336520         p = prime_factors(PiPart(n))
%o A336520         ret.append(Select(p, prev))
%o A336520         prev.extend(p)
%o A336520     return ret
%o A336520 print(A336520List(39))
%o A336520 # Query function of _David A. Corneth_ to determine if a(n) is prime.
%o A336520 def LcmPiPart(n):
%o A336520     return lcm([PiPart(n) for n in (1..n)])
%o A336520 def is_an_prime(n):
%o A336520     lcmpi = LcmPiPart(n - 1)
%o A336520     lm, m = 1, PiPart(n)
%o A336520     while lm != m:
%o A336520         lm, m = m, lcm(lcmpi, m) // lcmpi
%o A336520     return m > 1
%Y A336520 Cf. A090897, A336519 (variant).
%K A336520 nonn,base
%O A336520 1,1
%A A336520 _Peter Luschny_, Aug 22 2020