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.

3, 2, 53, 379, 58979, 161923, 2643383, 1746893, 6971, 5, 17, 1499, 11, 1555077581737, 297707, 4733, 37, 126541, 2130276389911155737, 1429, 71971, 383, 61, 1559, 29, 193, 12073, 698543, 157, 20289606809, 23687, 1249, 59, 2393, 251, 101, 15827173, 82351, 661

Offset

1,1

Comments

Inspired by a comment of Mario Cortés in A090897, who suggests that 1 might not appear in this sequence.

Differs from A336519 for n = 4, 16, 73, 83, 90, ....

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

Links

Table of n, a(n) for n=1..39.

Peter Luschny, Prime factorization for n = 1..100.

David A. Corneth, Explanation of a method to determine the primality of a(n). Comes with example and PARI program.

Example

[ 1] 3,          {3}                  -> 3;
[ 2] 14,         {2, 7}               -> 2;
[ 3] 159,        {3, 53}              -> 53;
[ 4] 2653,       {7, 379}             -> 379;
[ 5] 58979,      {58979}              -> 58979;
[ 6] 323846,     {2, 161923}          -> 161923;
[ 7] 2643383,    {2643383}            -> 2643383;
[ 8] 27950288,   {2, 1746893}         -> 1746893;
[ 9] 419716939,  {6971, 60209}        -> 6971;
[10] 9375105820, {2, 5, 1163, 403057} -> 5.

Prog

(SageMath)
def Select(item, Selected):
    return next((x for x in item if not (x in Selected)), 1)
def PiPart(n):
    return floor(pi * 10^(n * (n + 1) // 2 - 1)) % 10^n
def A336520List(len):
    prev = []; ret = []
    for n in range(1, len + 1):
        p = prime_factors(PiPart(n))
        ret.append(Select(p, prev))
        prev.extend(p)
    return ret
print(A336520List(39))
# Query function of David A. Corneth to determine if a(n) is prime.
def LcmPiPart(n):
    return lcm([PiPart(n) for n in (1..n)])
def is_an_prime(n):
    lcmpi = LcmPiPart(n - 1)
    lm, m = 1, PiPart(n)
    while lm != m:
        lm, m = m, lcm(lcmpi, m) // lcmpi
    return m > 1

Crossrefs

Cf. A090897, A336519 (variant).

Sequence in context: A104254 A336519 A231576 * A190286 A063513 A265635

Adjacent sequences: A336517 A336518 A336519 * A336521 A336522 A336523

Keyword

nonn,base

Author

Peter Luschny, Aug 22 2020

Status

approved