A245770 Prime sieve of Pi.

1, 9, 6, 5, 93, 84626, 3, 2, 502884, 69399, 5105820, 4944, 2, 816406286, 986, 4825342, 70, 9, 480865, 2, 66, 93, 55058, 25, 4081284, 174, 270, 85, 555964462, 4895, 9644288, 7566, 344, 28475648, 8, 65, 201, 45648, 23, 4543, 393, 2602, 412, 724, 660, 55, 74881, 20962, 25, 1715364, 892, 600

Offset: 1

Author

Gil Broussard, Aug 01 2014

Keywords

• nonn
• base

Examples

			Find the first occurrence of prime 2 in the digits of Pi (only 25 digits in this illustration):
3141592653589793238462643..., and replace it with a space:
314159 653589793238462643...  Repeat the process with 3:
14159 653589793238462643...  Then 5:
141 9 653589793238462643...  Then 7:
141 9 653589 93238462643...  Then 11, 13, 17, etc., until the first occurrence of every prime is eliminated from the digits of Pi.
1   9 6  5   93  84626  ...  Then consolidate gaps between the remaining digits into a single comma:
1,9,6,5,93,84626,...         to produce the first terms in the prime sieve of Pi.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..1029 (first 422 terms from Manfred Scheucher)
• Manfred Scheucher, Sage Script (Note: should also run in pure python with a few modifications)

Crossrefs

Cf. A000796, A246022, A248804 (Prime sieve of e), A248831 (Prime sieve of sqrt(2)).

Programs

Python

def arccot(x, unity):
    sum = xpower = unity // x
    n = 3
    sign = -1
    while 1:
        xpower = xpower // (x*x)
        term = xpower // n
        if not term:
            break
        sum += sign * term
        sign = -sign
        n += 2
    return sum
def pi(digits):
    unity = 10**(digits + 10)
    pi = 4 * (4*arccot(5, unity) - arccot(239, unity))
    return pi // 10**10
def primes(n):
    """ Returns  a list of primes < n """
    sieve = [True] * n
    for i in range(3,int(n**0.5)+1,2):
        if sieve[i]:
            sieve[i*i::2*i]=[False]*((n-i*i-1)/(2*i)+1)
    return [2] + [i for i in range(3,n,2) if sieve[i]]
a = pi(370)
b = primes(100000000)
y = str(a)
for x in b:
    if str(x) in y:
        y = y.replace(str(x)," ",1)#replace first occurrence only
        
while "  " in y:
    y = y.replace("  "," ")#replace long chains of spaces with a single space
z = y.split(" ")#split terms into a list
z = filter(None, z)#remove null terms
f = map(int,z)#convert to integers
print(f[0:-1])
# David Consiglio, Jr., Jan 03 2015

Extensions

a(14) corrected by Manfred Scheucher, May 25 2015