A248804 -id:A248804 - cp's OEIS frontend

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

Gil Broussard, Aug 01 2014

			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.

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)

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

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

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

A248831 Prime sieve of the square root of 2.

Original entry on oeis.org

1, 4, 1, 6, 309, 6, 8078, 1875, 69480, 66, 7, 32, 8462, 388, 87, 2764, 27350, 846, 12, 24, 6055850, 4, 999358, 41322266, 27505, 9995050, 527820605, 470, 5, 716059, 453459686, 285, 86408, 5, 87, 4508, 627995, 8968, 396546, 808, 640620, 5, 50, 502, 599, 2

Offset: 1

Jared Kish, Oct 15 2014

Algorithm: see A245770 (prime sieve of Pi).

Manfred Scheucher, Table of n, a(n) for n = 1..431
Wikipedia, Square root of 2
Manfred Scheucher, Sage Script (Note: should also run in pure python with a few modifications)

Cf. A002193 (square root of 2), A245770 (prime sieve of Pi), A248804 (prime sieve of e).

def sqroot(a, digits):
    a = a * (10**(2*digits))
    x_prev = 0
    x_next = 1 * (10**digits)
    while x_prev != x_next:
        x_prev = x_next
        x_next = (x_prev + (a // x_prev)) >> 1
    return x_next
def 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 = sqroot(2,300)#300 digits is arbitrary - lengthen for more digits
b = primes(10000000)#make sure to scan primes up to longest term in sequence
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 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)
# David Consiglio, Jr., Jan 03 2015

More terms from David Consiglio, Jr., Dec 02 2014, Jan 03 2015

More terms from Manfred Scheucher, May 25 2015

A257835 Whole number sieve of e.

Original entry on oeis.org

28, 2, 2, 5, 6, 9, 24, 9, 9, 57, 9, 6, 6, 3, 5, 2, 6, 27, 6, 3, 0, 99, 7, 35, 6, 0, 0, 59, 30, 28, 4, 6, 33, 75, 25, 4, 0, 40, 7, 6, 8, 4, 5, 4, 7, 5, 5, 70, 0, 4, 75, 49, 2, 3, 9, 7, 3, 4, 15, 6, 25, 9, 44, 5, 0, 4, 6, 6, 9, 3, 2, 99, 35, 2, 33, 5, 69, 8, 62, 30, 8

Offset: 1

Manfred Scheucher, Jun 05 2015

Find the first occurrence of 0 (the first whole number) in the digits of e (only 35 digits in this illustration):
27182818284590452353602874713526625..., and replace it with a space:
2718281828459 452353602874713526625...  Repeat the process with the next whole number, 1:
27 8281828459 452353602874713526625...  Then 2:
7 8281828459 452353602874713526625...  Then 3:
7 8281828459 452 53602874713526625...  Then 4,5,6,7, etc., until the first occurrence of every counting number is eliminated from the digits of e.
        28      2    02      5  6  ...  Then consolidate gaps between the remaining digits into a single comma:
28,2,2,5,6,9,24,9,9,57,9,6,6,3,5,2,...  to produce the first terms in the whole number sieve of e.

Manfred Scheucher, Table of n, a(n) for n = 1..536
Manfred Scheucher, Sage Script

Cf. A001113, A247747, A248804, A258640, A258481.

A246022 Prime sieve of Champernowne constant A033307.

Original entry on oeis.org

1, 4, 6, 10, 12, 1, 516, 18, 202122, 242, 28, 30, 3, 3, 6, 3, 94041, 444546, 4849, 52535455, 58, 60, 6, 646566, 686, 0, 72, 74, 78, 8, 848, 888990, 9495, 98, 0, 102, 104, 6, 108, 1, 112, 1141, 16, 201, 22, 26, 1, 29130, 3, 4, 6, 138, 1, 411, 44, 47148, 150, 531

Offset: 1

N. J. A. Sloane, Aug 14 2014

Start with the decimal expansion of the Champernowne constant A033307:
12345678910111213141516171819202122232425262728293031323334353637383940\
   4142434445464748495051525354555657...
Find the first occurrence of the prime 2 and replace it with a space:
1 3456789101112131415161718192021222324252627282930...
1  456789101112131415161718192021222324252627282930...  Repeat the process with 3
1  4 6789101112131415161718192021222324252627282930...  Then 5
1  4 6 89101112131415161718192021222324252627282930...  Then 7
1  4 6 8910  12  141516  18  202122  24252627282930...  Then 11, 13, 17, etc., until the first occurrence of every prime is eliminated. 89, 41, 52627, will disappear.
Then consolidate gaps between the remaining digits into a single comma:
1, 4, 6, 10, 12, 1, 516, 18, 202122, 242, 28, 30, 3, 3, 6, ...

Manfred Scheucher, Table of n, a(n) for n = 1..393
Manfred Scheucher, Sage Script

Suggested by A245770, A248804, A247861. Cf. A033307, A007376.

More terms from Manfred Scheucher, Jun 05 2015

A318708 Terms resulting from application of a prime sieve to the digits of the decimal expansions of the positive integers.

Original entry on oeis.org

1, 4, 6, 8, 9, 10, 1, 1, 14, 1, 16, 1, 18, 0, 1, 4, 6, 8, 9, 0, 1, 4, 6, 8, 9, 40, 4, 4, 44, 4, 46, 4, 48, 49, 0, 1, 4, 6, 8, 9, 60, 6, 6, 64, 6, 66, 6, 68, 69, 0, 1, 4, 6, 8, 9, 80, 81, 8, 8, 84, 8, 86, 8, 88, 90, 91, 9, 9, 94, 9, 96, 9, 98, 99, 100, 10, 10, 104, 10, 106, 10, 108, 0, 1, 4

Offset: 1

Ctibor O. Zizka, Sep 01 2018

Definition of sieving over the digits of k: Erase each digit 2 in the decimal expansion of k, then consolidate the remaining digits. Erase each digit 3 in what remains from the previous step, then consolidate the remaining digits. Repeat the procedure with 5, 7, ..., largest prime <= last consolidated remainder. What remains then becomes a term of the sequence. If there are no remaining digits after the procedure, this number disappears and is not a term.
Consolidation means the removal of all empty places at each step of the sieving process. Example: k = 1225; erasing all 2's in 1225 results in 1__5, which consolidates to 15; erasing all 3's in 15 results in 15; erasing all 5's in 15 results in 1_, which consolidates to 1. So for k = 1225 the result after sieving is 1. Example: k = 10101; erasing all 2's, ..., 97's results in 10101; erasing 101's in 10101 results in ___01, which consolidates to the last consolidated remainder 01. As there is no prime <= 01 to sieve with, the result for k = 10101 after sieving is 1.
Largest number of a sieve <= last consolidated remainder.
This sequence sieve is: {primes}. There could be other sieve definitions: {binary numbers}, {even numbers}, {odd numbers}, {triangular numbers}, predefined set of numbers like {0,3,11,27}, etc.

			n = 113
p_1 = 2, no occurrence of 2 in 113
p_2 = 3, 1 occurrence of 3 in 113, erase 3, remains 11
p_3 = 5, no occurrence of 5 in 11
p_4 = 7, no occurrence of 7 in 11
p_5 = 11, 1 occurrence of 11 in 11, no remainder
number 113 disappears and is not a member of the seq.
n = 114
p_1 = 2, no occurrence of 2 in 114
p_2 = 3, no occurrence of 3 in 114
p_3 = 5, no occurrence of 5 in 114
p_4 = 7, no occurrence of 7 in 114
p_5 = 11, 1 occurrence of 11 in 114, erase 11, remains 4
number 4 is a member of the seq.

Cf. A245770, A248831, A248804, A000027, A062115.

upto[n_] := Block[{s = ToString /@ Range[n]}, Do[s = StringReplace[s, ToString[p] -> ""], {p, Prime@ Range@ PrimePi@ n}]; ToExpression@ DeleteCases[s, ""]]; upto[115] (* Giovanni Resta, Sep 01 2018 *)

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A245770 Prime sieve of Pi.

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A248831 Prime sieve of the square root of 2.

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A257835 Whole number sieve of e.

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A246022 Prime sieve of Champernowne constant A033307.

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A318708 Terms resulting from application of a prime sieve to the digits of the decimal expansions of the positive integers.

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