A153031 -id:A153031 - cp's OEIS frontend

A242973 Positions in both e and Pi where both digits in the same position are prime.

1, 5, 9, 16, 17, 18, 25, 29, 30, 34, 40, 54, 64, 65, 74, 77, 84, 90, 92, 94, 100, 103, 112, 115, 124, 132, 136, 137, 138, 143, 144, 159, 178, 179, 180, 195, 204, 211, 217, 236, 242, 253, 275, 283, 286, 293, 302, 303, 305, 307, 317, 321, 326, 334, 339, 344, 347

Offset: 1

Philip Mizzi, May 28 2014

			Pi = 3.1415926535897932384626...
.....|....|...|......|||........
_e = 2.7182818284590452353602...

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A052055, A073302, A153031.

Module[{digs=350,p,e,th},p=RealDigits[Pi,10,digs][[1]];e=RealDigits[E,10,digs][[1]];th = Thread[{p,e}];Position[If[AllTrue[#,PrimeQ],1,0]&/@th,1]]//Flatten (* Harvey P. Dale, Jan 28 2023 *)

\p 1000
e=Vec(Str(exp(1)/10)); p=Vec(Str(Pi/10)); for(n=1, #e-9, if(isprime(eval(e[n+2])) && isprime(eval(p[n+2])), print1(n", "))) \\ Jens Kruse Andersen, Jul 23 2014

Definition clarified by Harvey P. Dale, Jan 28 2023

A242975 Positions in e and Pi where the digit at each position is equal and prime.

Original entry on oeis.org

17, 18, 34, 40, 100, 143, 275, 326, 334, 365, 412, 420, 453, 501, 504, 507, 610, 622, 642, 743, 825, 840, 841, 864, 866, 875, 878, 898, 920, 926, 941, 948, 956, 963, 1009, 1054, 1059, 1078, 1147, 1158, 1180, 1203, 1283, 1292, 1306, 1338, 1355, 1362, 1407, 1469

Offset: 1

Philip Mizzi, May 28 2014

			Pi = 3.1415926535897932384626...
......................||........
_e = 2.7182818284590452353602...

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A052055, A153031, A242973.

Module[{nn=1500,th},th=Thread[{RealDigits[Pi,10,nn][[1]],RealDigits[ E,10,nn] [[1]]}];Position[th,?(#[[1]]==#[[2]]&&PrimeQ[#[[1]]]&), 1,Heads->False]]//Flatten (* _Harvey P. Dale, Aug 10 2017 *)

\p 1500
e=Vec(Str(exp(1))); p=Vec(Str(Pi)); for(i=3, #e-9, if(e[i]==p[i] && isprime(eval(e[i])), print1(i-1", "))) \\ Jens Kruse Andersen, Jul 23 2014

A343422 Number of digits of earliest prime encountered at each digit n of the decimal expansion of Pi.

Original entry on oeis.org

1, 5, 2, 7, 1, 13, 1, 3, 1, 1, 1, 2, 2, 1, 4, 1, 1, 1, 3057, 6, 3490, 1, 3, 2, 1, 1, 2, 1, 1, 1, 20, 1, 1, 1, 9, 4, 2, 2, 2, 1, 4, 7, 6329, 1, 53, 3, 1, 1, 1, 19128, 1, 1, 4, 1, 2, 2, 1, 12, 39, 45, 35, 1, 30, 1, 1, 1, 1, 4834, 24, 341, 86, 127, 127, 1, 143

Offset: 1

Bill McEachen, Aug 21 2021

The underlying approach is an alternate way to spawn primes from Pi (and other irrational values) compared to A005042. Generally speaking, there should be a prime for every known digit (sequence is likely infinite, use -1 for any term without solution). By its construction, every prime will not be encountered, and primes will be repeated, especially 2,3,5 and 7. Large primes will be seen within the prime sequence. Note that concatenations with leading 0 will duplicate that of the subsequent concatenation having nonzero leading digit.

The corresponding primes are: 3, 14159, 41, 1592653, 5, 9265358979323, 2, 653, 5, 3, 5, 89, 97, 7, 9323, 3, 2, 3, ....

			The first term is the trivial prime 3, having length=1 digit, so a(1)=1.
The next evaluation starts at digit 1:  1 is not prime, 14 is composite, 141 is composite, 1415 is composite, but 14159 is prime, so a(2)=5.
The next evaluation starts at digit 4:  4 is composite, 41 is prime, so a(3)=2.
The 33rd and 34th digits of Pi are 0 and 2, and "02" converts to 2, a 1-digit prime.  Thus, a(33) = 1.

RosettaCode, Digits of Pi

Cf. A005042, A073264, A153031.

lista(p) = {default(realprecision, p); my(x=Pi, nb=#Str(x), d=digits(floor(x*10^(nb-1)))); for (i=1, #d, my(k=i, j=d[i]); while (! ispseudoprime(j), k++; if (k>#d, j=0; break, j = 10*j+d[k])); if (j==0, break, print1(#Str(j), ", ")););} \\ Michel Marcus, Sep 15 2021

from sympy import S, isprime
pi_digits = str(S.Pi.n(10**5+1)).replace(".", "")[:-1]
def a(n):
    s, k = pi_digits[n-1], 1
    while not isprime(int(s)):
        s, k = s + pi_digits[n-1+k], k + 1
    return len(str(int(s)))
print([a(n) for n in range(1, 19)]) # Michael S. Branicky, Aug 21 2021

a(A153031(n)) = 1. - Michel Marcus, Aug 22 2021

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A242973 Positions in both e and Pi where both digits in the same position are prime.

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A242975 Positions in e and Pi where the digit at each position is equal and prime.

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Author

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Examples

Links

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Mathematica

PARI

A343422 Number of digits of earliest prime encountered at each digit n of the decimal expansion of Pi.

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