A198018 Yet unseen primes occurring within the first 1,2,3,4,... digits of Pi, A000796 (ordered according to position of last, then initial digit).

3, 31, 41, 5, 314159, 14159, 4159, 59, 2, 1592653, 653, 53, 141592653589, 89, 415926535897, 5926535897, 6535897, 35897, 5897, 97, 7, 358979, 58979, 79, 589793, 9265358979323, 9323, 23, 93238462643, 462643, 643, 43, 433, 41592653589793238462643383, 89793238462643383, 38462643383, 2643383, 383, 83

Offset

1,1

Comments

Consider the first, then the first two, then the first three, ..., terms of A000796, i.e., decimal digits of Pi. Look whether the concatenation of a certain number of subsequent digits yields a prime which did not yet occur earlier (and thus necessarily involves the last of the considered digits). If so, add this prime to the sequence.

Contains A005042 as a subsequence.

Links

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

Example

The first digit of the sequence is the prime a(1)=3.
The first two digits, "3.1", yield the prime a(2)=31.
In "3.14" there are no more primes. In "3.141" there is the prime a(3)=41.
In "3.1415" there is the prime a(4)=5.
In "3.14159" we have the primes 314159, 14159, 4159 and 59.

Prog

(PARI) {my(PI=digits(Pi\.1^30), seen=[]); for(i=1, #PI-1, for(j=1, i, my(p=fromdigits(PI[j..i])); !isprime(p) || setsearch(seen, p) || print1(p", ") || seen=setunion(seen, [p])))} \\ edited to use current PARI syntax by Andrew Howroyd and M. F. Hasler, May 10 2021
(PARI) {my(a=List()); for(m=0, precision(.)-3, my(pi=Pi\.1^m, p); for(k=0, m, !isprime(p=pi%10^(m-k+1)) && setsearch(Set(a), p) && listput(a, p))); a} \\ M. F. Hasler, May 10 2021

Crossrefs

Cf. A198019 ("new" primes ordered w.r.t. their size instead of starting position).

Cf. A000796, A005042, A047658.

Sequence in context: A090151 A198187 A198019 * A211003 A068331 A177104

Adjacent sequences: A198015 A198016 A198017 * A198019 A198020 A198021

Keyword

nonn,base,dumb

Author

M. F. Hasler, Oct 20 2011

Status

approved