A334259 Self-locating numbers within the Copeland-Erdős constant: numbers k such that the string k is at the 0-indexed position k in the decimal digits of the concatenation of the prime numbers as a decimal sequence.

37, 3790, 4991, 38073, 908979, 8378611, 62110713, 87126031, 8490820681, 9514920697, 24717215429, 784191725098, 836390891918

Offset: 1

Author

Soren Telfer, Apr 20 2020

Keywords

• nonn
• base
• more

Comments

This is inspired by the self-locating digits in Pi (A057680). Similar to A064810, these digits are 0-indexed, whereas in A057680 the sequence is 1-indexed.

The first two terms of the 1-indexed sequence are 8031711 and 648967141. - Giovanni Resta, Apr 22 2020

Examples

			37 is a term because the 3 digit of 37 appears in the 37th 0-indexed position of the Copeland-Erdős constant.
		

Crossrefs

Cf. A006880, A033308, A057680, A064810.

Programs

Mathematica

q=23; p=3; dq=2; dn=dp=1; L={}; n=-1; pP=nP=10; While[++n < 10^6, If[n == nP, nP *= 10; dn++]; While[ q pP, pP *= 10; dp++]; q = q pP + p; dq += dp]; If[n == Floor[ q/10^(dq - dn)], Print@ AppendTo[L, n]]; q = Mod[q, 10^(--dq)]]; L (* Giovanni Resta, Apr 21 2020 *)

Python

import sympy
from sympy import sieve
def digits_at(ss, n):
    ''' Extracts len(str(n)) digits at position n.'''
    t = len(str(n))
    s = ss[n:n+t]
    if s == '':
        return -1
    return int(s)
def self_locating(ss, n):
    return digits_at(ss,n) == n
SS = ""
for p in sieve.primerange(2, 100000):
    SS += str(p)
for i in range(100000):
    if self_locating(SS, i):
        print(i,end=",")

Extensions

a(3)-a(13) from Giovanni Resta, Apr 22 2020