A371895 a(n) is the least k>0 such that n*k contains the n-th prime as a substring.

2, 15, 5, 18, 22, 22, 25, 24, 26, 29, 21, 31, 32, 31, 98, 96, 27, 34, 88, 355, 13, 36, 21, 79, 39, 39, 189, 383, 376, 371, 41, 41, 416, 41, 426, 42, 425, 43, 43, 433, 419, 431, 237, 44, 266, 433, 45, 465, 464, 458, 83, 46, 423, 417, 468, 47, 472, 468, 47, 469, 103, 473, 488, 486, 202, 481, 348, 496, 63, 499

Offset: 1

Author

Giorgos Kalogeropoulos, Apr 11 2024

Keywords

• nonn
• base
• look

Comments

From Robert Israel, Jun 06 2024: (Start)

If n is divisible by 2*10^k or 5*10^k, a(n) >= prime(n)*10^(k+1)/n.

Let n = 2^b * 5^c * k with k coprime to 10, and suppose prime(n) has d digits. If b <= c, then f(n) < 10^d * 2^(c-b). If b > c, then f(n) < 10^d * 5^(b-c).

a(10^k) = prime(10^k).

a(2*10^k) = 5 * prime(2*10^k).

a(5*10^k) = 2 * prime(5*10^k). (End)

Examples

			a(8) = 24 because 24 is the least positive integer such that 24*8 = 192 contains the prime(8) = 19.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A086064.

Programs

Maple

f:= proc(n) local t,k;
  t:= convert(ithprime(n),string);
  for k from 1 do
     if StringTools:-Search(t, convert(n*k,string)) > 0 then return k fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Jun 05 2024

Mathematica

a[n_]:=(k=1;While[!StringContainsQ[ToString[n*k],ToString@Prime@n],k++];k); Array[a,70]

PARI

a(n) = my(k=1, s=Str(prime(n))); while(#strsplit(Str(k*n), s) < 2, k++); k; \\ Michel Marcus, Apr 11 2024

Python

from sympy import prime
from itertools import count
def a(n): t=str(prime(n)); return next(k for k in count(1) if t in str(n*k))
print([a(n) for n in range(1, 71)]) # Michael S. Branicky, Apr 11 2024

Python

# faster for initial segment of sequence
from sympy import nextprime
from itertools import count, islice
def agen(): # generator of terms
    pn = 2
    for n in count(1):
        t = str(pn)
        yield next(k for k in count(1) if t in str(n*k))
        pn = nextprime(pn)
print(list(islice(agen(), 70))) # Michael S. Branicky, Apr 11 2024