A371895 - OEIS

%I #19 Jun 07 2024 14:09:23

%S 2,15,5,18,22,22,25,24,26,29,21,31,32,31,98,96,27,34,88,355,13,36,21,

%T 79,39,39,189,383,376,371,41,41,416,41,426,42,425,43,43,433,419,431,

%U 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

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

%C From _Robert Israel_, Jun 06 2024: (Start)

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

%C 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).

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

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

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

%H Robert Israel, <a href="/A371895/b371895.txt">Table of n, a(n) for n = 1..10000</a>

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

%p f:= proc(n) local t,k;

%p   t:= convert(ithprime(n),string);

%p   for k from 1 do

%p      if StringTools:-Search(t, convert(n*k,string)) > 0 then return k fi

%p   od

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Jun 05 2024

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

%o (Python)

%o from sympy import prime

%o from itertools import count

%o def a(n): t=str(prime(n)); return next(k for k in count(1) if t in str(n*k))

%o print([a(n) for n in range(1, 71)]) # _Michael S. Branicky_, Apr 11 2024

%o (Python) # faster for initial segment of sequence

%o from sympy import nextprime

%o from itertools import count, islice

%o def agen(): # generator of terms

%o     pn = 2

%o     for n in count(1):

%o         t = str(pn)

%o         yield next(k for k in count(1) if t in str(n*k))

%o         pn = nextprime(pn)

%o print(list(islice(agen(), 70))) # _Michael S. Branicky_, Apr 11 2024

%o (PARI) a(n) = my(k=1, s=Str(prime(n))); while(#strsplit(Str(k*n), s) < 2, k++); k; \\ _Michel Marcus_, Apr 11 2024

%Y Cf. A000040, A086064.

%K nonn,base,look

%O 1,1

%A _Giorgos Kalogeropoulos_, Apr 11 2024