A344937 a(n) is the largest k such that when strings of zeros of lengths t = 1..k are inserted between every pair of adjacent digits of prime(n), the resulting numbers are all primes.

1, 1, 1, 3, 0, 0, 0, 1, 2, 0, 0, 2, 2, 1, 2, 4, 0, 1, 0, 2, 4, 0, 0, 1, 1, 2, 0, 3, 0, 0, 0, 1, 0, 0, 2, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset

5,4

Comments

Initially, except for n = 1..4, similar to A290174, but the two sequences differ from n = 28 onwards.

Links

Table of n, a(n) for n=5..91.

Example

For n = 8: prime(8) = 19 and the numbers 109, 1009 and 10009 are all prime, while 100009 is not. Thus it is possible to insert strings of zeros of lengths 1, 2 and 3 between all adjacent digits of 19 such that the resulting number is prime. Since 3 is the largest length of such a string in case of 19, a(8) = 3.

Mathematica

Table[k=0; While[PrimeQ@FromDigits@Flatten@Riffle[IntegerDigits@Prime@n, {Table[0, k]}], k++]; k-1, {n, 5, 100}] (* Giorgos Kalogeropoulos, Jun 03 2021 *)

Prog

(PARI) eva(n) = subst(Pol(n), x, 10)
insert_zeros(num, len) = my(d=digits(num), v=[]); for(k=1, #d-1, v=concat(v, concat([d[k]], vector(len)))); v=concat(v, d[#d]); eva(v)
a(n) = my(p=prime(n), ip=p); for(k=1, oo, ip=insert_zeros(p, k); if(!ispseudoprime(ip), return(k-1)))
(Python)
from sympy import isprime, prime
def insert_zeros(n, k): return int(("0"*k).join(list(str(n))))
def a(n):
  pn, k = prime(n), 1
  while isprime(insert_zeros(pn, k)): k += 1
  return k - 1
print([a(n) for n in range(5, 92)]) # Michael S. Branicky, Jun 03 2021

Crossrefs

Cf. A290174, A344637, A344936.

Sequence in context: A089313 A052998 A290174 * A354061 A318921 A114516

Adjacent sequences: A344934 A344935 A344936 * A344938 A344939 A344940

Keyword

nonn,base

Author

Felix Fröhlich, Jun 03 2021

Status

approved