A344937 -id:A344937 - cp's OEIS frontend

A344936 a(n) is the smallest prime p such that a string s of n zeros can be inserted between all adjacent digits of p simultaneously such that the resulting number is also prime and is also prime for each s of length k with 0 < k < n.

11, 19, 19, 71, 98689, 130049597, 78736136153

Offset: 1

Felix Fröhlich, Jun 03 2021

a(n) is the smallest prime p such that A344937(i) >= n, where i is the index of p in A000040.

			For n = 5: 98689, 908060809, 9008006008009, 90008000600080009, 900008000060000800009 and 9000008000006000008000009 are all prime. Since 98689 is the smallest prime where strings of zeros of successive lengths up to 5 can be inserted between all adjacent digits such that each resulting number is also prime, a(5) = 98689.

Cf. A000040, A344637, A344937.

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

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) = forprime(p=10, , for(k=1, n, if(!ispseudoprime(eva(insert_zeros(p, k))), break, if(k==n, return(p)))))

from sympy import isprime, nextprime
def insert_zeros(n, k): return int(("0"*k).join(list(str(n))))
def ok(p, n): return all(isprime(insert_zeros(p, k)) for k in range(1, n+1))
def a(n, startat=11):
  p = startat
  while True:
    if ok(p, n): return p
    p = nextprime(p)
print([a(n) for n in range(1, 6)]) # Michael S. Branicky, Jun 03 2021

a(7) from Michael S. Branicky, Jun 11 2021

A341899 a(n) is the smallest prime p > 10 such that when strings of n zeros are inserted between every pair of adjacent digits the result is also a prime.

Original entry on oeis.org

11, 19, 17, 13, 13, 23, 17, 17, 31, 13, 23, 41, 127, 61, 23, 13, 13, 67, 53, 89, 19, 227, 17, 29, 61, 151, 31, 37, 107, 53, 1741, 263, 167, 23, 31, 89, 61, 13, 43, 241, 53, 347, 1319, 19, 79, 419, 521, 19, 809, 677, 97, 97, 1223, 89, 13, 79, 67, 257, 17, 499

Offset: 1

Felix Fröhlich, Jun 04 2021

First differs from A306920 at n = 13.

a(n) = A306920(n) if A306920(n) is < 100, i.e., is a two-digit number.

			For n = 13: Inserting 13 zeros between all adjacent digits of 127 gives 10000000000000200000000000007, which is prime. Since 127 is the smallest prime where inserting exactly 13 zeros between all adjacent digits results in a number that is also prime, a(13) = 127.

Cf. A306920, A344637, A344936, A344937.

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) = forprime(p=10, , if(ispseudoprime(insert_zeros(p, n)), return(p)))

cp's OEIS Frontend

A344936 a(n) is the smallest prime p such that a string s of n zeros can be inserted between all adjacent digits of p simultaneously such that the resulting number is also prime and is also prime for each s of length k with 0 < k < n.

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