A304246 Numbers that yield a prime whenever a '1' is inserted between any two digits.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 21, 31, 33, 37, 49, 63, 67, 69, 79, 81, 91, 99, 103, 109, 117, 123, 151, 163, 181, 193, 211, 213, 231, 241, 279, 309, 319, 363, 367, 391, 411, 427, 429, 453, 457, 459, 501, 513, 519, 547, 571, 601, 613, 621, 631, 697, 703, 709, 721, 729, 777, 787, 801, 811, 817, 879, 903, 951, 981, 987

Offset

1,3

Comments

The single-digit terms voidly satisfy the condition: no '1' can be inserted anywhere, so all possible insertions yield a prime.

Motivated by sequence A164329 which is the analog for inserting 0.

Compare to A068673 where 1 is prefixed or appended, and to A068679 where 1 is prefixed, appended or inserted anywhere - which is therefore the intersection between this sequence and A068673.

See also A050711 where 1 is inserted between all adjacent digits. - R. J. Mathar, Feb 28 2020

Links

Table of n, a(n) for n=1..71.

Example

21 is in the sequence, because if '1' is inserted between "any" pair consecutive digits (the only possibility being to insert it between the first and second digit, which yields 211), the result is always prime. The definition does not require the term itself to be prime.
103 is in the sequence because inserting 1 between the first and second, or between the second and third digit, would yield 1103 or 1013, respectively, which are both prime.

Maple

filter:= proc(n) local j, t;
  for j from 1 to ilog10(n) do
    if not isprime(10*n-9*(n mod 10^j)+10^j) then return false fi
  od;
  true
end proc:
select(filter, [$0..1000]); # Robert Israel, Jun 01 2018

Prog

(PARI) is(n)=!for(k=1, logint(n+!n, 10), isprime(10*n-n%10^k*9+10^k)||return)
(Magma) [0] cat [k:k in [1..1000]| forall{i:i in [1..#Intseq(k)-1]| IsPrime(Seqint(Reverse(v[1..i] cat [1] cat v[i+1..#v]))) where v is Reverse(Intseq(k)) }]; // Marius A. Burtea, Feb 09 2020

Crossrefs

Cf. A164329 (prime when 0 is inserted anywhere), A216169 (subset of composite terms), A215417 (subset of primes), A159236 (prime when 0 is inserted between all digits).

Cf. A068679 (1 is prefixed, appended or inserted anywhere), A069246 (primes among these), A068673 (1 is prefixed, or appended).

Cf. A158594 (3 is prefixed, appended or inserted anywhere), A215419 (primes among these).

Cf. A069832 (7 is prefixed, appended or inserted anywhere), A215420 (primes among these), A068677 (7 is prefixed or appended).

Cf. A069833 (9 is prefixed, appended or inserted anywhere), A215421 (primes among these).

Cf. A158232 (13 is prefixed or appended).

Cf. A304243 (2 is prefixed or prime(k+2) is inserted after the k-th digit), A304244 (prime(k) is inserted after the k-th digit), A304245 (prime(k+1) is inserted after the k-th digit, k > 1, or '2' after the first digit).

Sequence in context: A033088 A307498 A034294 * A271837 A290950 A032577

Adjacent sequences: A304243 A304244 A304245 * A304247 A304248 A304249

Keyword

nonn,base

Author

M. F. Hasler, Jun 01 2018

Status

approved