A164329 -id:A164329 - cp's OEIS frontend

A304248 Numbers that yield a prime whenever a '3' is inserted between any two digits.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 17, 19, 23, 29, 31, 37, 41, 43, 49, 61, 73, 79, 89, 97, 101, 103, 121, 127, 167, 173, 181, 209, 211, 233, 239, 247, 251, 271, 283, 299, 307, 331, 343, 359, 361, 373, 391, 437, 439, 473, 491, 497, 509, 523, 533, 547, 551, 599

Offset: 1

M. F. Hasler, Jun 01 2018

Motivated by existing sequences defined in a similar way for other digits (e.g., A164329 for digit 0), subsequence A158594 = intersection of this and A068674 ('3' is prefixed or appended), and others: cf. cross-references.

			121 is in the sequence because it yields a prime when a digit 3 is inserted after the first or after the second digit, which yields the prime 1321 or 1231, respectively. The term itself does not need to be prime.
The single-digit numbers 0..9 are in the sequence because they satisfy the condition voidly: nothing can be inserted, so no insertion yields a nonprime, so all possible insertions always yield a prime.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

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), A304246 (1 is inserted anywhere).
Cf. A304247 (2 is inserted anywhere).
Cf. A158594 (3 is prefixed, appended or inserted anywhere), A215419 (primes among these), A068674 (3 is prefixed or appended).
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).

[0] cat [k:k in [1..600]| forall{i:i in [1..#Intseq(k)-1]| IsPrime(Seqint(Reverse(v[1..i] cat [3] cat v[i+1..#v]))) where v is Reverse(Intseq(k))}]; // Marius A. Burtea, Feb 09 2020

Select[Range[0,600],AllTrue[FromDigits/@Table[Insert[IntegerDigits[#],3,n],{n,2,IntegerLength[ #]}],PrimeQ]&] (* Harvey P. Dale, Nov 06 2022 *)

is(n, p=3, L=logint(n+!n, 10)+1, d, P)=!for(k=1, L-1, isprime((d=divrem(n, P=10^(L-k)))[2]+(10*d[1]+p)*P)||return)

A332261 Numbers that yield a prime whenever a '4' is inserted between any two digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 21, 37, 39, 43, 49, 51, 57, 61, 63, 67, 73, 91, 97, 109, 129, 147, 153, 159, 171, 187, 199, 211, 223, 237, 241, 247, 259, 267, 333, 349, 357, 363, 409, 421, 423, 441, 447, 457, 463, 493, 517, 537, 541, 543, 549, 571, 579, 583, 627, 649, 681

Offset: 1

Eric Fox, Feb 08 2020

For single-digit terms, the condition is voidly satisfied: nothing can be inserted.

			10281 is in this sequence because 1(4)0281, 10(4)281, 102(4)81, and 1028(4)1 are all prime.

Cf. A164329, A304246, A304247, A304248, A050714.

a:=[]; for k in [1..700] do s:=0; v:=Reverse(Intseq(k)); for i in [1..#v-1] do vv:=v[1..i] cat [4] cat v[i+1..#v]; p:=Seqint(Reverse(vv)); if not IsPrime(p) then break; else s:=s+1; end if; end for;  if s eq #v-1 then Append(~a,k); end if; end for; [0] cat a; // Marius A. Burtea, Feb 09 2020

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A304248 Numbers that yield a prime whenever a '3' is inserted between any two digits.

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