A304244 Numbers that yield a prime when prime(k) is inserted after the k-th digit, for any k >= 1, k < number of digits.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 17, 23, 27, 29, 41, 51, 53, 77, 81, 83, 87, 89, 99, 101, 149, 191, 239, 251, 287, 317, 353, 359, 419, 473, 497, 509, 527, 533, 611, 677, 743, 797, 809, 821, 887, 893, 941, 983, 1037, 1043, 1277, 1421, 1841, 1853, 1973, 1979, 2543

Offset: 1

Author

M. F. Hasler, May 21 2018

Keywords

• nonn
• base
• fini
• full

Comments

The primes to insert are: 2 (after the first digit), 3 (after the second digit, if there are at least three), etc.

Inspired by A304243 and analog sequences given in cross-references.

The sequence is finite: if insertion of 3 after the second digit yields a prime, then the sum of digits must be congruent to 1 or 2 (mod 3). However, insertion of 2 after the first digit also must yield a prime, so only the second case is possible. But then, insertion of a digit 7 cannot yield a prime, so no term can have 5 digits or more. (Sequence A304243 circumvents this restriction by excluding 3 from the primes to insert, but it is still finite for a similar reason occurring later.)

Examples

			The 1-digit numbers 0..9 are included since the condition is voidly satisfied: Nothing can be inserted, therefore each of the resulting numbers is prime.
17 is in the sequence because 127 is prime.
101 is in the sequence because 1201 and 1031 are prime.
		

Links

• M. F. Hasler, Table of n, a(n) for n = 1..100 (complete sequence).

Crossrefs

Cf. A304243 (2 is prefixed or prime(k+2) is inserted after the k-th digit).

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

Programs

PARI

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