A309460 a(n) is the smallest number k such that there are n pairs (d,e) of (not necessarily distinct) divisors of k such that the concatenation d_e is a prime.

1, 4, 3, 9, 6, 27, 12, 18, 24, 36, 102, 42, 63, 90, 204, 120, 132, 156, 84, 198, 126, 312, 168, 210, 528, 720, 396, 594, 336, 660, 252, 672, 462, 420, 792, 990, 1170, 1560, 504, 2352, 756, 630, 840, 2688, 2436, 2100, 1596, 1008, 1092, 924, 2574, 1512, 2016

Offset

1,2

Comments

The same prime may arise in more than one way, and if so all occurrences are counted.

For n = 34, a(34) = 420: Divisors(420) = {1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420}. We obtain 34 prime numbers 11, 13, 17, 23, 31, 37, 41, 43, 47, 53, 61, 67, 71, 73, 101, 103, 107, 127, 151, 157, 211, 281, 283, 307, 353, 421, 421, 521, 601, 607, 701, 1021, 1051, 4201, as follows: 1_1, 1_3, 1_7, 2_3, 3_1, 3_7, 4_1, 4_3, 4_7, 5_3, 6_1, 6_7, 7_1, 7_3, 10_1, 10_3, 10_7, 12_7_ 15_1, 15_7, 21_1, 28_1, 28_3, 30_7, 35_3, 4_21, 42_1, 5_21, 60_1, 60_7, 70_1, 10_21, 105_1, 420_1. The prime number 421 is counted twice since it is obtained as 4_21 and 42_1.

For each n there may be several numbers m for which the divisors create n prime numbers, we take a(n) to be smallest such m.

For n = 1 we can take the prime numbers m = p = 9*s -1 (see A061242). By concatenating the divisors we obtain the numbers 1_1, 1_p, p_1, p_p. It is easy to see that only 1_1 = 11 is prime. Thus there are infinitely many numbers m for which the concatenation produces a single prime number.

For n = 2 get the numbers m = 4, 8, 10, 16, 19, 22, 25, 32, 37, 38, 43, 58, 67, 68, 73, 74, 79, .... . This includes the primes 19, 43, 67, 73, 79, ... of the form p = 6s + 1. For these only one of the numbers 1_p or p_1 is prime.

For n = 3 we get the numbers m = 3, 7, 13, 14, 20, 26, 31, 35, 44, 46, 50, 62, 64, 65, 77, 82, 97, 103, 109, .... The prime numbers 3, 7, 13, 31, 97, 103, 109, ... have the property that 1_p and p_1 are simultaneously prime.

Conjecture 1: a(n) always exists.

Conjecture 2: For any n there are infinitely many choices for m.

Links

Table of n, a(n) for n=1..53.

Example

For k = 1, Divisors(1) = {1} and 1_1 = 11, so a(1) = 1.
For k = 4, Divisors(4) = {1, 2, 4} and 1_1 = 11, 4_1 = 41, so a(2) = 4.
For k = 3, Divisors(3) = {1, 3} and 1_1 = 11, 1_3 = 13 , 3_1 = 31,so a(3) = 3.
For k = 9, Divisors(9) = {1, 3, 9} and 1_1 = 11, 1_3 = 13 , 1_9 = 19, 3_1 = 31,so a(4) = 9.
For k = 27, Divisors(27) = {1, 3, 9, 27} and 1_1 = 11, 1_3 = 13 , 1_9 = 19, 1_27 = 127, 3_1 = 31, 27_1 = 271 so a(6) = 27.

Prog

(Magma) m:=1; sol:=[]; for u in [1..53] do for k in [1..10000] do d:=Divisors(k); nr:=0; for i, j in [1..#d] do if IsPrime(d[i]*10^#Intseq(d[j])+d[j]) then nr:=nr+1; end if; end for; if nr eq u then sol[m]:=k; m:=m+1; break; end if; end for; end for; sol;

Crossrefs

Cf. A000040, A061242, A309219.

Sequence in context: A073238 A010655 A123596 * A200361 A222471 A180858

Adjacent sequences: A309457 A309458 A309459 * A309461 A309462 A309463

Keyword

nonn,base

Author

Marius A. Burtea, Aug 03 2019

Status

approved