A284643 Numbers k such that concat(k, d(k)) and concat(d(k), k) are both prime, where d(k) is the number of divisors of k.

1, 121, 169, 361, 1849, 14161, 17689, 25921, 41209, 51529, 52441, 64009, 69169, 76729, 124609, 187489, 247009, 436921, 776161, 851929, 935089, 1207801, 1951609, 2582449, 3179089, 3829849, 4363921, 4481689, 5094049, 6911641, 7382089, 7722841, 8116801, 8743849, 9186961

Offset

1,2

Comments

It appears that all terms are squares. - Michel Marcus, Apr 02 2017

All terms are odd squares, because k and d(k) must be odd. - Robert Israel, Apr 04 2017

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

d(121) = 3 and 1213 and 3121 are both prime.

Maple

with(numtheory): P:=proc(q) local a, b, n; for n from 1 by 2 to q do
a:=n*10^(ilog10(tau(n))+1)+tau(n); b:=tau(n)*10^(ilog10(n)+1)+n;
if isprime(a) and isprime(b) then print(n); fi; od; end: P(10^9);
# Alternative:
filter:= proc(k) local d; d:= numtheory:-tau(k); isprime(d + 10^(1+ilog10(d))*k) and isprime(k+10^(1+ilog10(k))*d) end proc:
select(filter, [seq(k^2, k=1..10^4, 2)]); # Robert Israel, Apr 04 2017

Mathematica

cnc[x_, y_] := FromDigits[Join @@ IntegerDigits@{x, y}]; Select[ Range[1, 5000, 2]^2, (t = DivisorSigma[0, #]; PrimeQ@ cnc[t, #] && PrimeQ@ cnc[#, t]) &] (* Giovanni Resta, Apr 04 2017 *)

Prog

(PARI) is(k)={my(v1=digits(k), v2=digits(numdiv(k))); isprime(fromdigits(concat(v1, v2)))&&isprime(fromdigits(concat(v2, v1)))};
findFirstTerms(q)={my(ans:list, k=1); ans=List(); q=max(1, q); for(c=1, q, while(!is(k), k++); listput(ans, k); k++); Vec(ans)} \\ R. J. Cano, Mar 31 2017

Crossrefs

Cf. A000005, A065998.

Sequence in context: A352221 A240775 A346316 * A074730 A268519 A364778

Adjacent sequences: A284640 A284641 A284642 * A284644 A284645 A284646

Keyword

nonn,base

Author

Paolo P. Lava, Mar 31 2017

Status

approved