A355700 - OEIS

%I #100 Aug 15 2022 08:34:11

%S 17,41,1787,37,1011749,-1,8070191,18919,31783046759,-1,107,13588859,

%T -1,906611171779,106661882252960131,9883,40487203,173127971,

%U 5664484284773,696222287901816728317439,-1,101,97453813,-1,28631342754671,15215869393552811003,629657070248572792452284791790843,-1,32233,123323,334444555566656663,122334444555553,122334444555566566663,1223334444555556666677767777,22333444455555666666777777788888898999999989,-1

%N Triangle T(n,k), n > 2, 1 < k < n, read by rows, where T(n,k) is in base n, the smallest prime consisting of digits d from a set of k nonzero consecutive digits, d times each, or -1 if no such number exists.

%C All terms T(n,k) != -1, 2 < n <= 10 and 1 < k < n are emirps.

%H Chris K. Caldwell and G. L. Honaker, Jr., <a href="https://primes.utm.edu/curios/page.php?short=32233">Prime Curios! 33223</a>

%H Chris K. Caldwell and G. L. Honaker, Jr., <a href="https://primes.utm.edu/curios/page.php?number_id=7878">Prime Curios! 123323</a>

%H Chris K. Caldwell and G. L. Honaker, Jr., <a href="https://math.utm.edu/curios/page.php?number_id=7881">Prime Curios! 122334444555553</a>

%e T(3,2) = 17 = 122_3 is prime and has base-3 digits only from the set {1,2} with k=2 digits. In 122_3, the digit 1 occurs one time and the digit 2 two times. No lesser prime satisfies this.

%e T(10,3) = 123323 is prime and has only one 1, two 2's, three 3's, and the numbers 122333 = 71 * 1723, 123233 = 11 * 17 * 659 less than 123323 with the same digits are not prime.

%e Triangle begins:

%e   17;

%e   41,    1787;

%e   37, 1011749, -1;

%e   ...

%e With the rows in base-n expansion:

%e n/k,2,        3,          4,                5,              6, ...

%e 3, 122;

%e 4, 221,       123323;

%e 5, 122,       224333444, -1;

%e 6, 444545555, 223331,     22333444554555,   -1;

%e 7, 212,       223334444, -1,                 122333445544555, 122333444455555666666;

%e 8, 23233,     232344443,  1224334443,        122333544554545, 223334444555556666777766777, -1;

%e 9, 122,       223334444, -1,                 122333444555455,

%e   122333444455566666655, 22333444455555666666777778887888788, -1;

%e 10, 32233,    123323,     334444555566656663, 122334444555553,    122334444555566566663, 1223334444555556666677767777, 22333444455555666666777777788888898999999989, -1;

%e 11, 33232,    223343444, -1, ...

%K sign,base,tabl

%O 3,1

%A _Jean-Marc Rebert_, Jul 14 2022