Primes whose base c expansion is also the base b expansion of a prime.

by M. F. Hasler, Jan.2014 - under construction : seq’s in yellow = submitted as draft, green = published

b \c	2	3	4	5	6	7	8	9	10	other	(b,c)
2	(A000040)	73, 521, 577,... = A235265	2,5,7,11,31,37... = A235461 ⊂ A077718	5,31,131,151,... = A235462 ⊂ A077719	... = A235263 ⊂ A077720	... = A235464 ⊂ A077721	... = A235465 ⊂ A077722	739,811,6571,... = A235466 ⊂ A077723	11,101,10111,... = A089971 ⊂ A020449	(10,13): A090712	(13,10): A235110
3	2,7,11,13,41,... = A235266	(A000040)	2,89,137,149,... = A235467	... = A235468	2,13,43,73,... = A235469	2,7,107,401,... = A235470	2,17,73,521,... = A235471	2,11,19,83,... = A235472	A089981	(10,17): A090713	(17,10): A235126
4	A152079	2,43,61,67,97... = A235473	(A000040)	...13,41,43,61... = A235615	...7,19,37,79... = A235616	...17,59,71,73... = A235617	...11,19,67,89... = A235618	... = A235619	A090707	(10,19): A090714	(19,10): A235144
5	2,7,11,13,19,... = A235475	... = A231474	2,3,11,29,31,... = A235474	(A000040)	... = A235626	...7,17,23,31... = A235627	...17,19,73,89... = A235628	...11,19,29,31... = A235629	A090708	(10,31): A090715	A235640 ?
6	3,5,7,11,17,19... =A235476	... = A231476	2,3,5,13,17,... = A235624	2,3,11,31,71,... = A235625	(A000040)	...5,19,61,89... = A235637	...13,17,29,37... = A235638	...5,19,23,41... = A235639	A090709	(10,32): A090716	A235641 ?
7	2, 31, 47, 59,... = A235477	... = A231477	2,3,11,23,29... = A235634	...,5,13,17,23... = A235635	2,3,5,17,47,... = A235636	(A000040)	...5,19,53,89... = A235622	...19,41,59,97... = A235621	A090710	(10,64): A090717...	A235642 ?
8	... = A235478	2,7,13,31,37... = A231478	2,3,7,11,19,29... = A235633	2,3,11,13,31... = A235632	2,3,5,11,13,23... = A235631	2,3,5,17,47,... = A235630	(A000040)	...19,41,59,97... = A235620	A235394 (by RGWv)	...(10,512): A090720	A235643 ?
9	11,13,19,41,... = A235479	2,5,7,11,17,19... = A235480	2,3,29,41,61,... = A235481	2,3,7,11,17,19... = A235482	...13,17,29;59... = A231481	2,5,7,11,17,19... = A231479	...17,37,53,79... = A231480	(A000040)	A235395 (by RGWv)	(10,1024): A090721	A235644 ?
10	A065720 ⊂ A036952	A065721	A065722	A065723	A065724	A065725	A065726	A065727	(A000040)	(10,11): A090711	(11,10): A091924

*since primes < min(b,c) are always in the sequence, some initial terms are omitted above, at the profit of one more larger term.

Useful PARI code:
is(p,b=2,c=3)=vecmax(d=digits(p,c))<b&&isprime(vector(#d,i,b^(#d-i))*d~)&&isprime(p)
r(p,b,c)=sum(i=1,#p=digits(p,c),p[i]*b^(#p-i)) \\ “rebase” from base c to base b

forprime(p=1,999,is(p,b,c)&&print1(p","))

forprime(p=1,999,is(p,b,c)&&print1(r(p,b,c)",")) \\ for c < b