cp's OEIS Frontend

A064893 Smallest number with "binary potency" of n. Blocks of at least n 0's must be inserted between the bits of a(n) to "dilute" it into a nonprime.

%I A064893 #35 Aug 12 2023 05:14:26
%S A064893 1,2,3,29,149,4079,4088027,1887647351,355898535581
%N A064893 Smallest number with "binary potency" of n. Blocks of at least n 0's must be inserted between the bits of a(n) to "dilute" it into a nonprime.
%C A064893 Are there values for all a(n)? Is the sequence monotonic?
%e A064893 1 is nonprime (potency 0) so a(0) = 1.
%e A064893 2 = 10 -> 100 = 4, potency 1, so a(1) = 2.
%e A064893 3 = 11 -> 101 = 5 -> 1001 = 9, potency 2 so a(2) = 3.
%e A064893 Potency 3 first appears for 29, so a(3) = 29; etc.
%e A064893 4088027 dilutes into 5858670956869, 10540679875665334793, 20632314388123853242044433, 41873360515671700575496732442657 and 86420918502629375433474712237244678209, all prime. But the next dilution, 179810732934100666625066494484898891551473793, is composite.
%o A064893 (PARI) isok(k, n) = for (i=1, n, my(b=binary(k), list=List()); for (j=1, #b-1, listput(list, b[j]); for (k=1, i, listput(list, 0););); listput(list, b[#b]); my(x=fromdigits(Vec(list), 2), isp=ispseudoprime(x)); if ((i<n) && !isp, return(0)); if ((i==n) && isp, return(0));); return(1);
%o A064893 a(n) = if (n==0, 1, my(k=2); while (!isok(k, n), k=nextprime(k+1)); k); \\ _Michel Marcus_, Aug 12 2023
%Y A064893 Cf. A000040, A064891, A064892.
%K A064893 base,nonn,nice,more
%O A064893 0,2
%A A064893 _Marc LeBrun_, Oct 10 2001
%E A064893 a(6) from _Don Reble_ and Fred W. Helenius (fredh(AT)ix.netcom.com), Oct 11 2001
%E A064893 a(7) from _Hans Havermann_, Oct 13 2001
%E A064893 a(8) from _Michael S. Branicky_, Jun 30 2023