A337123 - OEIS

%I #16 Jan 08 2021 20:39:40

%S 0,1,2,1,4,5,7,5,9,8,12,7,11,7,11,9,14,10,18,11,21,7,9,11,16,4,8,9,7,

%T 12,18,13,14,11,10,9,18,7,12,10,18,12,22,5,11,13,16,13,22,8,9,16,13,9,

%U 13,14,10,11,10,10,20,15,9,10,13,8,22,10,10,10,12,13

%N a(n) is the number of primes p in the n-digit "signed nonadjacent form" such that p has 3 or fewer nonzero digits.

%C Sign nonadjacent form notation is defined by the publications listed in the reference.

%C We use abbreviation SNF for "signed nonadjacent form" notation.

%C This is an expansion of A337124 to include 2 and primes in the form of 2^k+1 and 2^k-1.

%D Joerg Arndt, Matters Computational - Ideas, Algorithms, Source Code, 2011, Springer, pp. 61-62.

%H H. Prodinger, <a href="http://www.emis.de/journals/INTEGERS/papers/a8/a8.Abstract.html">On binary representations of integers with digits -1,0,1</a>, Integers 0 (2000), #A08.

%e There is only one number in single-digit SNF notation, which is 1 and 1 is not a prime.  So a(1)=0;

%e There is only one number in the two-digit SNF notation, which is 10 = 2 base 10 and it is a prime with one nonzero digit.  So a(2)=1;

%e There are three numbers in three digits SNF notation: 10T = 3 base 10, 100 = 4 base 10, and 101 = 5 base 10.  There are two prime numbers among 3, 4, and 5 and both of them have two nonzero digits. So a(3)=2;

%e ...

%e For seven-digit SNF numbers, 10T0T0T = 43 base 10 has 4 nonzero digits (excluded); 10T000T = 47 base 10 has 3 nonzero digits (included). Thereafter 10T0101 = 53: 4 digits, excluded; 1000T0T = 59: 3 digits, included; 1000T01 = 61: 3 digits, included; 100010T = 67: 3 digits, included; 100100T = 71: 3 digits, included; 1001001 = 73, 3 digits, included; 101000T = 79: 3 digits, included; 101010T = 83, 4 digits, excluded.  In total, 7 numbers fit the definition.  So a(7)=7.

%t Table[s1=2^(n-1);ct=0; If[n>1, If[PrimeQ[s1+1],ct++]; If[PrimeQ[s1-1],ct++]; If[n>=5, Do[s2=2^i; If[PrimeQ[s1+s2+1],ct++]; If[PrimeQ[s1+s2-1],ct++]; If[PrimeQ[s1-s2+1],ct++]; If[PrimeQ[s1-s2-1],ct++], {i,2,n-3}]]]; ct, {n,1,72}]

%Y Cf. A334913, A337124.

%K base,nonn

%O 1,3

%A _Lei Zhou_, Aug 17 2020