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A337123
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a(n) is the number of primes p in the n-digit "signed nonadjacent form" such that p has 3 or fewer nonzero digits.
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1
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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, 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, 13, 14, 10, 11, 10, 10, 20, 15, 9, 10, 13, 8, 22, 10, 10, 10, 12, 13
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OFFSET
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1,3
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COMMENTS
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Sign nonadjacent form notation is defined by the publications listed in the reference.
We use abbreviation SNF for "signed nonadjacent form" notation.
This is an expansion of A337124 to include 2 and primes in the form of 2^k+1 and 2^k-1.
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REFERENCES
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Joerg Arndt, Matters Computational - Ideas, Algorithms, Source Code, 2011, Springer, pp. 61-62.
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LINKS
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EXAMPLE
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There is only one number in single-digit SNF notation, which is 1 and 1 is not a prime. So a(1)=0;
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;
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;
...
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.
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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base,nonn
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AUTHOR
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STATUS
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approved
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