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A295609
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a(n) = least prime number p such that p AND n = n (where AND denotes the binary AND operator).
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5
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2, 3, 2, 3, 5, 5, 7, 7, 11, 11, 11, 11, 13, 13, 31, 31, 17, 17, 19, 19, 23, 23, 23, 23, 29, 29, 31, 31, 29, 29, 31, 31, 37, 37, 43, 43, 37, 37, 47, 47, 41, 41, 43, 43, 47, 47, 47, 47, 53, 53, 59, 59, 53, 53, 127, 127, 59, 59, 59, 59, 61, 61, 127, 127, 67, 67
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OFFSET
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0,1
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COMMENTS
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For any n > 0: gcd(A109613(n), A062383(n)) = 1, hence, by Dirichlet's theorem on arithmetic progressions, we have a prime number, say p, of the form A109613(n) + k * A062383(n) with k > 0; this prime number satisfies p AND n = n; also a(0) = 2, hence the sequence is well defined for any n >= 0.
a(n) = n iff n is prime.
Each prime number appears 2*k times in this sequence for some k > 0.
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LINKS
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FORMULA
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For any k > 1, a(2*k) = a(2*k+1).
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EXAMPLE
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a(42) = 42 + A295335(42) = 42 + 1 = 43.
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MATHEMATICA
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Table[Block[{p = 2}, While[BitAnd[p, n] != n, p = NextPrime@ p]; p], {n, 0, 65}] (* Michael De Vlieger, Nov 26 2017 *)
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PROG
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(PARI) avoid(n, i) = if (i, if (n%2, 2*avoid(n\2, i), 2*avoid(n\2, i\2)+(i%2)), 0) \\ (i+1)-th number k such that k AND n = 0
a(n) = for (i=0, oo, my (k=avoid(n, i)); if (isprime(n+k), return (n+k)))
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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