A206852 -id:A206852 - cp's OEIS frontend

A209332 a(n) is the minimal positive number k such that n<+>k is prime or 0 if no such number exists (operation <+> defined in A206853).

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 3, 5, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 0, 3, 1, 1, 1, 1, 2, 4, 0, 3, 2, 1, 0, 4, 1, 1, 1, 1, 1, 2, 1, 1, 0, 5, 0, 3, 2, 1, 0, 7, 2, 2, 1, 1, 2, 1, 0, 8, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 0, 7, 1, 2, 1, 1, 3, 2, 1, 1, 0, 3, 0, 4, 3

Offset: 1

Vladimir Shevelev, Mar 06 2012

Numbers n for which a(n) = 1 form sequence A208982.
a(n) = 0 for n = 25, 33, 37, 47,... (A209333).
A simple sufficient condition for a(n) = 0 (which is proved by induction) is that n<+>k is not prime up to the moment that n<+>k is even and n<+>(k+1)-n<+>k = 2^t, where t >= m+1 and m defined by the condition 2^m <= n < 2^(m+1).
Conjecture: for even n, a(n) > 0.

Cf. A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A206853, A206960, A209085, A208982, A209085, A209333.

A210566 Primes not expressed in form n<+>4, where operation <+> defined in A206853.

Original entry on oeis.org

2, 3, 5, 7, 23, 37, 53, 101, 103, 131, 149, 151, 167, 181, 229, 257, 263, 277, 293, 311, 359, 373, 389, 421, 439, 487, 503, 599, 613, 631, 641, 643, 647, 661, 677, 727, 743, 757, 769, 773, 821, 823, 853, 887, 919, 983, 997, 1013, 1031, 1061, 1063

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Mar 22 2012

Or primes p such that, for any nonnegative integer n

Cf. A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A206853, A206960, A208982, A209544, A209554.

hammingDistance[a_, b_] := Count[IntegerDigits[BitXor[a, b], 2], 1]; (* binary Hamming distance *) vS[a_,b_] := NestWhile[#+1&, a, hammingDistance[a,#] =!= b&]; (* vS[a_,b_] is the least c>=a,such that the binary Hamming distance D(a,c)=b. vS[a,b] is Vladimir's a<+>b *) A210566 = Map[Prime[#]&, Complement[Range[Max[#]], #]&[Map[PrimePi[#]&, Union[Map[#[[2]]&, Cases[Map[{PrimeQ[#],#}&[vS[#,4]]&, Range[7000]],{True,}]]]]]] (* _Peter J. C. Moses, Apr 02 2012 *)

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A209332 a(n) is the minimal positive number k such that n<+>k is prime or 0 if no such number exists (operation <+> defined in A206853).

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