A208982 -id:A208982 - cp's OEIS frontend

A209085 a(n) is the next larger than A208982(n) number with mutual Hamming distance 1.

3, 3, 7, 5, 7, 7, 11, 11, 13, 31, 17, 19, 19, 23, 23, 23, 31, 31, 29, 31, 31, 37, 47, 41, 43, 43, 47, 47, 47, 53, 59, 59, 61, 127, 67, 67, 71, 71, 71, 79, 73, 79, 79, 79, 83, 83, 89, 127, 97, 103, 101, 103, 103, 107, 107, 109, 127, 113, 127, 127, 127, 127

Offset: 1

Vladimir Shevelev, Mar 05 2012

All terms are prime by construction of A208982, and prime p occurs k times iff 2^k||p+1. In particular, every prime of the form 4*k+1 occurs 1 time. Thus all odd primes are in the sequence.

			Since 2^4 divides 48, but 2^5 not divides, i.e., 2^4||48, then 47 occurs four times.

Cf. A206853, A208982.

A209544 Primes not expressed in form n<+>2, where operation <+> defined in A206853.

Original entry on oeis.org

3, 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353, 401, 409, 433, 449, 457, 521, 569, 577, 593, 601, 617, 641, 673, 761, 769, 809, 857, 881, 929, 937, 953, 977, 1009, 1033, 1049, 1097, 1129, 1153, 1193, 1201, 1217, 1249, 1289, 1297, 1321

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Mar 10 2012

Trivially every odd prime is expressed in form n<+>1 (cf. A208982).

Are these related to A141174, A045390 or A007519? - R. J. Mathar, Mar 13 2012

Cf. A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A206853, A206960, A007519, A209554, A182187.

For n>=2, a(n) = A007519(n-1). - Vladimir Shevelev, Apr 18 2012

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).

Original entry on oeis.org

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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cp's OEIS Frontend

A209085 a(n) is the next larger than A208982(n) number with mutual Hamming distance 1.

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A209544 Primes not expressed in form n<+>2, where operation <+> defined in A206853.

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

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Mathematica