A205510 -id:A205510 - cp's OEIS frontend

A208982 Numbers n such that the next larger number with mutual Hamming distance 1 is prime.

1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 16, 17, 18, 19, 21, 22, 23, 27, 28, 29, 30, 36, 39, 40, 41, 42, 43, 45, 46, 52, 57, 58, 60, 63, 65, 66, 67, 69, 70, 71, 72, 75, 77, 78, 81, 82, 88, 95, 96, 99, 100, 101, 102, 105, 106, 108, 111, 112, 119, 123, 125, 126, 129, 130, 136, 137, 138, 147, 148, 149, 150

Offset: 1

Vladimir Shevelev, Mar 04 2012

If p is prime, then p-1 is in the sequence.

Using the prime number theorem in arithmetic progressions k*n+b with gcd(k,b)=1 and its uniformity over k<=exp(c*sqrt(log(x))), one can prove that the counting function of a(n)<=x is equivalent to 2*x/log(x), as x tends to infinity.

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

isok(n) = my(nextn = n+1); while (hammingweight(bitxor(n, nextn)) != 1, nextn++); isprime(nextn); \\ Michel Marcus, Jul 01 2014

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

A182336 a(n) is the least m>=n, such that the Hamming distance D(n,m)=4.

Original entry on oeis.org

15, 14, 13, 12, 11, 10, 9, 8, 19, 18, 17, 16, 17, 16, 16, 17, 31, 30, 29, 28, 27, 26, 25, 24, 33, 32, 32, 33, 32, 33, 34, 35, 47, 46, 45, 44, 43, 42, 41, 40, 51, 50, 49, 48, 49, 48, 48, 49, 63, 62, 61, 60, 59, 58, 57, 56, 64, 65, 66, 67, 68, 69, 70, 71, 79, 78, 77, 76, 75, 74

Offset: 0

Vladimir Shevelev, Apr 25 2012

Or (see comment in A206853) a(n)=n<+>4.

Conjecture: for n > 96, n + 1 <= a(n) <= 9n/8 + 1. - Charles R Greathouse IV, Apr 25 2012

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A206853, A206960, A207063, A209544, A209554, A007814, A086799, A182187, A182209.

hamming(n)=my(v=binary(n));sum(i=1,#v,v[i])
a(n)=my(k=n);while(hamming(bitxor(n,k++))!=4,);k \\ Charles R Greathouse IV, Apr 25 2012

Terms corrected by Charles R Greathouse IV, Apr 25 2012

A345985 Hamming distance between prime(n) and prime(n+1) in base 10.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 3, 1, 1, 2, 1, 1, 1

Offset: 1

N. J. A. Sloane, Jul 09 2021

			Prime(4) = 7, prime(5) = 11, the words 7 and 11  are at Hamming distance 2 apart, so a(4) = 2.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A205510.

\\ abs Hamming distance in decimal digits
dhd(j,k)={my(dj=digits(j),dk=digits(k),s=0);s=abs(#dj-#dk);for(i=1,min(#dj,#dk),s+=(dj[i]!=dk[i]));s};
a345985(limit)={my(pp=2);forprime(p=3,limit,print1(dhd(p,pp),", ");pp=p)};
a345985(prime(85)) \\ Hugo Pfoertner, Jul 09 2021

A374178 a(n) is the least prime p such that p and the next prime > p have exactly n common 1's in their binary expansion.

Original entry on oeis.org

2, 5, 19, 29, 181, 359, 379, 983, 3821, 3581, 8179, 16111, 81901, 98297, 131059, 131063, 524269, 917471, 3145679, 4128763, 16744423, 16776187, 58720223, 117440381, 266330047, 468713467, 536870879, 1073741309, 2147483629, 4294967231, 8589410287, 33285865469

Offset: 1

Hugo Pfoertner, Jul 08 2024

1

			  a(n): 2       5         19           29              181
   np   3       7         23           31              191
      [1 0]  [1 0 1]  [1 0 0 1 1]  [1 1 1 0 1]  [1 0 1 1 0 1 0 1]
      [1 1]  [1 1 1]  [1 0 1 1 1]  [1 1 1 1 1]  [1 0 1 1 1 1 1 1]
       ^      ^   ^    ^     ^ ^    ^ ^ ^   ^    ^   ^ ^   ^   ^
  n:   1        2          3            4               5

Alois P. Heinz, Table of n, a(n) for n = 1..100

Cf. A000120, A000788, A007088, A061712, A205510.

from sympy import nextprime
def A374178(n):
    p = 2
    while (q:=nextprime(p)):
        if (p&q).bit_count() == n:
            return p
        p = q # Chai Wah Wu, Jul 08 2024

A207460 Let a(1) = 4. For n > 1, a(n) is the least number greater than a(n-1) such that the Hamming distance D(a(n-1),a(n)) = 4.

Original entry on oeis.org

4, 11, 16, 31, 35, 44, 49, 62, 70, 73, 82, 93, 97, 110, 112, 127, 143, 145, 158, 162, 173, 176, 191, 199, 200, 211, 220, 224, 239, 241, 254, 286, 290, 301, 304, 319, 327, 328, 339, 348, 352, 367, 369, 382, 398, 400, 415, 419

Offset: 1

Vladimir Shevelev, Feb 18 2012

All terms are odious (A000069).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000225, A205509, A205510, A205511, A205302, 205649, A205533, A122565, A206852, A206853, A206960, A207016, A207017.

nxt[n_]:=Module[{k=n+1},While[HammingDistance[PadLeft[IntegerDigits[ n,2],IntegerLength[ k,2]],IntegerDigits[k,2]]!=4,k++];k]; NestList[ nxt,4,50] (* Harvey P. Dale, Nov 07 2020 *)

A207472 Let a(1) = 5. For n > 1, a(n) is the least number greater than a(n-1) such that the Hamming distance D(a(n-1),a(n)) = 5.

Original entry on oeis.org

5, 26, 33, 62, 66, 93, 96, 127, 135, 152, 163, 188, 192, 223, 225, 254, 270, 273, 294, 313, 320, 351, 353, 382, 390, 409, 418, 445, 449, 478, 480, 511, 543, 545, 574, 578, 605, 608, 639, 647, 664, 675, 700, 704, 735, 737, 766, 782, 785, 806, 825, 832, 863, 865

Offset: 1

Vladimir Shevelev, Feb 18 2012

Odious and evil terms are alternating (cf. A000069, A001969).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000069, A000225, A001969, A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A206853, A206960, A207016, A207017.

a[1] = 5; a[n_] := a[n] = Module[{k = a[n - 1], m = a[n-1] + 1}, While[DigitCount[BitXor[k, m], 2, 1] != 5, m++]; m]; Array[a, 100] (* Amiram Eldar, Aug 06 2023 *)

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 *)

A293424 Hamming distance between two consecutive semiprimes.

Original entry on oeis.org

1, 4, 2, 1, 1, 3, 2, 4, 2, 5, 2, 1, 2, 1, 2, 5, 1, 1, 3, 2, 1, 7, 1, 4, 3, 5, 3, 2, 1, 2, 2, 2, 1, 4, 2, 3, 2, 1, 3, 2, 1, 6, 1, 2, 3, 2, 1, 4, 2, 2, 2, 1, 5, 3, 4, 2, 2, 2, 3, 1, 5, 3, 2, 1, 2, 2, 5, 1, 2, 1, 3, 2, 1, 2, 6, 2, 2, 3, 3, 1, 2, 8, 2, 4, 1, 3, 1, 2, 5, 1, 1, 3, 1, 2, 2, 1, 4, 1, 4, 2, 6, 1, 2, 1, 3

Offset: 1

Robert G. Wilson v, Oct 08 2017

The least semiprime whose Hamming distance between it and its successor semiprime is k: 4, 9, 15, 6, 26, 123, 62, 254, 511, 3071, 2047, 8189, 32765, 16382, 98303, 65531, 393215, 262142, 1572863, 2621438, 1048574, 16777207, 8388607, 50331647, 33554429, 134217721, 268435451, etc.

Not surprisingly, the above are often the largest semiprime < 2^j.

			a(1) = 1 because the semiprimes 4 & 6, 100_2 & 110_2 have a Hamming distance of 1;
a(2) = 4 because the semiprimes 6 & 9, 110_2 & 1001_2 have a Hamming distance of 4;
a(3) = 2 because the semiprimes 9 & 10, 1001_2 & 1010_2 have a Hamming distance of 2; etc.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001358 (semiprimes), A205510 (between consecutive primes).

semiprimes:= select(t -> numtheory:-bigomega(t)=2, [$4..1023]):
L:=map(t -> convert(t+1024,base,2), semiprimes):
map(t -> 11 - numboccur(0,t), L[2..-1]-L[1..-2]); # Robert Israel, Oct 08 2017
# alternative
read("transforms") :
A293424 := proc(n)
    local s1,s2 ;
    s1 := A001358(n) ;
    s2 := A001358(n+1) ;
    XORnos(s1,s2) ;
    wt(%) ;
end proc: # R. J. Mathar, Jan 06 2018

Count[ IntegerDigits[ BitXor[ #[[1]], #[[2]]], 2], 1] & /@ Partition[ Select[ Range@330, PrimeOmega@# == 2 &], 2, 1]

lista(nn) = my(v = select(x->bigomega(x)==2, vector(nn, k, k))); vector(#v-1, k, norml2(binary(bitxor(v[k], v[k+1])))); \\ Michel Marcus, Oct 11 2017

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cp's OEIS Frontend

A208982 Numbers n such that the next larger number with mutual Hamming distance 1 is prime.

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

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A182336 a(n) is the least m>=n, such that the Hamming distance D(n,m)=4.

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A345985 Hamming distance between prime(n) and prime(n+1) in base 10.

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A374178 a(n) is the least prime p such that p and the next prime > p have exactly n common 1's in their binary expansion.

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A207460 Let a(1) = 4. For n > 1, a(n) is the least number greater than a(n-1) such that the Hamming distance D(a(n-1),a(n)) = 4.

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A207472 Let a(1) = 5. For n > 1, a(n) is the least number greater than a(n-1) such that the Hamming distance D(a(n-1),a(n)) = 5.

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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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A293424 Hamming distance between two consecutive semiprimes.

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