A061712 -id:A061712 - cp's OEIS frontend

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.

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

A381405 a(0) = 0; for n > 0, a(n) is the smallest unused number such that a(n) AND a(n-1) = 0, where AND is the binary AND operation, while the binary weight of a(n) does not equal that of a(n-1).

Original entry on oeis.org

0, 1, 6, 8, 3, 4, 9, 2, 5, 16, 7, 24, 32, 10, 21, 34, 13, 18, 37, 64, 11, 20, 35, 12, 19, 36, 25, 66, 28, 33, 14, 17, 38, 65, 22, 40, 23, 72, 39, 80, 15, 48, 67, 60, 128, 26, 68, 27, 96, 29, 98, 129, 30, 97, 130, 41, 86, 136, 49, 78, 144, 42, 85, 138, 53, 74, 132, 43, 84, 139, 52, 75, 148, 99, 140, 51, 76, 147, 44, 83, 160, 31, 192, 45, 82, 141, 50, 77, 146, 101

Offset: 0

Scott R. Shannon, Feb 22 2025

In the first 100000 terms the fixed points are 0, 1, 28, 76, 543, 1580, although more likely exist.

			a(3) = 8 = 1000_2 as 8 is unused and a(2) = 6 = 110_2, and 1000_2 AND 110_2 = 0 while the binary weights of 8 and 6 are 1 and 2 respectively.

Scott R. Shannon, Table of n, a(n) for n = 0..10000

Cf. A000120, A129760, A061712, A057168, A381406.

A381406 a(0) = 0; for n > 0, a(n) is the smallest unused number such that a(n) OR a(n-1) = 2^k - 1, where OR is the binary OR operation and k>=1, while the binary weight of a(n) does not equal that of a(n-1).

Original entry on oeis.org

0, 1, 3, 2, 5, 7, 4, 11, 6, 13, 10, 15, 8, 23, 9, 14, 17, 30, 19, 12, 27, 20, 31, 16, 47, 18, 29, 22, 43, 21, 46, 25, 39, 24, 55, 26, 45, 50, 61, 34, 63, 28, 51, 44, 59, 36, 91, 37, 58, 69, 62, 33, 94, 35, 60, 67, 124, 71, 56, 79, 48, 95, 32, 127, 38, 57, 70, 121, 54, 41, 86, 107, 52, 75, 117, 42, 53, 74, 119, 40, 87, 104, 151, 105, 118, 73, 126, 49, 78, 115

Offset: 0

Scott R. Shannon, Feb 22 2025

The fixed points begin 0, 1, 10, 315, 413, 415, 1551, 1559, 1797; there are likely infinitely more.

			a(4) = 5 = 101_2 as 5 is unused and a(3) = 2 = 10_2, and 101_2 OR 10_2 = 111_2 = 2^3 - 1, while the binary weights of 5 and 2 are 2 and 1 respectively.

Scott R. Shannon, Table of n, a(n) for n = 0..10000

Cf. A000120, A086799, A061712, A057168, A381405.

A145576 a(n) is the smallest prime with both exactly an n number of 0's and exactly an n number of 1's in its binary representation. a(n) = 0 if no such prime exists.

Original entry on oeis.org

2, 0, 37, 139, 541, 2141, 8287, 33119, 131519, 525247, 2098687, 8391679, 33561599, 134242271, 536895487, 2147548159, 8590061567, 34360196863, 137439412223, 549756861439, 2199026663423, 8796097216447, 35184380411903

Offset: 1

Leroy Quet, Oct 13 2008

			a(3) = 37 = 100101 (base 2) is the smallest prime with three 0's and three 1's in its binary representation. - _R. J. Mathar_, Oct 14 2008

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A066195, A061712.

A000120 := proc(n) local d; add(d,d=convert(n,base,2)) ; end: A080791 := proc(n) local d,dgs; dgs := convert(n,base,2) ; nops(dgs)-add(d,d=dgs) ; end: A070939 := proc(n) max(1,ilog2(n)+1) ; end: A145576 := proc(n) local p,pbin; p := nextprime(2^(2*n-1)-1); while true do pbin := A070939(p) ; if pbin > 2*n then RETURN(0) ; elif pbin = 2*n then if A000120(p) = n and A080791(p) = n then RETURN(p) ; fi; fi; p := nextprime(p) ; od: end: seq(A145576(n),n=1..30) ; # R. J. Mathar, Oct 14 2008
# Alternative:
F:= proc(n) local c,x;
      c:= [$n+1..2*n-2];
      do
        x:= 2^(2*n-1)+1+add(2^(2*n-1-c[i]),i=1..n-2);
        if isprime(x) then return x fi;
        c:= combinat:-prevcomb(c, 2*n-2)
      od
end proc:
2, 0, seq(F(n),n=3..30); # Robert Israel, Sep 24 2017

Table[SelectFirst[Prime@ Apply[Range, PrimePi@{2^(2 (n - 1)) + 1, 2^(2 n) - 1}], Union@ DigitCount[#, 2] == {n} &] /. k_ /; MissingQ@ k -> 0, {n, 12}] (* Michael De Vlieger, Sep 24 2017 *)

Extended by R. J. Mathar and Ray Chandler, Oct 14 2008

A177858 Triangle in which row n gives the number of primes <= 2^n having k 1's in their binary representation, k=1..n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 0, 1, 3, 4, 2, 1, 1, 3, 6, 4, 4, 0, 1, 3, 9, 9, 8, 0, 1, 1, 3, 12, 13, 20, 0, 5, 0, 1, 4, 12, 23, 31, 8, 14, 4, 0, 1, 4, 16, 29, 48, 24, 38, 9, 3, 0, 1, 4, 18, 42, 73, 52, 72, 29, 17, 1, 0, 1, 4, 21, 53, 111, 80, 151, 81, 52, 5, 5, 0, 1, 4, 23, 62, 152, 158, 256, 186

Offset: 1

T. D. Noe, May 14 2010

Every row begins with 1 because 2 is the only prime having one 1 in its binary representation. A row ends in 1 or 0, depending on whether 2^n-1 is prime or composite. The sum of terms in row n is A007053(n).

T. D. Noe, Rows n=1..30, flattened

Cf. A061712 (least prime having n 1's)

nn=20; cnt=Table[0,{nn}]; Flatten[Table[Do[p=Prime[i]; c=Total[IntegerDigits[p,2]]; cnt[[c]]++, {i, 1+PrimePi[2^(n-1)], PrimePi[2^n]}]; Take[cnt,n], {n,nn}]]

A278477 Primes that set a new record for the Hamming weight.

Original entry on oeis.org

2, 3, 7, 23, 31, 127, 383, 991, 2039, 3583, 6143, 8191, 63487, 129023, 131071, 522239, 524287, 1966079, 4128767, 14680063, 33546239, 67108351, 201064447, 260046847, 536739839, 1073479679, 2147483647, 8581545983, 16911433727

Offset: 1

Joerg Arndt, Nov 23 2016

The Mersenne primes (A000668) are a subsequence.

Robert Israel, Table of n, a(n) for n = 1..3301

Cf. A000668, A061712, A211997.

M:= 40: # to use A061712(1..M)
A061712:= proc(n) local d,c,cands;
  for d from 0 do
    cands:= map(t -> 2^(n+d)-1 - add(2^(n-1+d-j), j=t),
        combinat:-choose([$1..n-2+d], d));
    for c in cands do if  isprime(c) then return c fi od
  od
end proc:
A061712(1):= 2:
R:= map(A061712, [$1..M]):
R[select(t -> R[t] < `if`(isprime(2^(M+1)-1), 2^(M+1)-1, 2^(M+2)+2^M-1) and R[t] = min(R[t..-1]), [$1..nops(R)])]; # Robert Israel, Nov 23 2016

{my(h=0);forprime(p=2,10^11,my(t=hammingweight(p));if(t>h,print1(p,", ");h=t));}

A374179 a(n) is the least prime p such that the binary expansions of p and of the next prime q > p differ at exactly n positions, and p and q have the same binary length.

Original entry on oeis.org

2, 11, 47, 139, 157, 191, 1151, 1531, 3067, 7159, 20479, 36857, 49139, 98299, 360439, 917503, 1310719, 786431, 6291449, 5242877, 20971507, 58720253, 83886053, 201326557, 335544301, 402653171, 3489660919, 1879048183, 5368709117, 25769803751, 21474836479, 77309411323

Offset: 1

Hugo Pfoertner, Jul 09 2024

			  a(n): 2       11           47               139                157
   np   3       13           53               149                163
      [1 0]  [1 0 1 1]  [1 0 1 1 1 1]  [1 0 0 0 1 0 1 1]  [1 0 0 1 1 1 0 1]
      [1 1]  [1 1 0 1]  [1 1 0 1 0 1]  [1 0 0 1 0 1 0 1]  [1 0 1 0 0 0 1 1]
         ^      ^ ^        ^ ^   ^            ^ ^ ^ ^          ^ ^ ^ ^ ^
  n:     1       2            3                  4                 5

Cf. A000120, A000788, A007088, A061712, A205510, A374178.

from sympy import nextprime
def A374179(n):
    p, pb = 2, 2
    while (q:=nextprime(p)):
        if pb==(qb:=q.bit_length()) and (p^q).bit_count() == n:
            return p
        p, pb = q, qb  # Chai Wah Wu, Jul 10 2024

A140330 Smallest nonprime with Hamming weight n (i.e., with exactly n 1's when written in binary).

Original entry on oeis.org

1, 6, 14, 15, 55, 63, 247, 255, 511, 1023, 2047, 4095, 12287, 16383, 32767, 65535, 196607, 262143, 983039, 1048575, 2097151, 4194303, 8388607, 16777215, 33554431, 67108863, 134217727, 268435455, 536870911, 1073741823, 3221225471

Offset: 1

Jonathan Vos Post, May 26 2008

Apart from the first term, identical to A089226. - R. J. Mathar, May 31 2008

			a(10) = 1023 because 1023 base 2 = 1111111111 which has 10 1's and 1023 = 3 * 11 * 31 is nonprime.

Cf. A061712.

a(n) = MIN{k such that k is in A018252 and A000120(k) = n}.

More terms from R. J. Mathar, May 31 2008

A354480 a(n) is the smallest decimal palindrome with Hamming weight n (i.e., with exactly n 1's when written in binary).

Original entry on oeis.org

0, 1, 3, 7, 77, 55, 111, 191, 383, 767, 5115, 11711, 15351, 30703, 81918, 97279, 744447, 978879, 1570751, 3665663, 8387838, 66911966, 66322366, 132111231, 199212991, 389545983, 939474939, 3204444023, 3220660223, 11542724511, 34258485243, 33788788733, 34292629243

Offset: 0

Ilya Gutkovskiy, Jun 02 2022

Cf. A000120, A000225, A002113, A061712, A062388, A089226, A089998, A089999, A102029, A114477.

from itertools import count, islice, product
def pals(startd=1): # generator for base-10 palindromes
    for d in count(startd):
        for p in product("0123456789", repeat=d//2):
            if d//2 > 0 and p[0] == "0": continue
            left = "".join(p); right = left[::-1]
            for mid in [[""], "0123456789"][d%2]:
                yield int(left + mid + right)
def a(n):
    for p in pals(startd=len(str(2**n-1))):
        if bin(p).count("1") == n:
            return p
print([a(n) for n in range(33)]) # Michael S. Branicky, Jun 02 2022

a(21)-a(32) from Michael S. Branicky, Jun 02 2022

A362979 Square array, read by descending antidiagonals: row n lists the primes whose base-2 representation has exactly n ones, starting from n=3.

Original entry on oeis.org

7, 11, 23, 13, 29, 31, 19, 43, 47, 311

Offset: 3

Clark Kimberling, May 11 2023

			Corner:
  n=3:    7    11    13    19    37   41     67    73    97
  n=4:   23    29    43    53    71   83     89   101   113
  n=5:   31    47    59    61    79   103   107   109   151
  n=6:  311   317   347   349   359   373   461   467   571
The first four primes in row n=3 have these base-2 representations, respectively: 111, 1011, 1101, 10011.

Cf. A000040, A014499.

Cf. A019434 (row 2), A061712 (column 1), A081091 (row 3), A095077 (row 4).

t[n_] := Count[IntegerDigits[Prime[n], 2], 1]  (* A014499 *)
u = Table[t[n], {n, 1, 200}];
p[n_] := Flatten[Position[u, n]]
w = TableForm[Table[Prime[p[n]], {n, 3, 16}]]

New offset and edited by Michel Marcus, Jan 19 2024

cp's OEIS Frontend

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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A381405 a(0) = 0; for n > 0, a(n) is the smallest unused number such that a(n) AND a(n-1) = 0, where AND is the binary AND operation, while the binary weight of a(n) does not equal that of a(n-1).

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A381406 a(0) = 0; for n > 0, a(n) is the smallest unused number such that a(n) OR a(n-1) = 2^k - 1, where OR is the binary OR operation and k>=1, while the binary weight of a(n) does not equal that of a(n-1).

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A145576 a(n) is the smallest prime with both exactly an n number of 0's and exactly an n number of 1's in its binary representation. a(n) = 0 if no such prime exists.

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A177858 Triangle in which row n gives the number of primes <= 2^n having k 1's in their binary representation, k=1..n.

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Mathematica

A278477 Primes that set a new record for the Hamming weight.

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PARI

A374179 a(n) is the least prime p such that the binary expansions of p and of the next prime q > p differ at exactly n positions, and p and q have the same binary length.

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Python

A140330 Smallest nonprime with Hamming weight n (i.e., with exactly n 1's when written in binary).

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A354480 a(n) is the smallest decimal palindrome with Hamming weight n (i.e., with exactly n 1's when written in binary).

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