A014234 -id:A014234 - cp's OEIS frontend

A265326 n-th prime minus its binary reversal.

1, 0, 0, 0, -2, 2, 0, -6, -6, 6, 0, -4, 4, -10, -14, 10, 4, 14, -30, -42, 0, -42, -18, 12, 30, 18, -12, 0, 18, 42, 0, -62, -8, -70, -20, -82, -28, -34, -62, -8, -26, 8, -62, 62, 34, -28, 8, -28, 28, 62, 82, -8, 98, 28, 0, -186, -84, -210, -60

Offset: 1

Max Barrentine, Dec 07 2015

a(n) = 0 iff A000040(n) is in A016041. - Altug Alkan, Dec 07 2015

The graph consists of a succession of parallelograms. The parallelograms end when there is a long run of mostly positive terms followed by a long run of mostly negative terms. The places where the successive parallelograms end are the primes just before a power of 2: 3, 7, 13, 31, 61, 127, 251, 509, 1021, 2039, 4093, 8191, 16381, 32749, ..., which are terms with indices 2, 4, 6, 11, 18, 31, 54, 97, 172, 309, 564, 1028, 1900, 3512, 6542, 12251, 23000, 43390, 82025, ... (see A014234 and A007053). - N. J. A. Sloane, May 29 2016

			n=5: prime(5) = 11_10 = 1011_2, reversing gives 1101_2 = 13_10, so a(5) = 11-13 = -2.

N. J. A. Sloane, Table of n, a(n) for n = 1..23000, May 29 2016 [First 10000 terms from _Robert Israel_]
Rémy Sigrist, Colored scatterplot of the first 82025 terms (corresponding to the prime numbers < 2^20) (where the color is function of A000040(n) mod 8)
N. J. A. Sloane, Table of n, a(n) for n = 1..78498
N. J. A. Sloane and Brady Haran, Amazing Graphs, Numberphile video (2019)

Cf. A055945, A068396, A098957, A007053, A014234.

revdigs:= proc(n) local L, j;
  L:= convert(n,base,2);
  add(L[-j]*2^(j-1),j=1..nops(L))
end proc:
map(t -> t - revdigs(t),  select(isprime, [2,seq(i,i=3..1000,2)])); # Robert Israel, Dec 08 2015

Table[# - FromDigits[Reverse@ IntegerDigits[#, 2], 2] &@ Prime@ n, {n, 60}] (* Michael De Vlieger, Dec 09 2015 *)

a098957(n) = my(v=binary(prime(n)), s); forstep(i=#v, 1, -1, s+=s+v[i]); s
a(n) = prime(n) - a098957(n); \\ Altug Alkan, Dec 07 2015

a(n) = A000040(n) - A098957(n).

a(n) = A055945(A000040(n)). - Michel Marcus, Dec 08 2015

A377435 Number of perfect-powers x in the range 2^n <= x < 2^(n+1).

Original entry on oeis.org

1, 0, 1, 2, 3, 3, 5, 7, 8, 11, 16, 24, 32, 42, 61, 82, 118, 166, 231, 322, 453, 635, 892, 1253, 1767, 2487, 3505, 4936, 6959, 9816, 13850, 19538, 27578, 38933, 54972, 77641, 109668, 154922, 218879, 309277, 437047, 617658, 872968, 1233896, 1744153, 2465547, 3485478

Offset: 0

Gus Wiseman, Nov 04 2024

nonn

Perfect-powers (A001597) are numbers with a proper integer root, complement A007916.

Also the number of perfect-powers with n bits.

			The perfect-powers in each prescribed range (rows):
    1
    .
    4
    8    9
   16   25   27
   32   36   49
   64   81  100  121  125
  128  144  169  196  216  225  243
  256  289  324  343  361  400  441  484
  512  529  576  625  676  729  784  841  900  961 1000
Their binary expansions (columns):
  1  .  100  1000  10000  100000  1000000  10000000  100000000
             1001  11001  100100  1010001  10010000  100100001
                   11011  110001  1100100  10101001  101000100
                                  1111001  11000100  101010111
                                  1111101  11011000  101101001
                                           11100001  110010000
                                           11110011  110111001
                                                     111100100

The union of all numbers counted is A001597, without powers of two A377702.
The version for squarefree numbers is A077643.
These are the first differences of A188951.
The version for prime-powers is A244508.
For primes instead of powers of 2 we have A377432, zeros A377436.
Not counting powers of 2 gives A377467.
The version for non-perfect-powers is A377701.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect-powers, differences A053289.
A007916 lists the non-perfect-powers, differences A375706.
A081676 gives the greatest perfect-power <= n.
A131605 lists perfect-powers that are not prime-powers.
A377468 gives the least perfect-power > n.
Cf. A000015, A013597, A014210, A014234, A023055, A031218, A045542, A052410, A065514, A069623, A216765, A345531, A377434.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[Length[Select[Range[2^n,2^(n+1)-1],perpowQ]],{n,0,15}]

from sympy import mobius, integer_nthroot
def A377435(n):
    if n==0: return 1
    def f(x): return int(1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    return f((1<Chai Wah Wu, Nov 05 2024

For n != 1, a(n) = A377467(n) + 1.

a(26)-a(46) from Chai Wah Wu, Nov 05 2024

A377701 Number of non-perfect-powers x in the range 2^n < x < 2^(n+1).

Original entry on oeis.org

0, 1, 3, 6, 13, 29, 59, 121, 248, 501, 1008, 2024, 4064, 8150, 16323, 32686, 65418, 130906, 261913, 523966, 1048123, 2096517, 4193412, 8387355, 16775449, 33551945, 67105359, 134212792, 268428497, 536861096, 1073727974, 2147464110, 4294939718, 8589895659

Offset: 0

Gus Wiseman, Nov 05 2024

nonn

Non-perfect-powers (A007916) are numbers without a proper integer root.

Also the number of non-perfect-powers with n bits.

			The non-perfect-powers in each range (rows):
   .
   3
   5  6  7
  10 11 12 13 14 15
  17 18 19 20 21 22 23 24 26 28 29 30 31
Their binary expansions (columns):
  .  11  101  1010  10001
         110  1011  10010
         111  1100  10011
              1101  10100
              1110  10101
              1111  10110
                    10111
                    11000
                    11010
                    11100
                    11101
                    11110
                    11111

The union of all numbers counted is A007916.
For squarefree numbers we have A077643.
For prime-powers we have A244508.
For primes instead of powers of 2 we have A377433, ones A029707.
For perfect-powers we have A377467, for primes A377432, zeros A377436.
A000225(n) counts the interval from A000051(n) to A000225(n+1).
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect-powers, differences A053289, seconds A376559.
A007916 lists the non-perfect-powers, differences A375706, seconds A376562.
A081676 gives the greatest perfect-power <= n.
A131605 lists perfect-powers that are not prime-powers.
A377468 gives the least perfect-power > n.
Cf. A000015, A013597, A014210, A014234, A023055, A045542, A052410, A061398, A304521, A377434, A377435, A377702.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Table[Length[Select[Range[2^n+1, 2^(n+1)-1],radQ]],{n,0,15}]

from sympy import mobius, integer_nthroot
def A377701(n):
    def f(x): return int(x-1+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    return f((1<Chai Wah Wu, Nov 06 2024

a(n) = 2^n-1-A377467(n). - Pontus von Brömssen, Nov 06 2024

Offset corrected by, and a(16)-a(33) from Pontus von Brömssen, Nov 06 2024

A104090 Largest prime <= 5^n.

Original entry on oeis.org

5, 23, 113, 619, 3121, 15619, 78121, 390581, 1953109, 9765619, 48828113, 244140613, 1220703073, 6103515623, 30517578121, 152587890563, 762939453109, 3814697265523, 19073486328109, 95367431640599, 476837158203071, 2384185791015571, 11920928955078089

Offset: 1

Cino Hilliard, Mar 03 2005

Cf. A014234, A054321, A104088, A104089, A104091-A104094, A003618, A104096.

Cf. A000040, A000351, A007917.

NextPrime[5^Range[30] + 1, -1] (* Paolo Xausa, Oct 29 2024 *)

g(n,b) = for(x=0,n,print1(precprime(b^x)","))

a(n) = A007917(A000351(n)). - Paolo Xausa, Oct 29 2024

A226178 Exponents n such that 2^n - previous_prime(2^n) = next_prime(2^n) - 2^n.

Original entry on oeis.org

2, 6, 12, 76, 181, 1099, 1820, 9229

Offset: 1

Jean-François Alcover, May 30 2013

The differences next_prime(2^n) - 2^n are respectively: 1, 3, 3, 15, 165, 1035, 663, 2211.

If it exists, a(9) > 10000. - Hugo Pfoertner, Feb 06 2021

			2^6 = 64, next prime = 67, previous prime = 61, 67-64 = 64-61 = 3, hence 6 is in the sequence.

Cf. A000079, A014210, A014234, A117387, A145025.

Cf. A013597, A013603, A340707.

Reap[Do[m = 2^n; p = NextPrime[m, -1]; q = NextPrime[m]; If[p + q == 2*m, Print[n]; Sow[n]], {n, 2, 10^4}]][[2, 1]]

isok(n) = my(p=2^n); p-precprime(p-1) == nextprime(p+1) - p; \\ Michel Marcus, Oct 02 2019

for(n=2,1100,my(p2=2^n,pn=nextprime(p2),pp=p2-pn+p2);if(ispseudoprime(pp),if(precprime(p2)==pp,print1(n,", ")))) \\ Hugo Pfoertner, Feb 06 2021

from itertools import count, islice
from sympy import isprime, nextprime
def A226178_gen(): # generator of terms
    return filter(lambda n:isprime(r:=((k:=1<A226178_list = list(islice(A226178_gen(),5)) # Chai Wah Wu, Aug 08 2022

A340707(a(n)) = 0. - Hugo Pfoertner, Feb 06 2021

Offset 1 from Michel Marcus, Oct 02 2019

a(8) from Hugo Pfoertner, Feb 05 2021

A333877 a(n) is the largest prime 2^(n-1) <= p < 2^n maximizing the Hamming weight of all primes in this interval.

Original entry on oeis.org

3, 7, 13, 31, 61, 127, 251, 509, 1021, 2039, 4093, 8191, 16381, 32749, 65519, 131071, 262139, 524287, 1048573, 2097143, 4194301, 8388587, 16777213, 33546239, 67108859, 134217467, 260046847, 536870909, 1073741567, 2147483647, 4294967291, 8589934583, 16911433727

Offset: 2

Hugo Pfoertner, Apr 08 2020

nonn

This differs from A014234 at n=1 and then first at n=16: a(16) = 65519 != 65521 = A014234(16). - Alois P. Heinz, Apr 25 2020

Chai Wah Wu, Table of n, a(n) for n = 2..998

Cf. A014234, A091937, A091938, A333876, A333879.

a:= proc(n) option remember; local i, p;
      for i from 0 do p:= max(select(isprime, map(l-> add(l[j]*
        2^(j-1), j=1..n), combinat[permute]([1$(n-i),0$i]))));
        if p>0 then break fi
      od; p
    end:
seq(a(n), n=2..30);  # Alois P. Heinz, Apr 23 2020

a[n_] := a[n] = MaximalBy[{#, DigitCount[#, 2, 1]}& /@ Select[Range[ 2^(n-1), 2^n-1], PrimeQ], Last][[-1, 1]];
Table[Print[n, " ", a[n]]; a[n], {n, 2, 30}] (* Jean-François Alcover, Nov 09 2020 *)

for(n=2, 30, my(hmax=0,pmax); forprime(p=2^(n-1), 2^n, my(h=hammingweight(p)); if(h>=hmax,pmax=p;hmax=h)); print1(pmax,", "))

from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A333877(n):
    for i in range(n-1,-1,-1):
        q = 2**n-1
        for d in multiset_permutations('0'*i+'1'*(n-1-i)):
            p = q-int(''.join(d),2)
            if isprime(p):
                return p # Chai Wah Wu, Apr 08 2020

A168157 Number of 0's in the matrix whose lines are the binary expansion of the first n primes.

Original entry on oeis.org

1, 1, 4, 4, 9, 10, 19, 21, 22, 23, 23, 37, 40, 42, 43, 45, 46, 47, 69, 72, 76, 78, 81, 84, 88, 91, 93, 95, 97, 100, 100, 136, 141, 145, 149, 152, 155, 159, 162, 165, 168, 171, 172, 177, 181, 184, 187, 188, 191, 194, 197, 198, 201, 202, 263, 268, 273, 277, 282, 287

Offset: 1

M. F. Hasler, Nov 21 2009

The matrix is to be taken of minimal size, i.e., have n lines and the number of columns needed to write the n-th prime in the last line, A035100(n). Otherwise said, there is no zero column except for n=1 (prime(1) = 2 = 10[2] in binary).
The number of zeros in the last line of the matrix is given by A035103(n).
One has a(n)=a(n-1) iff n = A059305(k) for some k, i.e. prime(n) is a Mersenne prime A000668(k) = A000225(A000043(k)).
If prime(n)=2^2^k+1 is a Fermat prime (A019434), n>2, then one has a(n)=a(n-1)+n-1+2^k-1.
More generally, the "big jumps" a(n+1) > a(n)+n happen whenever a column is added, i.e. when prime(n) = A014234(k) <=> prime(n+1) = A104080(k) for some k,n>1.

			a(4)=4 is the number of zeros in the matrix [010] /* = 2 in binary */ [011] /* = 3 in binary */ [101] /* = 5 in binary */ [111] /* = 7 in binary */

A168157(n)=n*#binary(prime(n))-sum(i=1,n,norml2(binary(prime(i))))

a(n)=n*A035100(n)-A095375(n).

A298817 a(n) is the binary XOR of all n-bit prime numbers.

Original entry on oeis.org

0, 1, 2, 6, 23, 59, 99, 203, 469, 807, 1615, 3349, 2266, 4576, 14042, 25002, 89193, 131215, 135904, 814531, 885682, 60842, 3969154, 3370892, 6742296, 14350136, 42766902, 97565102, 444197631, 515121776, 2085329975, 2091732354, 7999937231, 14794305847

Offset: 1

Alex Ratushnyak, Jan 26 2018

XOR is the binary exclusive-or operator.
a(1)=0 for compatibility with similar sequences, and because 0 and 1 are not primes.
Note the sequence s(n)-a(n), where s(n)=A298816(n) is the binary XOR of all n-bit squares, begins: 1, -1, 2, 3, -14, -38, -87, -175, -20, -230, -1258, -2352, 3819, 9957, -1525, -9925, 31932, 21654, 264124, 226521, 405022, 2495526, 944510, 8579700, 15679080, 49342536, -35092149, -19209773, -131473914. The distribution of negative and positive terms does not look random: runs of negative terms are followed by runs of positive terms.

			There are two 4-bit primes, namely 11 and 13.  a(4) = (11 XOR 13) = 6.

Lars Blomberg, Table of n, a(n) for n = 1..41

Cf. A000040, A007088, A070939, A035100, A298816.

Cf. also A014234, A104080.

a(n) = {my(x = 0); for (k=2^(n-1), 2^n-1, if (isprime(k), x = bitxor(x, k));); x;} \\ Michel Marcus, Jan 27 2018

from sympy import nextprime
n = x = L = 2
print('0', end=',')
while L < 27:
    nextn = nextprime(n)
    if (nextn ^ n) > n:  # if lengths of binary representations are different
        print(str(x), end=',')
        x = 0
        prevL = L
        L = len(bin(nextn))-2
        for j in range(prevL, L-1):  print('0', end=',')
    n = nextn
    x ^= n

a(30)-a(34) from Lars Blomberg, Nov 10 2018

A352942 Let p = prime(n); a(n) = number of primes q with same number of binary digits as p that can be obtained from p by changing one binary digit.

Original entry on oeis.org

1, 1, 1, 1, 0, 0, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 2, 3, 2, 0, 1, 1, 0, 2, 1, 2, 3, 1, 1, 4, 1, 0, 1, 1, 0, 1, 3, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 2, 0, 3, 2, 1, 1, 2, 2, 1, 1, 0, 3, 0, 0, 2, 2, 0, 2, 2, 2, 3, 2, 2, 0, 2, 0, 1, 2, 0, 1

Offset: 1

Michael S. Branicky, May 11 2022

a(n) is also the degree of prime(n) in the graph P(A070939(prime(n)), 2), defined in A145667.

			prime(1) = 2, in binary 10, has one neighbor 11 in P(2, 2), so a(1) = 1.
prime(14) = 43, in binary 101011, has neighbors 101001 (41), 101111 (47), 111011 (59), so a(14) = 3.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Binary analog of A125002.

Cf. A000040, A004676, A070939, A104080, A014234, A137985, A145667, A353738.

a:= n-> (p-> nops(select(isprime, {seq(Bits[Xor]
        (p, 2^i), i=0..ilog2(p)-1)})))(ithprime(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, May 11 2022

A352942[n_] := Count[BitXor[#, 2^Range[0, BitLength[#] - 2]], _?PrimeQ] & [Prime[n]];
Array[A352942, 100] (* Paolo Xausa, Apr 23 2025 *)

from sympy import isprime, sieve
def neighs(s):
    digs = "01"
    ham1 = (s[:i]+d+s[i+1:] for i in range(len(s)) for d in digs if d!=s[i])
    yield from (h for h in ham1 if h[0] != '0')
def a(n):
    return sum(1 for s in neighs(bin(sieve[n])[2:]) if isprime(int(s, 2)))
print([a(n) for n in range(1, 88)])

a(n) = deg(prime(n)) in P(A070939(prime(n)), 2) (see A145667).

A356434 Prime nearest to 2^n. In case of a tie, choose the larger.

Original entry on oeis.org

2, 2, 5, 7, 17, 31, 67, 127, 257, 509, 1021, 2053, 4099, 8191, 16381, 32771, 65537, 131071, 262147, 524287, 1048573, 2097143, 4194301, 8388617, 16777213, 33554467, 67108859, 134217757, 268435459, 536870909, 1073741827, 2147483647, 4294967291, 8589934583

Offset: 0

Peter Munn, Aug 07 2022

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000

A117387 differs by preferring the smaller prime in the case of a tie, which occurs when n is in A226178.

Cf. A014210, A014234, A340707.

Join[{2,2},Table[Max[Nearest[{NextPrime[2^n,-1],NextPrime[2^n]},2^n]],{n,2,40}]] (* Harvey P. Dale, Feb 19 2023 *)

from sympy import prevprime, nextprime
def A356434(n): return (r if (m:=nextprime(k:=1< (k<<1)-(r:=prevprime(k)) else m) if n>1 else 2 # Chai Wah Wu, Aug 08 2022

a(0) = 2; for n >= 1, if A014210(n) + A014234(n) > 2^(n+1) then a(n) = A014234(n), otherwise a(n) = A014210(n).

cp's OEIS Frontend

A265326 n-th prime minus its binary reversal.

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A377435 Number of perfect-powers x in the range 2^n <= x < 2^(n+1).

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A377701 Number of non-perfect-powers x in the range 2^n < x < 2^(n+1).

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A104090 Largest prime <= 5^n.

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A226178 Exponents n such that 2^n - previous_prime(2^n) = next_prime(2^n) - 2^n.

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A333877 a(n) is the largest prime 2^(n-1) <= p < 2^n maximizing the Hamming weight of all primes in this interval.

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A168157 Number of 0's in the matrix whose lines are the binary expansion of the first n primes.

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