Jeremias M. Gomes - cp's OEIS frontend

A346408 Primes whose bitwise XOR of decimal digits is a prime.

2, 3, 5, 7, 13, 29, 31, 41, 43, 47, 61, 83, 103, 113, 131, 137, 139, 151, 157, 173, 193, 211, 223, 227, 233, 241, 263, 269, 277, 281, 311, 317, 337, 353, 367, 373, 379, 389, 397, 401, 409, 421, 443, 461, 467, 487, 557, 571, 577, 599, 601, 641, 647, 673, 683

Offset: 1

Jeremias M. Gomes, Jul 21 2021

			421 is a term because it is a prime whose bitwise XOR of digits is 7 which is also a prime.

Wikipedia, Bitwise operation

Cf. A000040, A346511 (XOR of digits of n).

b:= l-> `if`(l=[], 0, Bits[Xor](l[1], b(subsop(1=[][], l)))):
q:= n-> isprime(b(convert(n, base, 10))):
select(q, [ithprime(i)$i=1..160])[];  # Alois P. Heinz, Jul 21 2021

Select[Range[1000], PrimeQ[#] && PrimeQ[BitXor @@ IntegerDigits[#]] &] (* Amiram Eldar, Jul 21 2021 *)

def XOR(a, b):
    return a ^^ b
[n for n in (0..100) if (n in Primes() and reduce(XOR, map(lambda x: int(x), str(n))) in Primes())]

A346512 a(n) = bitwise XOR of decimal digits of primes.

Original entry on oeis.org

2, 3, 5, 7, 0, 2, 6, 8, 1, 11, 2, 4, 5, 7, 3, 6, 12, 7, 1, 6, 4, 14, 11, 1, 14, 0, 2, 6, 8, 3, 4, 3, 5, 11, 12, 5, 3, 4, 0, 5, 15, 8, 9, 11, 15, 1, 2, 3, 7, 9, 2, 8, 7, 6, 0, 7, 13, 4, 2, 11, 9, 8, 4, 3, 1, 5, 1, 7, 0, 14, 5, 15, 2, 7, 13, 8, 2, 13, 5, 13, 12

Offset: 1

Jeremias M. Gomes, Jul 21 2021

			a(10) = 2 XOR 9 = 11 as prime(10) = 29.

Mia Boudreau, Table of n, a(n) for n = 1..10000
Wikipedia, Bitwise operation

Cf. A000040, A346408, A346511 (XOR of digits of n), A003987 (Table of n XOR m read by antidiagonals).

b:= l-> `if`(l=[], 0, Bits[Xor](l[1], b(subsop(1=[][], l)))):
a:= n-> b(convert(ithprime(n), base, 10)):
seq(a(n), n=1..82);  # Alois P. Heinz, Jul 21 2021

Table[BitXor @@ IntegerDigits[Prime[n]], {n, 1, 100}] (* Amiram Eldar, Jul 21 2021 *)

a(n) = my(d=digits(prime(n)), k=0); for (i=1, #d, k= bitxor(k, d[i])); k; \\ Michel Marcus, Jul 21 2021

def XOR(a, b):
  return a ^^ b
[reduce(XOR, map(lambda x: int(x), str(p))) for p in (0..100) if p in Primes()]

a(n) = A346511(A000040(n)). - Mia Boudreau, Aug 06 2025

A346511 a(n) = bitwise XOR of decimal digits of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 3, 2, 1, 0, 7, 6, 5, 4, 11, 10, 4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 5, 4, 7, 6, 1, 0, 3, 2, 13, 12, 6, 7, 4, 5, 2, 3, 0, 1, 14, 15, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 8, 9, 10

Offset: 0

Jeremias M. Gomes, Jul 21 2021

			a(5) = 5.
a(12) = 1 XOR 2 = 3.
a(425) = 4 XOR 2 XOR 5 = 3.

Mia Boudreau, Table of n, a(n) for n = 0..10000
Wikipedia, Bitwise operation

char a(unsigned long long n){
 char p = 0;
 while (n > 0) {p ^= n % 10; n /= 10;}
 return p;}
 // Mia Boudreau, Aug 05 2025

b:= l-> `if`(l=[], 0, Bits[Xor](l[1], b(subsop(1=[][], l)))):
a:= n-> b(convert(n, base, 10)):
seq(a(n), n=0..82);  # Alois P. Heinz, Jul 21 2021

Table[BitXor @@ IntegerDigits[n], {n, 0, 100}] (* Amiram Eldar, Jul 21 2021 *)

a(n) = my(d=digits(n), k=0); for (i=1, #d, k= bitxor(k, d[i])); k; \\ Michel Marcus, Jul 21 2021

def XOR(a, b):
  return a ^^ b
[reduce(XOR, map(lambda x: int(x), str(n))) for n in (0..1000)]

A338375 Number of digits in (2n)! / (2^n * n!).

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 18, 19, 21, 22, 24, 26, 27, 29, 31, 32, 34, 36, 37, 39, 41, 43, 45, 46, 48, 50, 52, 54, 56, 58, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 104, 106, 108, 110

Offset: 1

Jeremias M. Gomes, Oct 23 2020

a(n) is the number of digits in double factorial of odd numbers (see A001147).

			For n = 7, (2*n-1)!! = 13!! = 135135 and the number of digits is 6.

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

Cf. A001147, A055642, A068398.

seq(1 + ilog10(doublefactorial(2*n-1)), n=1..100); # Robert Israel, Jan 15 2024

a[n_] := IntegerLength[(2 n - 1)!!]; Array[a, 65] (* Amiram Eldar, Oct 23 2020 *)

floor((log(factorial(2 * n) / ((2 ** n) * factorial(n))) / log(10))) + 1

a(n) = floor(log((2n)!/((2^n)*n!))/log(10))+1.

a(n) = A055642(A001147(n)).

A308250 Squares of automorphic numbers in base 12 (cf. A201919).

Original entry on oeis.org

0, 1, 49, 64, 2401, 21904, 118336, 5764801, 1297152256, 22880797696, 149429406721, 3114293149696, 33232930569601, 2203546857570304, 3418922510231809

Offset: 1

Jeremias M. Gomes, May 17 2019

			2401 = c37_14 and sqrt(c37_14) = 37_14. Hence 2401 is in the sequence.

Cf. A201919, A237583, A308250.

[(n * n) for n in (0..1000000) if (n * n).str(base = 14).endswith(n.str(base = 14))]

Equals A201919(n)^2.

A308249 Squares of automorphic numbers in base 12 (cf. A201918).

Original entry on oeis.org

0, 1, 16, 81, 4096, 6561, 263169, 1478656, 40960000, 205549569, 54988374016, 233605955584, 6263292059649, 303894740860929, 338531738189824, 170196776412774400, 709858175909625856, 18638643564726714369, 124592287100855910400, 2576097707358918017025, 479214351668445504864256

Offset: 1

Jeremias M. Gomes, May 17 2019

(sqrt(m))12 is a suffix of m_12. - _A.H.M. Smeets, Aug 09 2019

All terms k^2 in this sequence (except the trivials 0 and 1) have a square root k that is the suffix of one of the 12-adic numbers given by A259468 or A259469. From this, the sequence has an infinite number of terms. - A.H.M. Smeets, Aug 09 2019

			4096 = 2454_12 and sqrt(2454_12) = 54_12. Hence 4096 is in the sequence.

V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.

Cf. A201918, A259468, A259469.

dig = "0123456789AB"
def To12(n):
    s = ""
    while n > 0:
        s, n = dig[n%12]+s, n//12
    return s
n, m = 1, 0
print(n,m*m)
while n < 100:
    m = m+1
    m2, m1 = To12(m*m), To12(m)
    i, i2, i1 = 0, len(m2), len(m1)
    while i < i1 and (m2[i2-i-1] == m1[i1-i-1]):
        i = i+1
    if i == i1:
        print(n,m*m)
n = n+1 # A.H.M. Smeets, Aug 09 2019

[(n * n) for n in (0..1000000) if (n * n).str(base = 12).endswith(n.str(base = 12))]

Equals A201918(n)^2.

Terms a(16)..a(21) from A.H.M. Smeets, Aug 09 2019

A308248 Squares of automorphic numbers in base 6 (cf. A237583).

Original entry on oeis.org

0, 1, 9, 16, 81, 784, 6561, 18496, 1478656, 43046721, 281165824, 893352321, 5859137025, 41368305664, 405026597889, 1088266240000, 15965210931201, 36991307874304, 583272781383681, 1318789102698496

Offset: 1

Jeremias M. Gomes, May 17 2019

			784 = 3344_6 and sqrt(3344_6) = 44_6. Hence 784 is in the sequence.

Cf. A237583.

[(n * n) for n in (0..1000000) if (n * n).str(base = 6).endswith(n.str(base = 6))]

a(n) = A237583(n)^2. - Michel Marcus, May 17 2019

A319598 Numbers in base 10 that are palindromic in bases 2, 4, 8, and 16.

Original entry on oeis.org

0, 1, 3, 5, 4095, 4097, 12291, 20485, 21845, 16777215, 16777217, 16781313, 50331651, 50343939, 83886085, 83906565, 89458005, 89478485, 68702703615, 68719476735, 68719476737, 68736258049, 206158430211, 206208774147, 343597383685, 343602954245, 343681290245

Offset: 1

Jeremias M. Gomes, Sep 24 2018

Intersection of A006995, A014192, A029803, and A029730.

This sequence is infinite because it contains terms of the forms 4096^k-1 (k>=0) and 4096^k+1 (k>0). - Bruno Berselli, Sep 24 2018

			4095 = 111111111111_2 = 333333_4 = 7777_8 = FFF_16. Hence 4095 is in the sequence.

A.H.M. Smeets, Table of n, a(n) for n = 1..114 (first 70 terms from Rémy Sigrist)

Cf. A006995 (base 2), A014192 (base 4), A029803 (base 8), and A029730 (base 16).

palQ[n_, b_] := PalindromeQ[IntegerDigits[n, b]];
Reap[Do[If[palQ[n, 2] && palQ[n, 4] && palQ[n, 8] && palQ[n, 16], Print[n]; Sow[n]], {n, 0, 10^6}]][[2, 1]] (* Jean-François Alcover, Sep 25 2018 *)

[n for n in (0..100000) if Word(n.digits(2)).is_palindrome() and Word(n.digits(4)).is_palindrome() and Word(n.digits(8)).is_palindrome() and Word(n.digits(16)).is_palindrome()]

a(19)-a(27) from Rémy Sigrist, Nov 15 2018

A319609 Numbers in base 10 that are palindromic in bases 4, 8 and 16.

Original entry on oeis.org

0, 1, 2, 3, 5, 170, 4095, 4097, 8194, 12291, 20485, 21845, 696490, 699050, 16777215, 16777217, 16781313, 16785409, 16789505, 33554434, 33558530, 33562626, 33566722, 50331651, 50335747, 50339843, 50343939, 83886085, 83906565, 89458005, 89478485

Offset: 1

Jeremias M. Gomes, Sep 24 2018

			170 = 2222_4 = 252_8 = AA_16. Hence 170 is in the sequence.

Intersection of A014192, A029803 and A029730.

[n: n in [0..2*10^7] | Intseq(n, 4) eq Reverse(Intseq(n, 4)) and Intseq(n, 8) eq Reverse(Intseq(n, 8)) and Intseq(n, 16) eq Reverse(Intseq(n, 16))]; // Vincenzo Librandi, Sep 24 2018

palQ[n_, b_] := PalindromeQ[IntegerDigits[n, b]];
Reap[Do[If[palQ[n, 4] && palQ[n, 8] && palQ[n, 16], Print[n]; Sow[n]], {n, 0, 10^6}]][[2, 1]] (* Jean-François Alcover, Sep 25 2018 *)

[n for n in (0..100000) if Word(n.digits(4)).is_palindrome() and Word(n.digits(8)).is_palindrome() and Word(n.digits(16)).is_palindrome()]

A319585 Numbers in base 10 that are palindromic in bases 2, 8, and 16.

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 3951, 4095, 4097, 12291, 20485, 21845, 28679, 30039, 36873, 16187247, 16777215, 16777217, 16781313, 50331651, 50343939, 83886085, 83894277, 83906565, 83914757, 89458005, 89466197, 89478485, 89486677, 117440519, 117448711, 117460999

Offset: 1

Jeremias M. Gomes, Sep 23 2018

Intersection of A006995, A029803, and A029730.

			16187247 = 111101101111111101101111_2 = 75577557_8 = F6FF6F_16.

Cf. A006995 (base 2), A029803 (base 8), and A029730 (base 16).

[n: n in [0..2*10^7] | Intseq(n, 2) eq Reverse(Intseq(n, 2)) and Intseq(n, 8) eq Reverse(Intseq(n, 8)) and Intseq(n, 16) eq Reverse(Intseq(n, 16))]; // Vincenzo Librandi, Sep 24 2018

palQ[n_, b_] := PalindromeQ[IntegerDigits[n, b]];
Reap[Do[If[palQ[n, 2] && palQ[n, 8] && palQ[n, 16], Print[n]; Sow[n]], {n, 0, 10^6}]][[2, 1]] (* Jean-François Alcover, Sep 25 2018 *)

[n for n in (0..10000) if Word(n.digits(2)).is_palindrome() and Word(n.digits(8)).is_palindrome() and Word(n.digits(16)).is_palindrome()]

cp's OEIS Frontend

User: Jeremias M. Gomes

A346408 Primes whose bitwise XOR of decimal digits is a prime.

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A346512 a(n) = bitwise XOR of decimal digits of primes.

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A346511 a(n) = bitwise XOR of decimal digits of n.

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A338375 Number of digits in (2n)! / (2^n * n!).

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A308250 Squares of automorphic numbers in base 12 (cf. A201919).

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A308249 Squares of automorphic numbers in base 12 (cf. A201918).

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Python

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A308248 Squares of automorphic numbers in base 6 (cf. A237583).

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