A255401 -id:A255401 - cp's OEIS frontend

A222008 Primes of the form 4^k + 1 for some k > 0.

5, 17, 257, 65537

Offset: 1

Jonathan Sondow, Feb 04 2013

nonn

Same as Fermat primes 2^(2^m) + 1 for m >= 1. See A019434 for comments, etc.
Chebyshev showed that 3 is a primitive root of all primes of the form 2^(2*k) + 1 with k > 0. If the sequence is infinite, then Artin's conjecture ("every nonsquare integer n != -1 is a primitive root of infinitely many primes q") is true for n = 3.
As a(n) is a Fermat prime > 3, by Pépin's test a(n) has primitive root 3.
Conjecture: let p a prime number, a(n) is not congruent to p mod (p^2-3)/2. - Vincenzo Librandi, Jun 15 2014
This conjecture is false when p = a(n), but may be true for primes p != a(n). - Jonathan Sondow, Jun 15 2014
Primes p with the property that k-th smallest divisor of its squares p^2, for all 1 <= k <= tau(p^2), contains exactly k "1" digits in its binary representation. Corresponding values of squares p^2: 25, 289, 66049, 4295098369. Example: p = 257, set of divisors of p^2 = 66049 in binary representation: {1, 100000001, 10000001000000001}. Subsequence of A255401. - Jaroslav Krizek, Dec 21 2016

			4^1 + 1 = 5 is prime, so a(1) = 5. Also, 3^k == 3, 4, 2, 1 (mod 5) for k = 1, 2, 3, 4, resp., so 3 is a primitive root for a(1).

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 102, nr. 3.
P. L. Chebyshev, Theory of congruences. Elements of number theory, Chelsea, 1972, p. 306.

P. L. Chebyshev, Theorie der Congruenzen, Mayer & Mueller, 1889, p. 306-313.
R. Fueter, Über primitive Wurzeln von Primzahlen, Comment. Math. Helv., 18 (1946), 217-223, p. 217.
Wikipedia, Pépin's test
Index entries for sequences related to Artin's conjecture
Index entries for primes by primitive root

Cf. A005596, A019334, A019434, A090866, A221981.

Select[Table[4^k + 1, {k, 10^3}], PrimeQ] (* Michael De Vlieger, Dec 22 2016 *)

a(n) = A019434(n+1) for n > 0.

A354722 Composite numbers whose divisors have distinct binary weights (A000120).

Original entry on oeis.org

25, 39, 55, 57, 69, 87, 95, 111, 115, 119, 121, 123, 125, 141, 145, 159, 169, 177, 183, 185, 187, 201, 203, 205, 213, 215, 219, 221, 235, 237, 249, 253, 265, 289, 291, 299, 301, 303, 305, 319, 321, 323, 329, 335, 339, 355, 361, 365, 371, 377, 391, 393, 411, 413

Offset: 1

Amiram Eldar, Jun 04 2022

Without the restriction of composite numbers, 1 and all the odd primes would have been terms of this sequence.

Since 1 and 2 have the same binary weight, all the terms are odd.

			25 is a term since its divisors, 1, 5 and 25, have binary weights 1, 2 and 3, respectively.
55 is a term since its divisors, 1, 5, 11 and 55, have binary weights 1, 2, 3 and 5, respectively.

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

Cf. A000120, A027750.

Subsequences: A255401 and A354724.

bw[n_] := DigitCount[n, 2, 1]; q[n_] := CompositeQ[n] && UnsameQ @@ (bw /@ Divisors[n]); Select[Range[1, 400, 2], q]

isok(c) = {if ((c>1) && !isprime(c), my(d=divisors(c)); #Set(apply(hammingweight, d)) == #d;);} \\ Michel Marcus, Jun 04 2022

from sympy import divisors
def binwt(n): return bin(n).count("1")
def ok(n):
    binwts, divs = set(), 0
    for d in divisors(n, generator=True):
        b = binwt(d)
        if b in binwts: return False
        binwts.add(b)
        divs += 1
    return divs > 2
print([k for k in range(415) if ok(k)]) # Michael S. Branicky, Jun 04 2022

A354724 Numbers k whose ordered binary weights (A000120) of their divisors are the numbers 1 to A000005(k).

Original entry on oeis.org

1, 3, 5, 17, 25, 39, 57, 69, 145, 201, 257, 265, 289, 291, 323, 393, 579, 1075, 1083, 2307, 2645, 2875, 4205, 4503, 5555, 5593, 7955, 8815, 9399, 9401, 9519, 11033, 11155, 11407, 12297, 12455, 12711, 12909, 13205, 13281, 13611, 13737, 14001, 14915, 15879, 16629

Offset: 1

Amiram Eldar, Jun 04 2022

The subsequence of terms whose divisors have binary weights in order is A255401.

All terms are odd. Proof: All binary weights must be distinct but the binary weights of k and 2*k are equal. A contradiction. - David A. Corneth, Jun 04 2022

			3 is a term since its divisors, 1 and 3, have binary weights 1 and 2, respectively.
69 is a term since its divisors, 1, 3, 23 and 69, have binary weights 1, 2, 4 and 3, respectively.

Amiram Eldar, Table of n, a(n) for n = 1..2475

Cf. A000005, A000120, A354722.

A255401 is a subsequence.

bw[n_] := DigitCount[n, 2, 1]; q[n_] := Module[{d = Divisors[n]}, Union[bw /@ d] == Range[Length[d]]]; Select[Range[1, 10^4, 2], q]

from sympy import divisors
def binwt(n): return bin(n).count("1")
def ok(n):
    if n%2 == 0: return False
    binwts, divs = set(), 0
    for d in divisors(n, generator=True):
        b = binwt(d)
        if b in binwts: return False
        binwts.add(b)
        divs += 1
    return binwts == set(range(1, divs+1))
print([k for k in range(20000) if ok(k)]) # Michael S. Branicky, Jun 04 2022

A253204 a(1) = 1; for n>1, a(n) is a prime power p^h (h>=1) with the property that its k-th smallest divisor, for all 1 <= k <= tau(p^h), contains exactly k "1" digits in its binary representation.

Original entry on oeis.org

1, 3, 5, 17, 25, 257, 289, 65537, 66049, 4295098369

Offset: 1

Jaroslav Krizek, Mar 25 2015

Supersequence of A019434 (Fermat primes). Subsequence of A071593 and A255401.

Sequence is finite if there is only 5 Fermat primes (A019434).

			The divisors of 4295098369, expressed in base 2 and listed in ascending order as 1, 10000000000000001, 100000000000000100000000000000001, contain 1, 2 and 3, "1" digits, respectively.

Cf. A019434, A071593, A255401.

Set(Sort([1] cat [n: n in [2..1000000] | [&+Intseq(d, 2): d in Divisors(n)] eq [1..NumberOfDivisors(n)] and #(PrimeDivisors(n)) eq 1]));

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A222008 Primes of the form 4^k + 1 for some k > 0.

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A354722 Composite numbers whose divisors have distinct binary weights (A000120).

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A354724 Numbers k whose ordered binary weights (A000120) of their divisors are the numbers 1 to A000005(k).

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A253204 a(1) = 1; for n>1, a(n) is a prime power p^h (h>=1) with the property that its k-th smallest divisor, for all 1 <= k <= tau(p^h), contains exactly k "1" digits in its binary representation.

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