A052330 -id:A052330 - cp's OEIS frontend

A240521 a(n) = A050376(n)*A050376(n+1) where A050376(n) is the n-th number of the form p^(2^k) with p is prime and k >= 0.

6, 12, 20, 35, 63, 99, 143, 208, 272, 323, 437, 575, 725, 899, 1147, 1517, 1763, 2021, 2303, 2597, 3127, 3599, 4087, 4757, 5183, 5767, 6399, 6723, 7387, 8633, 9797, 10403, 11021, 11663, 12317, 13673, 15367, 16637, 17947, 19043, 20711, 22499, 23707, 25591

Offset: 1

Vladimir Shevelev, Apr 07 2014

nonn

Let m be an odd positive number. Let S_m denote the sequence {Product_{i=1..r} q_(n+t_i)}A050376%20and%20Sum">{n>=1}, where {q_i} is sequence A050376 and Sum{i=1..r} 2^(t_1 - t_i) is the binary representation of m, such that t_1 > t_2 > ... > t_r = 0. Note that {S_1, S_3, S_5, ...} is a partition of all integers > 1. Then S_1=A050376, which is obtained when we set r=1, t_1 = 0. [Formula made compatible with A240535 data by Peter Munn, Aug 10 2021]
This present sequence is S_3 in this partition. It is obtained when we set r=2, t_1=1, t_2=0.
S_m(n) = A052330(A030101(m)*2^(n-1)) = A329330(A050376(n), A052330(A030101(m))). - Peter Munn, Aug 10 2021
A minimal set of generators for A000379 as a group under A059897(.,.). - Peter Munn, Aug 11 2019

Eric Weisstein's World of Mathematics, Group.
Wikipedia, Generating set of a group.

Positions of 3's in A240535.
Sequences for other parts of the partition described in the first comment: A050376 (S_1), A240522 (S_5), A240524 (S_7), A240536 (S_9), A241024 (S_11), A241025 (S_13).
Cf. A000379, A030101, A052330, A059897, A329330.

from sympy import primepi, integer_nthroot
def A240521(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(integer_nthroot(x,1<Chai Wah Wu, Feb 18-19 2025

a(n) = A052330(3*2^(n-1)) = A329330(A050376(n), 6). - Peter Munn, Aug 10 2021

More terms from Peter J. C. Moses, Apr 18 2014

A329575 Numbers whose smallest Fermi-Dirac factor is 3.

Original entry on oeis.org

3, 12, 15, 21, 27, 33, 39, 48, 51, 57, 60, 69, 75, 84, 87, 93, 105, 108, 111, 123, 129, 132, 135, 141, 147, 156, 159, 165, 177, 183, 189, 192, 195, 201, 204, 213, 219, 228, 231, 237, 240, 243, 249, 255, 267, 273, 276, 285, 291, 297, 300, 303, 309, 321, 327, 336, 339, 345

Offset: 1

Peter Munn, Apr 27 2020

nonn

Every positive integer is the product of a unique subset of the terms of A050376 (sometimes called Fermi-Dirac primes). This sequence lists the numbers where the relevant subset includes 3 but not 2.
Numbers whose squarefree part is divisible by 3 but not 2.
Positive multiples of 3 that survive sieving by the rule: if m appears then 2m, 3m and 6m do not. Asymptotic density is 1/6.

			6 is the product of the following terms of A050376: 2, 3. These terms include 2, so 6 is not in the sequence.
12 is the product of the following terms of A050376: 3, 4. These terms include 3, but not 2, so 12 is in the sequence.
20 is the product of the following terms of A050376: 4, 5. These terms do not include 3, so 20 is not in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jan Snellman, Greedy Regular Convolutions, arXiv:2504.02795 [math.NT], 2025.

Cf. A007913, A036668, A050376, A059897, A223490, A307150, A339690.
Intersection of any 2 of A003159, A145204 and A325424; also subsequence of A028983.
Ordered 3rd quadrisection of A052330.

f[p_, e_] := p^(2^IntegerExponent[e, 2]); fdmin[n_] := Min @@ f @@@ FactorInteger[n]; Select[Range[350], fdmin[#] == 3 &] (* Amiram Eldar, Nov 27 2020 *)

isok(m) = core(m) % 6 == 3; \\ Michel Marcus, May 01 2020

from itertools import count
from sympy import integer_log
def A329575(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n+x
        for i in range(integer_log(x,9)[0]+1):
            i2 = 9**i
            for j in count(0,2):
                k = i2<x:
                    break
                m = x//k
                c -= (m-1)//6+(m-5)//6+2
        return c
    return 3*bisection(f,n,n) # Chai Wah Wu, Apr 10 2025

A223490(a(n)) = 3.
A007913(a(n)) == 3 (mod 6).
A059897(2, a(n)) = 2 * a(n).
A059897(3, a(n)) * 3 = a(n).
{a(n) : n >= 1} = {k : 3 * A307150(k) = 2 * k}.
A003159 = {a(n) / 3 : n >= 1} U {a(n) : n >= 1}.
A036668 = {a(n) / 3 : n >= 1} U {a(n) * 2 : n >= 1}.
A145204 \ {0} = {a(n) : n >= 1} U {a(n) * 2 : n >= 1}.
a(n) = 3*A339690(n). - Chai Wah Wu, Apr 10 2025

A096113 a(1) = 1, a(2) = 2; then all new products of subsets of pre-existing terms, then the first integer not present, and so on.

Original entry on oeis.org

1, 2, 3, 6, 4, 8, 12, 18, 24, 36, 48, 72, 144, 5, 10, 15, 16, 20, 30, 32, 40, 54, 60, 64, 80, 90, 96, 108, 120, 160, 180, 192, 216, 240, 270, 288, 320, 324, 360, 384, 432, 480, 540, 576, 648, 720, 768, 864, 960, 1080, 1152, 1296, 1440, 1536, 1620, 1728, 1920, 1944

Offset: 1

Amarnath Murthy, Jun 29 2004

nonn

Another rearrangement of the natural numbers.

Description: the iterative extension of the sequence is a loop over the steps: (i) Select the smallest integer not yet in the sequence and append it. (ii) Compute a set of all products of two or more distinct factors taken from the current, finite version of the sequence. (iii) Remove members from this set that are already in the sequence. Append the sorted list of the numbers in the set to the sequence. Return to (i). - R. J. Mathar, Feb 21 2009

			a(3) = 3 because all products of {1, 2} are already included. The only new product generated by {1, 2, 3} is 6, then 4 is the first integer which doesn't appear. Then {1, 2, 3, 6, 4} generates 8 (=2*4), 12 (=2*6=3*4), 18 (=3*6), 24 (=6*4=2*3*4), 36 (=2*3*6), 48 (=2*6*4), 72 (=3*6*4) and 144 (=2*3*6*4). Then the next term is 5. And so on.

R. J. Mathar, Table of n, a(n) for n = 1..385

Cf. A096111, A052330.

L[1]={1} L[n_]:=L[n]=Join[L[n-1], Complement[Union[Exp[Map[ Total,Log[Subsets[Delete[L[n-1],1]]]]]],L[n-1]],{n}] L[6]

Edited by Joel B. Lewis, Nov 15 2006

A182979 Fermi-Dirac representation of n. Let n have factorization p1^(2^e1) * p2^(2^e2) * ... * pr^(2^er), where each factor is in A050376. The number n is represented by a binary string that indicates which terms of A050376 appear in the factorization of n.

Original entry on oeis.org

0, 1, 10, 100, 1000, 11, 10000, 101, 100000, 1001, 1000000, 110, 10000000, 10001, 1010, 100000000, 1000000000, 100001, 10000000000, 1100, 10010, 1000001, 100000000000, 111, 1000000000000, 10000001, 100010, 10100, 10000000000000, 1011, 100000000000000, 100000001, 1000010, 1000000001, 11000, 100100

Offset: 1

Daniel Forgues, Feb 10 2011, Feb 13 2011

nonn

Every number has a unique representation as a product of terms from A050376. - N. J. A. Sloane, Feb 11 2011
The "Fermi-Dirac factorization" of n, i.e., the factorization of n into prime powers of the form p_k^(2^e_k), e_k >= 0, (A050376) allows each of those prime powers to be used at most once, since this corresponds to the binary representation of the exponents of the prime powers p^a of the "Bose-Einstein factorization" of n, i.e., the classic prime factorization of n. (Cf. A050376 comments.)
The prime powers of the form p_k^(2^e_k), e_k >= 0 (A050376) might be called "Fermi-Dirac primes" since they may appear at most once (thus raised to powers 0 or 1) in the "Fermi-Dirac factorization" of n. Compare with the classic prime factorization of n, which might be called the "Bose-Einstein factorization" of n, where the primes (which might be called "Bose-Einstein primes") may appear any number of times >= 0.
In the "Fermi-Dirac representation" of n, if a given prime power with powers of two as exponents does not appear in the factorization of n into prime powers with powers of two as exponents, we use 0 as a placeholder; otherwise, we use 1 to indicate that the given prime power with powers of two as exponents does appear in the "Fermi-Dirac factorization" of n.
In the base-b representation of n, we do not show the leading 0's, except for 0 where it is more convenient to show it than to show nothing. Similarly, for the "Fermi-Dirac representation" of n, we do not show the leading 0's, except for 0, which is the representation of 1, where it is more convenient to show it than to show nothing.
The limit of the supremum of the number of "binary digits" of the representation of n is asymptotic to the number of primes up to n, i.e., n/log(n), making this representation absolutely impractical!
See A052330 for the numbers having representation as 0, 1, 10, 11, 100, 101, 110, 111, ... which is an ordering of the positive integers. (Cf. OEIS Wiki page.)
Let n have factorization (f_r)^g_r * ... * (f_2)^g_2 * (f_1)^g_1, where f_i is the i-th prime power of the form p_k^(2^e_k), e_k >= 0 (A050376, A302778); then a(n) = Sum_{i=1..r} g_i * 2^(i-1).
The number of 1's in a(n) is the number of terms of A050376 dividing n with odd maximal exponent. For example, if n=96, then the maximal exponent of 2 that divides 96 is 5, for 3 it is 1, for 4 it is 2, for 16 it is 1. Thus only 2, 3 and 16 divide n with odd maximal exponents. Therefore, the number of 1's in a(96) is 3. Moreover, since 2=A050376(1), 3=A050376(2) and 16=A050376(9), then 1's appear in positions 1,2,9 from the right. - Vladimir Shevelev, Nov 02 2013

			"Fermi-Dirac factorizations" (cf. A050376 examples, here with exponents of "Fermi-Dirac primes" being either 0 or 1):
6 = 3*2 = 3^1*2^1, so a(6) = 11;
8 = 4*2 = 4^1*3^0*2^1, so a(8) = 101;
20 = 5*4 = 5^1*4^1*3^0*2^0, so a(20) = 1100;
24 = 4*3*2 = 4^1*3^1*2^1, so a(24) = 111;
27 = 9*3 = 9^1*7^0*5^0*4^0*3^1*2^0, so a(27) = 100010;
32 = 16*2 = 16^1*13^0*11^0*9^0*7^0*5^0*4^0*3^0*2^1, so a(32) = 100000001;
64 = 16*4 = 16^1*13^0*11^0*9^0*7^0*5^0*4^1*3^0*2^0, so a(64) = 100000100;
108 = 9*4*3 = 9^1*7^0*5^0*4^1*3^1*2^0, so a(108) = 100110;
120 = 5*4*3*2 = 5^1*4^1*3^1*2^1, so a(120) = 1111;
...

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, "Fermi-Dirac representation" of n
OEIS Wiki, Ordering of positive integers by increasing "Fermi-Dirac representation"

Cf. A050376, A052330, A052331, A302778.

nn=24; p=Select[Range[nn], PrimeQ]; Do[p=Select[Union[p,p^2], #<=nn&], {Floor[Log[2,Log[2,nn]]]}]; Table[m=n; FromDigits[Table[If[Mod[m,i]==0, m=m/i; 1, 0], {i,Reverse[p]}]],{n,nn}]

Let q_1,q_2,q_3,... be consecutive terms of A050376 and n = q_1^a_1 * q_2^a_2 *...* q_r^a_r, where a_i = 0 or 1. Then a(n) = a_1 + 10*a_2 + ... +10^(r-1)*a_r. For example, since 30 = 2^1 * 3^1 * 4^0 * 5^1, then a(30)= 1 + 10 + 1000 = 1011. - Vladimir Shevelev, Nov 02 2013

a(n) = A007088(A052331(n)). - Antti Karttunen, Apr 17 2018

Clearer definition from T. D. Noe, Feb 11 2011

Edited by N. J. A. Sloane, Jul 21 2018

A334110 The squares of squarefree numbers (A062503), ordered lexicographically according to their prime factors. a(n) = Product_{k in I} prime(k+1)^2, where I are the indices of nonzero binary digits in n = Sum_{k in I} 2^k.

Original entry on oeis.org

1, 4, 9, 36, 25, 100, 225, 900, 49, 196, 441, 1764, 1225, 4900, 11025, 44100, 121, 484, 1089, 4356, 3025, 12100, 27225, 108900, 5929, 23716, 53361, 213444, 148225, 592900, 1334025, 5336100, 169, 676, 1521, 6084, 4225, 16900, 38025, 152100, 8281, 33124, 74529, 298116, 207025, 828100, 1863225, 7452900, 20449, 81796, 184041

Offset: 0

Antti Karttunen and Peter Munn, May 01 2020

nonn

For the lexicographic ordering, the prime factors must be written in nonincreasing order. If we write the factors in nondecreasing order, we get a lexicographically ordered set with an order type that is greater than a natural number index - the resulting sequence does not include all qualifying numbers. (Note also that the symbols used for the lexicographic order are the prime numbers, not their digits.)
a(n) is the n-th power of 4 in the monoid defined in A331590.
Conjecture: a(n) is the position of the first occurrence of n in A334109.

			The initial terms are shown below, equated with the product of their prime factors to exhibit the lexicographic ordering. The list starts with 1, since 1 is factored as the empty product and the empty list is first in lexicographic order.
    1 = .
    4 = 2*2.
    9 = 3*3.
   36 = 3*3*2*2.
   25 = 5*5.
  100 = 5*5*2*2.
  225 = 5*5*3*3.
  900 = 5*5*3*3*2*2.
   49 = 7*7.
  196 = 7*7*2*2.
  441 = 7*7*3*3.

Cf. A000079, A019565 (square roots), A048675, A097248, A225546, A267116, A332382, A334109 (a left inverse).
Column 2 of A329332. Permutation of A062503.
After 1, the right children of the leftmost edge of A334860, or respectively, the left children of the rightmost edge of A334866.
Subsequences: A001248, A061742, A166329.
Subsequence of A052330.
A003961, A003987, A059897, A331590 are used to express relationship between terms of this sequence.

Array[If[# == 0, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[3^#]]] &, 51, 0] (* Michael De Vlieger, May 26 2020 *)

A334110(n) = { my(p=2,m=1); while(n, if(n%2, m *= p^2); n >>= 1; p = nextprime(1+p)); (m); };

a(n) = A019565(n)^2.
For n >= 1, a(A000079(n-1)) = A001248(n).
For all n >= 0, A334109(a(n)) = n.
a(n+k) = A331590(a(n), a(k)).
a(n XOR k) = A059897(a(n), a(k)), where XOR denotes bitwise exclusive-or, A003987.
a(n) = A225546(3^n).
a(2n) = A003961(a(n)).
a(2n+1) = 4 * a(2n).
a(2^k-1) = A061742(k).
A267116(a(n)) = 2.
A048675(a(n)) = 2n.
A097248(a(n)) = A332382(n) = A019565(2n).

A096115 If n = (2^k)-1, a(n) = a((n+1)/2) = k, if n = 2^k, a(n) = a(n-1)+1 = k+1, otherwise a(n) = (A000523(n)+1)*a(A035327(n-1)).

Original entry on oeis.org

1, 2, 2, 3, 6, 6, 3, 4, 12, 24, 24, 12, 8, 8, 4, 5, 20, 40, 40, 60, 120, 120, 60, 20, 15, 30, 30, 15, 10, 10, 5, 6, 30, 60, 60, 90, 180, 180, 90, 120, 360, 720, 720, 360, 240, 240, 120, 30, 24, 48, 48, 72, 144, 144, 72, 24, 18, 36, 36, 18, 12, 12, 6, 7, 42, 84, 84, 126

Offset: 1

Amarnath Murthy, Jun 30 2004

nonn

A fractal sequence. For k in range [1,(2^n)-1], a(2^n + k)/a(2^n - k) = n+1. Each n > 1 occurs 2*A045778(n) times in the sequence.

Permutation of A096111, i.e. a(n) = A096111(A122199(n)-1) [Note the different starting offsets]. Cf. A096113, A052330, A096114, A096116.

(define (A096115 n) (cond ((pow2? (+ n 1)) (+ 1 (A000523 n))) ((pow2? n) (+ 1 (A096115 (- n 1)))) (else (* (+ (A000523 n) 1) (A096115 (A035327 (- n 1)))))))
(define (pow2? n) (and (> n 0) (zero? (A004198bi n (- n 1)))))

Edited, extended and Scheme code added by Antti Karttunen, Aug 25 2006

A059900 Successively interleaved alternate bits in binary expansion of n gives vector of exponents in prime factorization of a(n).

Original entry on oeis.org

1, 2, 3, 6, 4, 8, 12, 24, 5, 10, 15, 30, 20, 40, 60, 120, 16, 32, 48, 96, 64, 128, 192, 384, 80, 160, 240, 480, 320, 640, 960, 1920, 9, 18, 27, 54, 36, 72, 108, 216, 45, 90, 135, 270, 180, 360, 540, 1080, 144, 288, 432, 864, 576, 1152, 1728, 3456, 720, 1440, 2160

Offset: 0

Marc LeBrun, Feb 07 2001

Every value appears once. Inverse of A059884. Similar to A052330 (they diverge at n=16) but assigns bits via interleaving versus numerical order.

			To find a(7): 7 = ...000111; ... 0 0 1 1 = 3 -> 2^3 = 8; ... 0 1 = 1 -> 3^1 = 3; so a(7) = 3*8 = 24.

Index entries for sequences that are permutations of the natural numbers

Cf. A059884, A052330.

A096114 a(1)=1, a(2)=2, a(32^k) = 32^k, a(32^k + i) = 32^k + a(32^k - i), for i in range [1, 32^k - 1].

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 10, 11, 9, 8, 7, 12, 19, 20, 21, 23, 22, 18, 16, 17, 15, 14, 13, 24, 37, 38, 39, 41, 40, 42, 46, 47, 45, 44, 43, 36, 31, 32, 33, 35, 34, 30, 28, 29, 27, 26, 25, 48, 73, 74, 75, 77, 76, 78, 82, 83, 81, 80, 79, 84, 91, 92, 93, 95, 94, 90, 88, 89, 87, 86, 85, 72

Offset: 1

Amarnath Murthy, Jun 29 2004

nonn

Inverse: A121664. Cf. A096111, A096113, A052330, A096116.

a = {1, 2}; Do[a = Join[a, {3*2^k}, 3*2^k + Reverse[a]], {k, 0, 4}]; a (* Ivan Neretin, Sep 04 2017 *)

Edited, extended and Scheme code added by Antti Karttunen, Aug 25 2006

A186287 a(n) is the denominator of the rational number whose "factorization" into terms of A186285 has the balanced ternary representation corresponding to n.

Original entry on oeis.org

1, 1, 2, 1, 1, 6, 3, 3, 2, 1, 1, 2, 1, 1, 30, 15, 15, 10, 5, 5, 10, 5, 5, 6, 3, 3, 2, 1, 1, 2, 1, 1, 6, 3, 3, 2, 1, 1, 2, 1, 1, 105, 105, 105, 35, 35, 35, 35, 35, 35, 21, 21, 21, 7, 7, 7, 7, 7, 7, 21, 21, 21, 7, 7, 7, 7, 7, 7, 15, 15, 15, 5, 5, 5, 5, 5, 5, 3, 3, 3, 1, 1, 1, 1, 1, 1, 3, 3, 3

Offset: 0

Daniel Forgues, Feb 17 2011

Denominators from the ordering of positive rational numbers by increasing balanced ternary representation of the "factorization" of positive rational numbers into terms of A186285 (prime powers with a power of three as exponent).

			The balanced ternary digits {-1,0,+1} are represented here as {2,0,1}.
   n BalTern A186286/A186287 (in reduced form)
   0      0  Empty product = 1 = 1/1, a(n) = 1
   1      1  2 = 2/1,                 a(n) = 1
   2     12  3*(1/2) = 3/2,           a(n) = 2
   3     10  3 = 3/1,                 a(n) = 1
   4     11  3*2 = 6 = 6/1,           a(n) = 1
   5    122  5*(1/3)*(1/2) = 5/6,     a(n) = 6
   6    120  5*(1/3) = 5/3,           a(n) = 3
   7    121  5*(1/3)*2 = 10/3,        a(n) = 3
...    ...
  41  12222  8*(1/7)*(1/5)*(1/3)*(1/2) = 8/210 = 4/105, a(n) = 105

Daniel Forgues, Table of n, a(n) for n = 0..9841
OEIS Wiki, Orderings of rational numbers

Cf. A186285, A186286, A185169, A052330.

The balanced ternary representation of n
    n = Sum(i=0..1+floor(log_3(2|n|)) n_i * 3^i, n_i in {-1,0,1},
  is taken as the representation of the "factorization" of the positive rational number c(n)/d(n) into terms from A186285
    c(n)/d(n) = Prod(i=0..1+floor(log_3(2|n|)) (A186285(i+1))^(n_i), where A186285(i+1) is the (i+1)th prime power with exponent being a power of 3. Then a(n) is the denominator, i.e., d(n).

A322822 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(2) = -1, f(n) = 0 if n is a Fermi-Dirac prime (A050376) > 2, and f(n) = A300840(n) for all other numbers.

Original entry on oeis.org

1, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 9, 3, 3, 10, 3, 11, 12, 13, 3, 7, 3, 14, 15, 16, 3, 9, 3, 17, 18, 19, 20, 21, 3, 22, 23, 11, 3, 12, 3, 24, 25, 26, 3, 27, 3, 28, 29, 30, 3, 15, 31, 16, 32, 33, 3, 34, 3, 35, 36, 37, 38, 18, 3, 39, 40, 20, 3, 21, 3, 41, 42, 43, 44, 23, 3, 45, 3, 46, 3, 47, 48, 49, 50, 24, 3, 25, 51, 52, 53, 54, 55, 27, 3, 56, 57, 58, 3, 29, 3, 30

Offset: 1

Antti Karttunen, Dec 29 2018

nonn

For all i, j: a(i) = a(j) => A322823(i) = A322823(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A050376, A052330, A052331, A302777, A300840, A322823.

Cf. also A322805, A322807.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
A050376list(up_to) = { my(v=vector(up_to), i=0); for(n=1,oo,if(A302777(n), i++; v[i] = n); if(i == up_to,return(v))); };
v050376 = A050376list(up_to);
A050376(n) = v050376[n];
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
A300840(n) = A052330(A052331(n)>>1);
A322822aux(n) = if((2==n),-1,if(A302777(n),0,A300840(n)));
v322822 = rgs_transform(vector(up_to,n,A322822aux(n)));
A322822(n) = v322822[n];

cp's OEIS Frontend

A240521 a(n) = A050376(n)*A050376(n+1) where A050376(n) is the n-th number of the form p^(2^k) with p is prime and k >= 0.

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A329575 Numbers whose smallest Fermi-Dirac factor is 3.

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A096113 a(1) = 1, a(2) = 2; then all new products of subsets of pre-existing terms, then the first integer not present, and so on.

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A182979 Fermi-Dirac representation of n. Let n have factorization p1^(2^e1) * p2^(2^e2) * ... * pr^(2^er), where each factor is in A050376. The number n is represented by a binary string that indicates which terms of A050376 appear in the factorization of n.

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A334110 The squares of squarefree numbers (A062503), ordered lexicographically according to their prime factors. a(n) = Product_{k in I} prime(k+1)^2, where I are the indices of nonzero binary digits in n = Sum_{k in I} 2^k.

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A096115 If n = (2^k)-1, a(n) = a((n+1)/2) = k, if n = 2^k, a(n) = a(n-1)+1 = k+1, otherwise a(n) = (A000523(n)+1)*a(A035327(n-1)).

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A059900 Successively interleaved alternate bits in binary expansion of n gives vector of exponents in prime factorization of a(n).

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A096114 a(1)=1, a(2)=2, a(3*2^k) = 3*2^k, a(3*2^k + i) = 3*2^k + a(3*2^k - i), for i in range [1, 3*2^k - 1].

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A096114 a(1)=1, a(2)=2, a(32^k) = 32^k, a(32^k + i) = 32^k + a(32^k - i), for i in range [1, 32^k - 1].