A277010 -id:A277010 - cp's OEIS frontend

A156552 Unary-encoded compressed factorization of natural numbers.

0, 1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 11, 32, 17, 10, 15, 64, 13, 128, 19, 18, 33, 256, 23, 12, 65, 14, 35, 512, 21, 1024, 31, 34, 129, 20, 27, 2048, 257, 66, 39, 4096, 37, 8192, 67, 22, 513, 16384, 47, 24, 25, 130, 131, 32768, 29, 36, 71, 258, 1025, 65536, 43, 131072, 2049, 38, 63, 68, 69, 262144

Offset: 1

Leonid Broukhis, Feb 09 2009

The primes become the powers of 2 (2 -> 1, 3 -> 2, 5 -> 4, 7 -> 8); the composite numbers are formed by taking the values for the factors in the increasing order, multiplying them by the consecutive powers of 2, and summing. See the Example section.
From Antti Karttunen, Jun 27 2014: (Start)
The odd bisection (containing even terms) halved gives A244153.
The even bisection (containing odd terms), when one is subtracted from each and halved, gives this sequence back.
(End)
Question: Are there any other solutions that would satisfy the recurrence r(1) = 0; and for n > 1, r(n) = Sum_{d|n, d>1} 2^A033265(r(d)), apart from simple variants 2^k * A156552(n)? See also A297112, A297113. - Antti Karttunen, Dec 30 2017

			For 84 = 2*2*3*7 -> 1*1 + 1*2 + 2*4 + 8*8 =  75.
For 105 = 3*5*7 -> 2*1 + 4*2 + 8*4 = 42.
For 137 = p_33 -> 2^32 = 4294967296.
For 420 = 2*2*3*5*7 -> 1*1 + 1*2 + 2*4 + 4*8 + 8*16 = 171.
For 147 = 3*7*7 = p_2 * p_4 * p_4 -> 2*1 + 8*2 + 8*4 = 50.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1024 terms from Antti Karttunen)
Hans Havermann, Factorization of the first 10000 terms, in format [[primes], [exponents]]
A puzzle by Sergey Orlov (in Russian)
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences computed from indices in prime factorization

One less than A005941.
Inverse permutation: A005940 with starting offset 0 instead of 1.
Cf. A000079, A000120, A001222, A052126, A054429, A061395, A064216, A064989, A003188, A243071, A243065-A243066, A244153, A243354, A112798, A125106, A056239, A161511.
Cf. also A297106, A297112 (Möbius transform), A297113, A153013, A290308, A300827, A323243, A323244, A323247, A324201, A324812 (n for which a(n) is a square), A324813, A324822, A324823, A324398, A324713, A324815, A324819, A324865, A324866, A324867.
Other related permutations: A253551, A253792, A253564, A253791, A277195, A297163, A297164, A297165, A297166, A302023, A305418, A322863, A322864.

Table[Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[ Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ n]], {n, 67}] (* Michael De Vlieger, Sep 08 2016 *)

a(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ David A. Corneth, Mar 08 2019

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n)))); \\ (based on the given recurrence) - Antti Karttunen, Mar 08 2019

# Program corrected per instructions from Leonid Broukhis. - Antti Karttunen, Jun 26 2014
# However, it gives correct answers only up to n=136, before corruption by a wrap-around effect.
# Note that the correct answer for n=137 is A156552(137) = 4294967296.
$max = $ARGV[0];
$pow = 0;
foreach $i (2..$max) {
@a = split(/ /, `factor $i`);
shift @a;
$shift = 0;
$cur = 0;
while ($n = int shift @a) {
$prime{$n} = 1 << $pow++ if !defined($prime{$n});
$cur |= $prime{$n} << $shift++;
}
print "$cur, ";
}
print "\n";
(Scheme, with memoization-macro definec from Antti Karttunen's IntSeq-library, two different implementations)
(definec (A156552 n) (cond ((= n 1) 0) (else (+ (A000079 (+ -2 (A001222 n) (A061395 n))) (A156552 (A052126 n))))))
(definec (A156552 n) (cond ((= 1 n) (- n 1)) ((even? n) (+ 1 (* 2 (A156552 (/ n 2))))) (else (* 2 (A156552 (A064989 n))))))
;; Antti Karttunen, Jun 26 2014

from sympy import primepi, factorint
def A156552(n): return sum((1<Chai Wah Wu, Mar 10 2023

From Antti Karttunen, Jun 26 2014: (Start)
a(1) = 0, a(n) = A000079(A001222(n)+A061395(n)-2) + a(A052126(n)).
a(1) = 0, a(2n) = 1+2*a(n), a(2n+1) = 2*a(A064989(2n+1)). [Compare to the entanglement recurrence A243071].
For n >= 0, a(2n+1) = 2*A244153(n+1). [Follows from the latter clause of the above formula.]
a(n) = A005941(n) - 1.
As a composition of related permutations:
a(n) = A003188(A243354(n)).
a(n) = A054429(A243071(n)).
For all n >= 1, A005940(1+a(n)) = n and for all n >= 0, a(A005940(n+1)) = n. [The offset-0 version of A005940 works as an inverse for this permutation.]
This permutations also maps between the partition-lists A112798 and A125106:
A056239(n) = A161511(a(n)). [The sums of parts of each partition (the total sizes).]
A003963(n) = A243499(a(n)). [And also the products of those parts.]
(End)
From Antti Karttunen, Oct 09 2016: (Start)
A161511(a(n)) = A056239(n).
A029837(1+a(n)) = A252464(n). [Binary width of terms.]
A080791(a(n)) = A252735(n). [Number of nonleading 0-bits.]
A000120(a(n)) = A001222(n). [Binary weight.]
For all n >= 2, A001511(a(n)) = A055396(n).
For all n >= 2, A000120(a(n))-1 = A252736(n). [Binary weight minus one.]
A252750(a(n)) = A252748(n).
a(A250246(n)) = A252754(n).
a(A005117(n)) = A277010(n). [Maps squarefree numbers to a permutation of A003714, fibbinary numbers.]
A085357(a(n)) = A008966(n). [Ditto for their characteristic functions.]
For all n >= 0:
a(A276076(n)) = A277012(n).
a(A276086(n)) = A277022(n).
a(A260443(n)) = A277020(n).
(End)
From Antti Karttunen, Dec 30 2017: (Start)
For n > 1, a(n) = Sum_{d|n, d>1} 2^A033265(a(d)). [See comments.]
More linking formulas:
A106737(a(n)) = A000005(n).
A290077(a(n)) = A000010(n).
A069010(a(n)) = A001221(n).
A136277(a(n)) = A181591(n).
A132971(a(n)) = A008683(n).
A106400(a(n)) = A008836(n).
A268411(a(n)) = A092248(n).
A037011(a(n)) = A010052(n) [conjectured, depends on the exact definition of A037011].
A278161(a(n)) = A046951(n).
A001316(a(n)) = A061142(n).
A277561(a(n)) = A034444(n).
A286575(a(n)) = A037445(n).
A246029(a(n)) = A181819(n).
A278159(a(n)) = A124859(n).
A246660(a(n)) = A112624(n).
A246596(a(n)) = A069739(n).
A295896(a(n)) = A053866(n).
A295875(a(n)) = A295297(n).
A284569(a(n)) = A072411(n).
A286574(a(n)) = A064547(n).
A048735(a(n)) = A292380(n).
A292272(a(n)) = A292382(n).
A244154(a(n)) = A048673(n), a(A064216(n)) = A244153(n).
A279344(a(n)) = A279339(n), a(A279338(n)) = A279343(n).
a(A277324(n)) = A277189(n).
A037800(a(n)) = A297155(n).
For n > 1, A033265(a(n)) = 1+A297113(n).
(End)
From Antti Karttunen, Mar 08 2019: (Start)
a(n) = A048675(n) + A323905(n).
a(A324201(n)) = A000396(n), provided there are no odd perfect numbers.
The following sequences are derived from or related to the base-2 expansion of a(n):
A000265(a(n)) = A322993(n).
A002487(a(n)) = A323902(n).
A005187(a(n)) = A323247(n).
A324288(a(n)) = A324116(n).
A323505(a(n)) = A323508(n).
A079559(a(n)) = A323512(n).
A085405(a(n)) = A323239(n).
The following sequences are obtained by applying to a(n) a function that depends on the prime factorization of its argument, which goes "against the grain" because a(n) is the binary code of the factorization of n, which in these cases is then factored again:
A000203(a(n)) = A323243(n).
A033879(a(n)) = A323244(n) = 2*a(n) - A323243(n),
A294898(a(n)) = A323248(n).
A000005(a(n)) = A324105(n).
A000010(a(n)) = A324104(n).
A083254(a(n)) = A324103(n).
A001227(a(n)) = A324117(n).
A000593(a(n)) = A324118(n).
A001221(a(n)) = A324119(n).
A009194(a(n)) = A324396(n).
A318458(a(n)) = A324398(n).
A192895(a(n)) = A324100(n).
A106315(a(n)) = A324051(n).
A010052(a(n)) = A324822(n).
A053866(a(n)) = A324823(n).
A001065(a(n)) = A324865(n) = A323243(n) - a(n),
A318456(a(n)) = A324866(n) = A324865(n) OR a(n),
A318457(a(n)) = A324867(n) = A324865(n) XOR a(n),
A318458(a(n)) = A324398(n) = A324865(n) AND a(n),
A318466(a(n)) = A324819(n) = A323243(n) OR 2*a(n),
A318467(a(n)) = A324713(n) = A323243(n) XOR 2*a(n),
A318468(a(n)) = A324815(n) = A323243(n) AND 2*a(n).
(End)

More terms from Antti Karttunen, Jun 28 2014

A332449 a(n) = A005940(1+(3*A156552(n))).

Original entry on oeis.org

1, 4, 9, 10, 25, 16, 49, 30, 21, 36, 121, 22, 169, 100, 81, 90, 289, 40, 361, 250, 225, 196, 529, 66, 55, 484, 105, 490, 841, 64, 961, 270, 441, 676, 625, 154, 1369, 1156, 1089, 750, 1681, 144, 1849, 1210, 39, 1444, 2209, 198, 91, 84, 1521, 1690, 2809, 120, 1225, 1470, 2601, 2116, 3481, 34, 3721, 3364, 1029, 810, 3025, 400

Offset: 1

Antti Karttunen, Feb 14 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A329609 (terms sorted into ascending order).
Cf. A000290, A003961, A005117 (positions of squares), A005940, A010052, A156552, A277010, A329603, A332450, A332451, A347119, A347120, A353267 [= A348717(a(n))], A353269, A353270 [= gcd(n, a(n))], A353271, A353272, A353273.
Cf. also A332223.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A332449(n) = A005940(1+(3*A156552(n)));

a(n) = A005940(1+(3*A156552(n))).
a(p) = p^2 for all primes p.
a(u) = A332451(u) and A010052(a(u)) = 1 for all squarefree numbers (A005117).
a(A003961(n)) = A003961(a(n)) = A005940(1+(6*A156552(n))).
From Antti Karttunen, Apr 10 2022: (Start)
a(n) = A347119(n) * A000290(A347120(n)) = A353270(n) * A353272(n).
a(A353269(n)) = 1 for all n.
(End)

A085357 Common residues of binomial(3n,n)/(2n+1) modulo 2: relates ternary trees (A001764) to the infinite Fibonacci word (A003849).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Paul D. Hanna, Jun 25 2003

nonn

The n-th runs of ones is given by: 3 - A003849(n) (infinite Fibonacci word) = A076662(n+1). Runs of zeros are given by: A085358 and are also directly related to the Fibonacci sequence. Coefficients of A(x)^3 are found in A085359.
a(n) = 0 iff some binary digit of n is 1 while the corresponding binary digit of 3*n is 0. - Robert Israel, Jul 12 2016
The Run Length Transform of [0,1,0,0,0,...], A063524, the characteristic function of 1. (See A227349 for the definition). - Antti Karttunen, Oct 15 2016

Robert Israel, Table of n, a(n) for n = 0..10000
J.-P. Allouche, F. v. Haeseler, H.-O. Peitgen, A. Petersen and G. Skordev, Automaticity of double sequences generated by one-dimensional linear cellular automata, Theoret. Comput. Sci. 186 (1997), 195-209.
J.-P. Allouche, F. v. Haeseler, H.-O. Peitgen and G. Skordev, Linear cellular automata, finite automata and Pascal's triangle, Discrete Appl. Math. 66 (1996), 1-22.
R. Bacher and C. Reutenauer, The number of right ideals of given codimension over a finite field, in Noncommutative Birational Geometry, Representations and Combinatorics, edited by Arkady. Berenstein and Vladimir. Retakha, Contemporary Mathematics, Vol. 592, 2013.
Paul Tarau, Emulating Primality with Multiset Representations of Natural Numbers, in Theoretical Aspects of Computing, ICTAC 2011, Lecture Notes in Computer Science, 2011, Volume 6916/2011, 218-238, DOI: 10.1007/978-3-642-23283-1_15.
Index entries for characteristic functions
Index entries for sequences computed with run length transform

Cf. A001764 (ternary trees), A085358 (runs of zeros), A076662 (runs of ones), A003849 (infinite Fibonacci word), A085359 (A(x)^3).
Cf. also A003714, A005940, A008966, A063524, A227349, A277010, A324964.
Absolute values of A132971.

[Binomial(3*n,n) mod 2: n in [0..100]]; // Vincenzo Librandi, Jul 09 2016

f:= proc(n) local L,Lp;
  L:= convert(n,base,2);
  Lp:= convert(3*n,base,2);
  if has(L-Lp[1..nops(L)],1) then 0 else 1 fi
end proc:
map(f, [$0..100]); # Robert Israel, Jul 12 2016

Table[Mod[Binomial[3 n, n], 2], {n, 0, 120}] (* Michael De Vlieger, Jul 08 2016 *)

A085357(n) = !bitand(n,n<<1); \\ Antti Karttunen, Aug 22 2019

def A085357(n): return int(not n&(n<<1)) # Chai Wah Wu, Jun 25 2025

G.f.: 1 + x*A(x)^3 = A(x) (Mod 2); a(n) = A001764(n) (Mod 2).
a(n) = binomial(3n, n) (mod 2). Characteristic function of Fibbinary numbers (i.e. a(n)=1 iff n is in A003714). - Benoit Cloitre, Nov 15 2003
Recurrence: a(0) = 1, a(2n) = a(4n+1) = a(n), a(4n+3) = 0.
a(n-2) = A000256(n)(mod 2), for n>2. - John M. Campbell, Jul 08 2016
a(n) = A000621(n+1)(mod 2). - John M. Campbell, Jul 15 2016
a(n) = A000625(n)(mod 2). - John M. Campbell, Jul 15 2016
a(n) = A008966(A005940(1+n)). [Follows from the Run Length Transform interpretation, see also A277010.] - Antti Karttunen, Oct 15 2016
a(n) = abs(A132971(n)) = abs(A008683(A005940(1+n))). - Antti Karttunen, May 30 2017

A277020 Unary-binary representation of Stern polynomials: a(n) = A156552(A260443(n)).

Original entry on oeis.org

0, 1, 2, 5, 4, 13, 10, 21, 8, 45, 26, 93, 20, 109, 42, 85, 16, 173, 90, 477, 52, 957, 186, 733, 40, 749, 218, 1501, 84, 877, 170, 341, 32, 685, 346, 3549, 180, 12221, 954, 7133, 104, 14269, 1914, 49021, 372, 28605, 1466, 5853, 80, 5869, 1498, 30685, 436, 61373, 3002, 23517, 168, 12013, 1754, 24029, 340, 7021, 682, 1365

Offset: 0

Antti Karttunen, Oct 07 2016

Sequence encodes Stern polynomials (see A125184, A260443) with "unary-binary method" where any nonzero coefficient c > 0 is encoded as a run of c 1-bits, separated from neighboring 1-runs by exactly one zero (this follows because A260442 is a subsequence of A073491). See the examples.

Terms which are not multiples of 4 form a subset of A003754, or in other words, each term is 2^k * {a term from a certain subsequence of A247648}, which subsequence remains to be determined.

			n    Stern polynomial       Encoded as              a(n)
                            "unary-binary" number   (-> decimal)
----------------------------------------------------------------
0    B_0(x) = 0                     "0"               0
1    B_1(x) = 1                     "1"               1
2    B_2(x) = x                    "10"               2
3    B_3(x) = x + 1               "101"               5
4    B_4(x) = x^2                 "100"               4
5    B_5(x) = 2x + 1             "1101"              13
6    B_6(x) = x^2 + x            "1010"              10
7    B_7(x) = x^2 + x + 1       "10101"              21
8    B_8(x) = x^3                "1000"               8
9    B_9(x) = x^2 + 2x + 1     "101101"              45

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

Cf. A087808 (a left inverse), A156552, A260443, A277189 (odd bisection).

Cf. also A000079, A000120, A000225, A002450, A002487, A003754, A073491, A247648, A260442, A277010, A277012, A276081.

(define (A277020 n) (A156552 (A260443 n)))
;; Another implementation, more practical to run:
(define (A277020 n) (list_of_numbers_to_unary_binary_representation (A260443as_index_lists n)))
(definec (A260443as_index_lists n) (cond ((zero? n) (list)) ((= 1 n) (list 1)) ((even? n) (cons 0 (A260443as_index_lists (/ n 2)))) (else (add_two_lists (A260443as_index_lists (/ (- n 1) 2)) (A260443as_index_lists (/ (+ n 1) 2))))))
(define (add_two_lists nums1 nums2) (let ((len1 (length nums1)) (len2 (length nums2))) (cond ((< len1 len2) (add_two_lists nums2 nums1)) (else (map + nums1 (append nums2 (make-list (- len1 len2) 0)))))))
(define (list_of_numbers_to_unary_binary_representation nums) (let loop ((s 0) (nums nums) (b 1)) (cond ((null? nums) s) (else (loop (+ s (* (A000225 (car nums)) b)) (cdr nums) (* (A000079 (+ 1 (car nums))) b))))))

a(n) = A156552(A260443(n)).
Other identities. For all n >= 0:
A087808(a(n)) = n.
A000120(a(n)) = A002487(n).
a(2n) = 2*a(n).
a(2^n) = 2^n.
a(A000225(n)) = A002450(n).

A277006 a(n) = A005940(1+A003714(n)); permutation of squarefree numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 15, 11, 14, 21, 35, 30, 13, 22, 33, 55, 42, 77, 70, 105, 17, 26, 39, 65, 66, 91, 110, 165, 143, 154, 231, 385, 210, 19, 34, 51, 85, 78, 119, 130, 195, 187, 182, 273, 455, 330, 221, 286, 429, 715, 462, 1001, 770, 1155, 23, 38, 57, 95, 102, 133, 170, 255, 209, 238, 357, 595, 390, 247, 374, 561, 935, 546, 1309, 910, 1365, 323

Offset: 0

Antti Karttunen, Oct 07 2016

nonn

Permutation of A005117 (squarefree numbers).

Antti Karttunen, Table of n, a(n) for n = 0..10946

Cf. A000040, A000045, A003714, A005117, A005940, A048675, A048679, A087808.

Cf. also A019565, A277010, A277196.

(define (A277006 n) (A005940 (+ 1 (A003714 n))))

a(n) = A005940(1+A003714(n)).
Other identities.
For n >= 0, A048675(a(n)) = A087808(A003714(n)) = A048679(n).
For n >= 1, a(A000045(n+1)) = A000040(n).

A332451 a(n) = A005940(1+A048724(A156552(n))).

Original entry on oeis.org

1, 4, 9, 6, 25, 16, 49, 10, 15, 36, 121, 54, 169, 100, 81, 14, 289, 24, 361, 150, 225, 196, 529, 250, 35, 484, 21, 294, 841, 64, 961, 22, 441, 676, 625, 90, 1369, 1156, 1089, 490, 1681, 144, 1849, 726, 375, 1444, 2209, 686, 77, 60, 1521, 1014, 2809, 40, 1225, 1210, 2601, 2116, 3481, 486, 3721, 3364, 735, 26, 3025, 400

Offset: 1

Antti Karttunen, Feb 15 2020

nonn

Cf. A000290, A003961, A005117 (gives the positions of squares), A005940, A008836, A010052, A048724, A156552, A277010, A293448, A332449, A332450.
Permutation of A028260.
Cf. A332460 for complementary sequence (after its initial 1).

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A048724(n) = bitxor(n, 2*n); \\ From A048724
A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A332451(n) = A005940(1+A048724(A156552(n)));

a(n) = A005940(1+A048724(A156552(n))).
a(p) = p^2 for all primes p.
For all squarefree numbers u, a(u) = A332449(u) and A010052(a(u)) = 1.
a(A003961(n)) = A003961(a(n)).
a(A293448(n)) = A293448(a(n)).
a(A332450(n)) = A332450(A003961(n)); A332450(a(n)) = A003961(A332450(n)).
A008836(a(n)) = +1 for all n.

cp's OEIS Frontend

A277195 Permutation of nonnegative integers: a(n) = A022290(A277010(n)).

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A156552 Unary-encoded compressed factorization of natural numbers.

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Mathematica

PARI

PARI

Perl

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A332449 a(n) = A005940(1+(3*A156552(n))).

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A085357 Common residues of binomial(3n,n)/(2n+1) modulo 2: relates ternary trees (A001764) to the infinite Fibonacci word (A003849).

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Magma

Maple

Mathematica

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A277020 Unary-binary representation of Stern polynomials: a(n) = A156552(A260443(n)).

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A277006 a(n) = A005940(1+A003714(n)); permutation of squarefree numbers.

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A332451 a(n) = A005940(1+A048724(A156552(n))).

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PARI