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

Author

Leonid Broukhis, Feb 09 2009

Keywords

• easy
• base
• nonn

Comments

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

Examples

			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.
		

Links

• 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

Crossrefs

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.

Programs

Mathematica

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 *)

PARI

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

PARI

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

Perl

# 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

Python

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

Formula

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)

Extensions

More terms from Antti Karttunen, Jun 28 2014