A252735 -id:A252735 - 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

A260443 Prime factorization representation of Stern polynomials: a(0) = 1, a(1) = 2, a(2n) = A003961(a(n)), a(2n+1) = a(n)*a(n+1).

Original entry on oeis.org

1, 2, 3, 6, 5, 18, 15, 30, 7, 90, 75, 270, 35, 450, 105, 210, 11, 630, 525, 6750, 245, 20250, 2625, 9450, 77, 15750, 3675, 47250, 385, 22050, 1155, 2310, 13, 6930, 5775, 330750, 2695, 3543750, 128625, 1653750, 847, 4961250, 643125, 53156250, 18865, 24806250, 202125, 727650, 143, 1212750, 282975, 57881250, 29645, 173643750, 1414875, 18191250, 1001

Offset: 0

Antti Karttunen, Jul 28 2015

The exponents in the prime factorization of term a(n) give the coefficients of the n-th Stern polynomial. See A125184 and the examples.
None of the terms have prime gaps in their factorization, i.e., all can be found in A073491.
Contains neither perfect squares nor prime powers with exponent > 1. A277701 gives the positions of the terms that are 2*square. - Antti Karttunen, Oct 27 2016
Many of the derived sequences (like A002487) have similar "Fir forest" or "Gaudian cathedrals" style scatter plot. - Antti Karttunen, Mar 21 2017

			n    a(n)   prime factorization    Stern polynomial
------------------------------------------------------------
0       1   (empty)                B_0(x) = 0
1       2   p_1                    B_1(x) = 1
2       3   p_2                    B_2(x) = x
3       6   p_2 * p_1              B_3(x) = x + 1
4       5   p_3                    B_4(x) = x^2
5      18   p_2^2 * p_1            B_5(x) = 2x + 1
6      15   p_3 * p_2              B_6(x) = x^2 + x
7      30   p_3 * p_2 * p_1        B_7(x) = x^2 + x + 1
8       7   p_4                    B_8(x) = x^3
9      90   p_3 * p_2^2 * p_1      B_9(x) = x^2 + 2x + 1

Alois P. Heinz, Table of n, a(n) for n = 0..4096 (first 1025 terms from Antti Karttunen)

Same sequence sorted into ascending order: A260442.
Cf. A000040, A000079, A000225, A001222, A002487, A003415, A003961, A005811, A007949, A046523, A056239, A073491, A090880, A097249, A101979, A125184, A178590, A186891, A206284, A277314, A277315, A277325, A277326, A277329, A277330, A277701, A277705, A277899, A278243, A278530, A278544, A284010, A284011.
Cf. also A048675, A277333 (left inverses).
Cf. A277323, A277324 (bisections), A277200 (even terms sorted), A277197 (first differences), A277198.
Cf. A277316 (values at primes), A277318.
Cf. A023758 (positions of squarefree terms), A101082 (of terms not squarefree), A277702 (positions of records), A277703 (their values).
Cf. A283992, A283993 (number of irreducible, reducible polynomials in range 1 .. n).
Cf. also A206296 (Fibonacci polynomials similarly represented).

b:= n-> mul(nextprime(i[1])^i[2], i=ifactors(n)[2]):
a:= proc(n) option remember; `if`(n<2, n+1,
      `if`(irem(n, 2, 'h')=0, b(a(h)), a(h)*a(n-h)))
    end:
seq(a(n), n=0..56);  # Alois P. Heinz, Jul 04 2024

a[n_] := a[n] = Which[n < 2, n + 1, EvenQ@ n, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &@ a[n/2], True, a[#] a[# + 1] &[(n - 1)/2]]; Table[a@ n, {n, 0, 56}] (* Michael De Vlieger, Apr 05 2017 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus
A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2)))); \\ After Charles R Greathouse IV's code for "ps" in A186891.
\\ Antti Karttunen, Oct 11 2016

from sympy import factorint, prime, primepi
from functools import reduce
from operator import mul
def a003961(n):
    F = factorint(n)
    return 1 if n==1 else reduce(mul, (prime(primepi(i) + 1)**F[i] for i in F))
def a(n): return n + 1 if n<2 else a003961(a(n//2)) if n%2==0 else a((n - 1)//2)*a((n + 1)//2)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 21 2017

;; Uses memoization-macro definec:
(definec (A260443 n) (cond ((<= n 1) (+ 1 n)) ((even? n) (A003961 (A260443 (/ n 2)))) (else (* (A260443 (/ (- n 1) 2)) (A260443 (/ (+ n 1) 2))))))
;; A more standalone version added Oct 10 2016, requiring only an implementation of A000040 and the memoization-macro definec:
(define (A260443 n) (product_primes_to_kth_powers (A260443as_coeff_list n)))
(define (product_primes_to_kth_powers nums) (let loop ((p 1) (nums nums) (i 1)) (cond ((null? nums) p) (else (loop (* p (expt (A000040 i) (car nums))) (cdr nums) (+ 1 i))))))
(definec (A260443as_coeff_list n) (cond ((zero? n) (list)) ((= 1 n) (list 1)) ((even? n) (cons 0 (A260443as_coeff_list (/ n 2)))) (else (add_two_lists (A260443as_coeff_list (/ (- n 1) 2)) (A260443as_coeff_list (/ (+ 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)))))))

a(0) = 1, a(1) = 2, a(2n) = A003961(a(n)), a(2n+1) = a(n)*a(n+1).
Other identities. For all n >= 0:
A001221(a(n)) = A277314(n). [#nonzero coefficients in each polynomial.]
A001222(a(n)) = A002487(n). [When each polynomial is evaluated at x=1.]
A048675(a(n)) = n.          [at x=2.]
A090880(a(n)) = A178590(n). [at x=3.]
A248663(a(n)) = A264977(n). [at x=2 over the field GF(2).]
A276075(a(n)) = A276081(n). ["at factorials".]
A156552(a(n)) = A277020(n). [Converted to "unary-binary" encoding.]
A051903(a(n)) = A277315(n). [Maximal coefficient.]
A277322(a(n)) = A277013(n). [Number of irreducible polynomial factors.]
A005361(a(n)) = A277325(n). [Product of nonzero coefficients.]
A072411(a(n)) = A277326(n). [And their LCM.]
A007913(a(n)) = A277330(n). [The squarefree part.]
A000005(a(n)) = A277705(n). [Number of divisors.]
A046523(a(n)) = A278243(n). [Filter-sequence.]
A284010(a(n)) = A284011(n). [True for n > 1. Another filter-sequence.]
A003415(a(n)) = A278544(n). [Arithmetic derivative.]
A056239(a(n)) = A278530(n). [Weighted sum of coefficients.]
A097249(a(n)) = A277899(n).
a(A000079(n)) = A000040(n+1).
a(A000225(n)) = A002110(n).
a(A000051(n)) = 3*A002110(n).
For n >= 1, a(A000918(n)) = A070826(n).
A007949(a(n)) is the interleaving of A000035 and A005811, probably A101979.
A061395(a(n)) = A277329(n).
Also, for all n >= 1:
A055396(a(n)) = A001511(n).
A252735(a(n)) = A061395(a(n)) - 1 = A057526(n).
a(A000040(n)) = A277316(n).
a(A186891(1+n)) = A277318(n). [Subsequence for irreducible polynomials].

More linking formulas added by Antti Karttunen, Mar 21 2017

A252464 a(1) = 0, a(2n) = 1 + a(n), a(2n+1) = 1 + a(A064989(2n+1)); also binary width of terms of A156552 and A243071.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 3, 3, 4, 5, 4, 6, 5, 4, 4, 7, 4, 8, 5, 5, 6, 9, 5, 4, 7, 4, 6, 10, 5, 11, 5, 6, 8, 5, 5, 12, 9, 7, 6, 13, 6, 14, 7, 5, 10, 15, 6, 5, 5, 8, 8, 16, 5, 6, 7, 9, 11, 17, 6, 18, 12, 6, 6, 7, 7, 19, 9, 10, 6, 20, 6, 21, 13, 5, 10, 6, 8, 22, 7, 5, 14, 23, 7, 8, 15, 11, 8, 24, 6, 7, 11, 12, 16, 9, 7, 25, 6, 7, 6, 26, 9, 27

Offset: 1

Antti Karttunen, Dec 20 2014

nonn

a(n) tells how many iterations of A252463 are needed before 1 is reached, i.e., the distance of n from 1 in binary trees like A005940 and A163511.

Similarly for A253553 in trees A253563 and A253565. - Antti Karttunen, Apr 14 2019

			From _Gus Wiseman_, Apr 02 2019: (Start)
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so a(n) is the size of the inner lining of the integer partition with Heinz number n, which is also the size of the largest hook of the same partition. For example, the partition with Heinz number 715 is (6,5,3), with diagram
  o o o o o o
  o o o o o
  o o o
which has inner lining
          o o
      o o o
  o o o
and largest hook
  o o o o o o
  o
  o
both of which have size 8, so a(715) = 8.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000040, A000079, A001221, A001222, A005940, A029837, A061395, A156552 (A005941), A163511, A243071, A252461, A252463, A252735, A252736, A252759, A253553, A253563, A253565, A297113, A297155, A297167, A324870, A324872.
Right edge of irregular triangle A265146.
Cf. also A246348.
Cf. A052126, A056239, A093641, A112798, A257990, A325133, A325134, A325135.

Table[If[n==1,1,PrimeOmega[n]+PrimePi[FactorInteger[n][[-1,1]]]]-1,{n,100}] (* Gus Wiseman, Apr 02 2019 *)

A061395(n) = if(n>1, primepi(vecmax(factor(n)[, 1])), 0);
A252464(n) = (bigomega(n) + A061395(n) - 1); \\ Antti Karttunen, Apr 14 2019

from sympy import primepi, primeomega, primefactors
def A252464(n): return primeomega(n)+primepi(max(primefactors(n)))-1 if n>1 else 0 # Chai Wah Wu, Jul 17 2023

a(1) = 0; for n > 1: a(n) = 1 + a(A252463(n)).
a(n) = A029837(1+A243071(n)). [a(n) = binary width of terms of A243071.]
a(n) = A029837(A005941(n)) = A029837(1+A156552(n)). [Also binary width of terms of A156552.]
Other identities. For all n >= 1:
a(A000040(n)) = n.
a(A001248(n)) = n+1.
a(A030078(n)) = n+2.
And in general, a(prime(n)^k) = n+k-1.
a(A000079(n)) = n. [I.e., a(2^n) = n.]
For all n >= 2:
a(n) = A001222(n) + A061395(n) - 1 = A001222(n) + A252735(n) = A061395(n) + A252736(n) = 1 + A252735(n) + A252736(n).
a(n) = A325134(n) - 1. - Gus Wiseman, Apr 02 2019
From Antti Karttunen, Apr 14 2019: (Start)
a(1) = 0; for n > 1: a(n) = 1 + a(A253553(n)).
a(n) = A001221(n) + A297167(n) = A297113(n) + A297155(n).
(End).

A087117 Number of zeros in the longest string of consecutive zeros in the binary representation of n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 1, 0, 3, 2, 1, 1, 2, 1, 1, 0, 4, 3, 2, 2, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 0, 5, 4, 3, 3, 2, 2, 2, 2, 3, 2, 1, 1, 2, 1, 1, 1, 4, 3, 2, 2, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 0, 6, 5, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 4, 3, 2, 2, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 4, 3, 3, 2, 2

Offset: 0

Reinhard Zumkeller, Aug 14 2003

The following four statements are equivalent: a(n) = 0; n = 2^k - 1 for some k > 0; A087116(n) = 0; A023416(n) = 0.

The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. Then a(k) is the maximum part of this composition, minus one. The maximum part is A333766(k). - Gus Wiseman, Apr 09 2020

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to binary expansion of n

Cf. A023416, A007088, A007814.
Cf. A025480, A038374, A090046, A090047, A090048, A090049, A090050.
Positions of zeros are A000225.
Positions of terms <= 1 are A003754.
Positions of terms > 0 are A062289.
Positions of first appearances are A131577.
The version for prime indices is A252735.
The proper maximum is A333766.
The version for minimum is A333767.
Maximum prime index is A061395.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Runs are counted by A124767.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Runs-resistance is A333628.
- Weakly decreasing compositions are A114994.
- Weakly increasing compositions are A225620.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
Cf. A029931, A048793, A061395, A087117, A228351, A328594, A333217, A333218, A333219, A333767, A333768.

import Data.List (unfoldr, group)
a087117 0       = 1
a087117 n
  | null $ zs n = 0
  | otherwise   = maximum $ map length $ zs n where
  zs = filter ((== 0) . head) . group .
       unfoldr (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2)
-- Reinhard Zumkeller, May 01 2012

A087117 := proc(n)
    local d,l,zlen ;
    if n = 0 then
        return 1 ;
    end if;
    d := convert(n,base,2) ;
    for l from nops(d)-1 to 0 by -1 do
        zlen := [seq(0,i=1..l)] ;
        if verify(zlen,d,'sublist') then
            return l ;
        end if;
    end do:
    return 0 ;
end proc; # R. J. Mathar, Nov 05 2012

nz[n_]:=Max[Length/@Select[Split[IntegerDigits[n,2]],MemberQ[#,0]&]]; Array[nz,110,0]/.-\[Infinity]->0 (* Harvey P. Dale, Sep 05 2017 *)

h(n)=if(n<2, return(0)); my(k=valuation(n,2)); if(k, max(h(n>>k), k), n++; n>>=valuation(n,2); h(n-1))
a(n)=if(n,h(n),1) \\ Charles R Greathouse IV, Apr 06 2022

a(n) = max(A007814(n), a(A025480(n-1))) for n >= 2. - Robert Israel, Feb 19 2017
a(2n+1) = a(n) (n>=1); indeed, the binary form of 2n+1 consists of the binary form of n with an additional 1 at the end - Emeric Deutsch, Aug 18 2017
For n > 0, a(n) = A333766(n) - 1. - Gus Wiseman, Apr 09 2020

A252736 a(1) = a(2) = 0; for n > 2: a(2n) = 1 + a(n), a(2n+1) = a(A064989(2n+1)).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 2, 0, 1, 1, 3, 0, 2, 0, 2, 1, 1, 0, 3, 1, 1, 2, 2, 0, 2, 0, 4, 1, 1, 1, 3, 0, 1, 1, 3, 0, 2, 0, 2, 2, 1, 0, 4, 1, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 3, 0, 1, 2, 5, 1, 2, 0, 2, 1, 2, 0, 4, 0, 1, 2, 2, 1, 2, 0, 4, 3, 1, 0, 3, 1, 1, 1, 3, 0, 3, 1, 2, 1, 1, 1, 5, 0, 2, 2, 3, 0, 2, 0, 3, 2, 1, 0, 4, 0, 2, 1, 4, 0, 2, 1, 2, 2, 1, 1, 4, 1, 1, 1, 2, 2, 3, 0, 6

Offset: 1

Antti Karttunen, Dec 21 2014

nonn

Consider the binary tree illustrated in A005940: If we start from any n, computing successive iterations of A252463 until 1 is reached (i.e., we are traversing level by level towards the root of the tree, starting from that vertex of the tree where n is located), a(n) gives the number of even numbers > 2 encountered on the path (i.e., excluding the 2 from the count but including the starting n if it was even).

The number of pairs in any factorization tree of n. For example, a possible factorization tree of 12 is 12 -> (4*3) -> (2*2)*3. There are 2 pairs in this factor tree: (4*3) and (2*2). Thus, a(12) - 1 = 3 - 1 = 2. - Melvin Peralta, Aug 29 2016

Antti Karttunen, Table of n, a(n) for n = 1..8192

Essentially one less than A001222.
Cf. A000120, A005940, A064989, A080791, A156552, A243071, A252464, A252735.
Cf. also A246370.

a[1] = a[2] = 0; a[n_] := a[n] = If[EvenQ@ n, 1 + a[n/2], a[Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n]]; Array[a, 120] (* Michael De Vlieger, Aug 30 2016 *)

a(1) = a(2) = 0; for n > 2: a(2n) = 1 + a(n), a(2n+1) =  a(A064989(2n+1)).
a(n) = A080791(A243071(n)). [Number of nonleading 0-bits in A243071(n).]
Other identities. For all n >= 2:
a(n) = A000120(A156552(n)) - 1. [One less than the binary weight of A156552(n).]
a(n) = A252464(n) - A252735(n) - 1.
a(n) = A001222(n) - 1.

A252462 Shift one instance of the largest prime one step towards smaller primes: a(1) = 1, a(2^n) = 2^(n-1), and for other numbers: a(n) = (n / prime(g)) * prime(g-1), where g = A061395(n), index of the greatest prime dividing n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 4, 6, 6, 7, 8, 11, 10, 9, 8, 13, 12, 17, 12, 15, 14, 19, 16, 15, 22, 18, 20, 23, 18, 29, 16, 21, 26, 25, 24, 31, 34, 33, 24, 37, 30, 41, 28, 27, 38, 43, 32, 35, 30, 39, 44, 47, 36, 35, 40, 51, 46, 53, 36, 59, 58, 45, 32, 55, 42, 61, 52, 57, 50, 67, 48, 71, 62, 45, 68, 49, 66, 73, 48, 54, 74, 79, 60

Offset: 1

Antti Karttunen, Dec 20 2014

nonn

Iterating from any n as a(n), a(a(n)), a(a(a(n))), etc. reaches 1 after A056239(n) iterations.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Variant: A252461.

Cf. A001222, A008578, A052126, A061395, A056239, A064989, A209229, A252735.

a252462[n_Integer] := Block[{a008578, a052126, a061395, a},
  a008578[x_] := If[x == 1, 1, Prime[x - 1]];
  a052126[x_] := If[x == 1, 1, x/FactorInteger[x][[-1]][[1]]];
  a061395[x_] := PrimePi[FactorInteger[x][[-1]][[1]]];
  a[1] = 1;
  a[x_] := a008578[a061395[x]]*a052126[x];
Array[a, n]]; a252462[84] (* Michael De Vlieger, Dec 21 2014 *)

(define (A252462 n) (if (= 1 n) n  (* (A008578 (A061395 n)) (A052126 n))))

a(1) = 1; for n>1: a(n) = A008578(A061395(n)) * A052126(n). [Compare to the similar formula given for A064989.]
a(n) = A008578(1 + A252735(n)) * A052126(n).
Other identities. For all n >= 1:
a(2^n) = 2^(n-1).
For n >= 2, A001222(a(n)) = A001222(n) - A209229(n). [Number of prime divisors decreases only when n is a power of 2 larger than 1.]

cp's OEIS Frontend

A285725 a(1) = 0; for n > 1, a(n) = A252735(n) - A000035(n).

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A156552 Unary-encoded compressed factorization of natural numbers.

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A260443 Prime factorization representation of Stern polynomials: a(0) = 1, a(1) = 2, a(2n) = A003961(a(n)), a(2n+1) = a(n)*a(n+1).

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A252464 a(1) = 0, a(2n) = 1 + a(n), a(2n+1) = 1 + a(A064989(2n+1)); also binary width of terms of A156552 and A243071.

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A087117 Number of zeros in the longest string of consecutive zeros in the binary representation of n.

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A252736 a(1) = a(2) = 0; for n > 2: a(2n) = 1 + a(n), a(2n+1) = a(A064989(2n+1)).

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A252462 Shift one instance of the largest prime one step towards smaller primes: a(1) = 1, a(2^n) = 2^(n-1), and for other numbers: a(n) = (n / prime(g)) * prime(g-1), where g = A061395(n), index of the greatest prime dividing n.