A005940 The Doudna sequence: write n-1 in binary; power of prime(k) in a(n) is # of 1's that are followed by k-1 0's.

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 15, 12, 25, 18, 27, 16, 11, 14, 21, 20, 35, 30, 45, 24, 49, 50, 75, 36, 125, 54, 81, 32, 13, 22, 33, 28, 55, 42, 63, 40, 77, 70, 105, 60, 175, 90, 135, 48, 121, 98, 147, 100, 245, 150, 225, 72, 343, 250, 375, 108, 625, 162, 243, 64, 17, 26, 39

Offset: 1

Author

N. J. A. Sloane and J. H. Conway

Keywords

• nonn
• easy
• nice
• tabf
• look

Comments

A permutation of the natural numbers. - Robert G. Wilson v, Feb 22 2005

Fixed points: A029747. - Reinhard Zumkeller, Aug 23 2006

The even bisection, when halved, gives the sequence back. - Antti Karttunen, Jun 28 2014

From Antti Karttunen, Dec 21 2014: (Start)

This irregular table can be represented as a binary tree. Each child to the left is obtained by applying A003961 to the parent, and each child to the right is obtained by doubling the parent:

1

|

...................2...................

3 4

5......../ \........6 9......../ \........8

/ \ / \ / \ / \

/ \ / \ / \ / \

/ \ / \ / \ / \

7 10 15 12 25 18 27 16

11 14 21 20 35 30 45 24 49 50 75 36 125 54 81 32

etc.

Sequence A163511 is obtained by scanning the same tree level by level, from right to left. Also in binary trees A253563 and A253565 the terms on level of the tree are some permutation of the terms present on the level n of this tree. A252464(n) gives the distance of n from 1 in all these trees.

A252737(n) gives the sum and A252738(n) the product of terms on row n (where 1 is on row 0, 2 on row 1, 3 and 4 on row 2, etc.). A252745(n) gives the number of nodes on level n whose left child is larger than the right child, A252750 the difference between left and right child for each node from node 2 onward.

(End)

-A008836(a(1+n)) gives the corresponding numerator for A323505(n). - Antti Karttunen, Jan 19 2019

(a(2n+1)-1)/2 [= A244154(n)-1, for n >= 0] is a permutation of the natural numbers. - George Beck and Antti Karttunen, Dec 08 2019

From Peter Munn, Oct 04 2020: (Start)

Each term has the same even part (equivalently, the same 2-adic valuation) as its index.

Using the tree depicted in Antti Karttunen's 2014 comment:

Numbers are on the right branch (4 and descendants) if and only if divisible by the square of their largest prime factor (cf. A070003).

Numbers on the left branch, together with 2, are listed in A102750.

(End)

According to Kutz (1981), he learned of this sequence from American mathematician Byron Leon McAllister (1929-2017) who attributed the invention of the sequence to a graduate student by the name of Doudna (first name Paul?) in the mid-1950's at the University of Wisconsin. - Amiram Eldar, Jun 17 2021

From David James Sycamore, Sep 23 2022: (Start)

Alternative (recursive) definition: If n is a power of 2 then a(n)=n. Otherwise, if 2^j is the greatest power of 2 not exceeding n, and if k = n - 2^j, then a(n) is the least m*a(k) that has not occurred previously, where m is an odd prime.

Example: Use recursion with n = 77 = 2^6 + 13. a(13) = 25 and since 11 is the smallest odd prime m such that m*a(13) has not already occurred (see a(27), a(29),a(45)), then a(77) = 11*25 = 275. (End)

The odd bisection, when transformed by replacing all prime(k)^e in a(2*n - 1) with prime(k-1)^e, returns a(n), and thus gives the sequence back. - David James Sycamore, Sep 28 2022

Examples

			From _N. J. A. Sloane_, Aug 22 2022: (Start)
Let c_i = number of 1's in binary expansion of n-1 that have i 0's to their right, and let p(j) = j-th prime.  Then a(n) = Product_i p(i+1)^c_i.
If n=9, n-1 is 1000, c_3 = 1, a(9) = p(4)^1 = 7.
If n=10, n-1 = 1001, c_0 = 1, c_2 = 1, a(10) = p(1)*p(3) = 2*5 = 10.
If n=11, n-1 = 1010, c_1 = 1, c_2 = 1, a(11) = p(2)*p(3) = 15. (End)
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..8192 (terms 1..1024 from Reinhard Zumkeller)
• Michael De Vlieger, 6 row Doudna Tree diagram as mentioned in Comments.
• Michael De Vlieger, Annotated fan-style binary tree showing 10 levels, with a color function where 2^m is shown in medium blue in row m, k < 2^m is darker blue, and k > 2^m is brighter green, with records in each row shown in red.
• Ronald E. Kutz, Two unusual sequences, Two-Year College Mathematics Journal, Vol. 12, No. 5 (1981), pp. 316-319.
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A103969. Inverse is A005941 (A156552).

Cf. A125106. [From Franklin T. Adams-Watters, Mar 06 2010]

Cf. A252737 (gives row sums), A252738 (row products), A332979 (largest on row).

Related permutations of positive integers: A163511 (via A054429), A243353 (via A006068), A244154, A253563 (via A122111), A253565, A332977, A334866 (via A225546).

A000120, A003602, A003961, A006519, A053645, A070939, A246278, A250246, A252753, A253552 are used in a formula defining this sequence.

Formulas for f(a(n)) are given for f = A000265, A003963, A007949, A055396, A056239.

Numbers that occur at notable sets of positions in the binary tree representation of the sequence: A000040, A000079, A002110, A070003, A070826, A102750.

Cf. also A000142, A001511, A002450, A112798, A252463, A252464, A252745, A252750, A324054, A324106, A323505, A323508.

Cf. A106737, A290077, A323915, A324052, A324054, A324055, A324056, A324057, A324058, A324114, A324335, A324340, A324348, A324349 for various number-theoretical sequences applied to (i.e., permuted by) this sequence.

k-adic valuation: A007814 (k=2), A337821 (k=3).

Positions of multiples of 3: A091067.

Primorial deflation: A337376 / A337377.

Sum of prime indices of a(n) is A161511, reverse version A359043.

A048793 lists binary indices, ranked by A019565.

A066099 lists standard comps, partial sums A358134 (ranked by A358170).

Cf. A001222, A029837, A059893, A241916, A242628, A253566, A359042.

Programs

Haskell

a005940 n = f (n - 1) 1 1 where
   f 0 y _          = y
   f x y i | m == 0 = f x' y (i + 1)
           | m == 1 = f x' (y * a000040 i) i
           where (x',m) = divMod x 2
-- Reinhard Zumkeller, Oct 03 2012
(Scheme, with memoization-macro definec from Antti Karttunen's IntSeq-library)
(define (A005940 n) (A005940off0 (- n 1))) ;; The off=1 version, utilizing any one of three different offset-0 implementations:
(definec (A005940off0 n) (cond ((< n 2) (+ 1 n)) (else (* (A000040 (- (A070939 n) (- (A000120 n) 1))) (A005940off0 (A053645 n))))))
(definec (A005940off0 n) (cond ((<= n 2) (+ 1 n)) ((even? n) (A003961 (A005940off0 (/ n 2)))) (else (* 2 (A005940off0 (/ (- n 1) 2))))))
(define (A005940off0 n) (let loop ((n n) (i 1) (x 1)) (cond ((zero? n) x) ((even? n) (loop (/ n 2) (+ i 1) x)) (else (loop (/ (- n 1) 2) i (* x (A000040 i)))))))
;; Antti Karttunen, Jun 26 2014

Maple

f := proc(n,i,x) option remember ; if n = 0 then x; elif type(n,'even') then procname(n/2,i+1,x) ; else procname((n-1)/2,i,x*ithprime(i)) ; end if; end proc:
A005940 := proc(n) f(n-1,1,1) ; end proc: # R. J. Mathar, Mar 06 2010

Mathematica

f[n_] := Block[{p = Partition[ Split[ Join[ IntegerDigits[n - 1, 2], {2}]], 2]}, Times @@ Flatten[ Table[q = Take[p, -i]; Prime[ Count[ Flatten[q], 0] + 1]^q[[1, 1]], {i, Length[p]}] ]]; Table[ f[n], {n, 67}] (* Robert G. Wilson v, Feb 22 2005 *)
Table[Times@@Prime/@(Join@@Position[Reverse[IntegerDigits[n,2]],1]-Range[DigitCount[n,2,1]]+1),{n,0,100}] (* Gus Wiseman, Dec 28 2022 *)

PARI

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, n%2 && (t*=p) || p=nextprime(p+1)); t } \\ M. F. Hasler, Mar 07 2010; update Aug 29 2014

PARI

a(n)=my(p=2, t=1); for(i=0,exponent(n), if(bittest(n,i), t*=p, p=nextprime(p+1))); t \\ Charles R Greathouse IV, Nov 11 2021

Python

from sympy import prime
import math
def A(n): return n - 2**int(math.floor(math.log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
print([b(n - 1) for n in range(1, 101)]) # Indranil Ghosh, Apr 10 2017

Python

from math import prod
from itertools import accumulate
from collections import Counter
from sympy import prime
def A005940(n): return prod(prime(len(a)+1)**b for a, b in Counter(accumulate(bin(n-1)[2:].split('1')[:0:-1])).items()) # Chai Wah Wu, Mar 10 2023

Formula

From Reinhard Zumkeller, Aug 23 2006, R. J. Mathar, Mar 06 2010: (Start)

a(n) = f(n-1, 1, 1)

where f(n, i, x) = x if n = 0,

= f(n/2, i+1, x) if n > 0 is even

= f((n-1)/2, i, x*prime(i)) otherwise. (End)

From Antti Karttunen, Jun 26 2014: (Start)

Define a starting-offset 0 version of this sequence as:

b(0)=1, b(1)=2, [base cases]

and then compute the rest either with recurrence:

b(n) = A000040(1+(A070939(n)-A000120(n))) * b(A053645(n)).

or

b(2n) = A003961(b(n)), b(2n+1) = 2 * b(n). [Compare this to the similar recurrence given for A163511.]

Then define a(n) = b(n-1), where a(n) gives this sequence A005940 with the starting offset 1.

Can be also defined as a composition of related permutations:

a(n+1) = A243353(A006068(n)).

a(n+1) = A163511(A054429(n)). [Compare the scatter plots of this sequence and A163511 to each other.]

This permutation also maps between the partitions as enumerated in the lists A125106 and A112798, providing identities between:

A161511(n) = A056239(a(n+1)). [The corresponding sums ...]

A243499(n) = A003963(a(n+1)). [... and the products of parts of those partitions.]

(End)

From Antti Karttunen, Dec 21 2014 - Jan 04 2015: (Start)

A002110(n) = a(1+A002450(n)). [Primorials occur at (4^n - 1)/3 in the offset-0 version of the sequence.]

a(n) = A250246(A252753(n-1)).

a(n) = A122111(A253563(n-1)).

For n >= 1, A055396(a(n+1)) = A001511(n).

For n >= 2, a(n) = A246278(1+A253552(n)).

(End)

From Peter Munn, Oct 04 2020: (Start)

A000265(a(n)) = a(A000265(n)) = A003961(a(A003602(n))).

A006519(a(n)) = a(A006519(n)) = A006519(n).

a(n) = A003961(a(A003602(n))) * A006519(n).

A007814(a(n)) = A007814(n).

A007949(a(n)) = A337821(n) = A007814(A003602(n)).

a(n) = A225546(A334866(n-1)).

(End)

a(2n) = 2*a(n), or generally a(2^k*n) = 2^k*a(n). - Amiram Eldar, Oct 03 2022

If n-1 = Sum_{i} 2^(q_i-1), then a(n) = Product_{i} prime(q_i-i+1). These are the Heinz numbers of the rows of A125106. If the offset is changed to 0, the inverse is A156552. - Gus Wiseman, Dec 28 2022

Extensions

More terms from Robert G. Wilson v, Feb 22 2005

Sign in a formula switched and Maple program added by R. J. Mathar, Mar 06 2010

Binary tree illustration and keyword tabf added by Antti Karttunen, Dec 21 2014