|
%I M0509 #200 Mar 10 2023 16:34:28
%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,
%T 75,36,125,54,81,32,13,22,33,28,55,42,63,40,77,70,105,60,175,90,135,
%U 48,121,98,147,100,245,150,225,72,343,250,375,108,625,162,243,64,17,26,39
%N 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.
%C A permutation of the natural numbers. - _Robert G. Wilson v_, Feb 22 2005
%C Fixed points: A029747. - _Reinhard Zumkeller_, Aug 23 2006
%C The even bisection, when halved, gives the sequence back. - _Antti Karttunen_, Jun 28 2014
%C From _Antti Karttunen_, Dec 21 2014: (Start)
%C 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:
%C 1
%C |
%C ...................2...................
%C 3 4
%C 5......../ \........6 9......../ \........8
%C / \ / \ / \ / \
%C / \ / \ / \ / \
%C / \ / \ / \ / \
%C 7 10 15 12 25 18 27 16
%C 11 14 21 20 35 30 45 24 49 50 75 36 125 54 81 32
%C etc.
%C 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.
%C 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.
%C (End)
%C -A008836(a(1+n)) gives the corresponding numerator for A323505(n). - _Antti Karttunen_, Jan 19 2019
%C (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
%C From _Peter Munn_, Oct 04 2020: (Start)
%C Each term has the same even part (equivalently, the same 2-adic valuation) as its index.
%C Using the tree depicted in Antti Karttunen's 2014 comment:
%C Numbers are on the right branch (4 and descendants) if and only if divisible by the square of their largest prime factor (cf. A070003).
%C Numbers on the left branch, together with 2, are listed in A102750.
%C (End)
%C 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
%C From _David James Sycamore_, Sep 23 2022: (Start)
%C 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.
%C 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)
%C 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
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Antti Karttunen, <a href="/A005940/b005940.txt">Table of n, a(n) for n = 1..8192</a> (terms 1..1024 from Reinhard Zumkeller)
%H Michael De Vlieger, 6 row <a href="/A005940/a005940.png">Doudna Tree diagram</a> as mentioned in Comments.
%H Michael De Vlieger, <a href="/A005940/a005940_2.png">Annotated fan-style binary tree showing 10 levels</a>, 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.
%H Ronald E. Kutz, <a href="http://www.jstor.org/stable/3027304">Two unusual sequences</a>, Two-Year College Mathematics Journal, Vol. 12, No. 5 (1981), pp. 316-319.
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F From _Reinhard Zumkeller_, Aug 23 2006, _R. J. Mathar_, Mar 06 2010: (Start)
%F a(n) = f(n-1, 1, 1)
%F where f(n, i, x) = x if n = 0,
%F = f(n/2, i+1, x) if n > 0 is even
%F = f((n-1)/2, i, x*prime(i)) otherwise. (End)
%F From _Antti Karttunen_, Jun 26 2014: (Start)
%F Define a starting-offset 0 version of this sequence as:
%F b(0)=1, b(1)=2, [base cases]
%F and then compute the rest either with recurrence:
%F b(n) = A000040(1+(A070939(n)-A000120(n))) * b(A053645(n)).
%F or
%F b(2n) = A003961(b(n)), b(2n+1) = 2 * b(n). [Compare this to the similar recurrence given for A163511.]
%F Then define a(n) = b(n-1), where a(n) gives this sequence A005940 with the starting offset 1.
%F Can be also defined as a composition of related permutations:
%F a(n+1) = A243353(A006068(n)).
%F a(n+1) = A163511(A054429(n)). [Compare the scatter plots of this sequence and A163511 to each other.]
%F This permutation also maps between the partitions as enumerated in the lists A125106 and A112798, providing identities between:
%F A161511(n) = A056239(a(n+1)). [The corresponding sums ...]
%F A243499(n) = A003963(a(n+1)). [... and the products of parts of those partitions.]
%F (End)
%F From _Antti Karttunen_, Dec 21 2014 - Jan 04 2015: (Start)
%F A002110(n) = a(1+A002450(n)). [Primorials occur at (4^n - 1)/3 in the offset-0 version of the sequence.]
%F a(n) = A250246(A252753(n-1)).
%F a(n) = A122111(A253563(n-1)).
%F For n >= 1, A055396(a(n+1)) = A001511(n).
%F For n >= 2, a(n) = A246278(1+A253552(n)).
%F (End)
%F From _Peter Munn_, Oct 04 2020: (Start)
%F A000265(a(n)) = a(A000265(n)) = A003961(a(A003602(n))).
%F A006519(a(n)) = a(A006519(n)) = A006519(n).
%F a(n) = A003961(a(A003602(n))) * A006519(n).
%F A007814(a(n)) = A007814(n).
%F A007949(a(n)) = A337821(n) = A007814(A003602(n)).
%F a(n) = A225546(A334866(n-1)).
%F (End)
%F a(2n) = 2*a(n), or generally a(2^k*n) = 2^k*a(n). - _Amiram Eldar_, Oct 03 2022
%F 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
%e From _N. J. A. Sloane_, Aug 22 2022: (Start)
%e 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.
%e If n=9, n-1 is 1000, c_3 = 1, a(9) = p(4)^1 = 7.
%e If n=10, n-1 = 1001, c_0 = 1, c_2 = 1, a(10) = p(1)*p(3) = 2*5 = 10.
%e If n=11, n-1 = 1010, c_1 = 1, c_2 = 1, a(11) = p(2)*p(3) = 15. (End)
%p 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:
%p A005940 := proc(n) f(n-1,1,1) ; end proc: # _R. J. Mathar_, Mar 06 2010
%t 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 *)
%t Table[Times@@Prime/@(Join@@Position[Reverse[IntegerDigits[n,2]],1]-Range[DigitCount[n,2,1]]+1),{n,0,100}] (* _Gus Wiseman_, Dec 28 2022 *)
%o (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
%o (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
%o (Haskell)
%o a005940 n = f (n - 1) 1 1 where
%o f 0 y _ = y
%o f x y i | m == 0 = f x' y (i + 1)
%o | m == 1 = f x' (y * a000040 i) i
%o where (x',m) = divMod x 2
%o -- _Reinhard Zumkeller_, Oct 03 2012
%o (Scheme, with memoization-macro definec from Antti Karttunen's IntSeq-library)
%o (define (A005940 n) (A005940off0 (- n 1))) ;; The off=1 version, utilizing any one of three different offset-0 implementations:
%o (definec (A005940off0 n) (cond ((< n 2) (+ 1 n)) (else (* (A000040 (- (A070939 n) (- (A000120 n) 1))) (A005940off0 (A053645 n))))))
%o (definec (A005940off0 n) (cond ((<= n 2) (+ 1 n)) ((even? n) (A003961 (A005940off0 (/ n 2)))) (else (* 2 (A005940off0 (/ (- n 1) 2))))))
%o (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)))))))
%o ;; _Antti Karttunen_, Jun 26 2014
%o (Python)
%o from sympy import prime
%o import math
%o def A(n): return n - 2**int(math.floor(math.log(n, 2)))
%o def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
%o print([b(n - 1) for n in range(1, 101)]) # _Indranil Ghosh_, Apr 10 2017
%o (Python)
%o from math import prod
%o from itertools import accumulate
%o from collections import Counter
%o from sympy import prime
%o 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
%Y Cf. A103969. Inverse is A005941 (A156552).
%Y Cf. A125106. [From _Franklin T. Adams-Watters_, Mar 06 2010]
%Y Cf. A252737 (gives row sums), A252738 (row products), A332979 (largest on row).
%Y Related permutations of positive integers: A163511 (via A054429), A243353 (via A006068), A244154, A253563 (via A122111), A253565, A332977, A334866 (via A225546).
%Y A000120, A003602, A003961, A006519, A053645, A070939, A246278, A250246, A252753, A253552 are used in a formula defining this sequence.
%Y Formulas for f(a(n)) are given for f = A000265, A003963, A007949, A055396, A056239.
%Y Numbers that occur at notable sets of positions in the binary tree representation of the sequence: A000040, A000079, A002110, A070003, A070826, A102750.
%Y Cf. also A000142, A001511, A002450, A112798, A252463, A252464, A252745, A252750, A324054, A324106, A323505, A323508.
%Y 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.
%Y k-adic valuation: A007814 (k=2), A337821 (k=3).
%Y Positions of multiples of 3: A091067.
%Y Primorial deflation: A337376 / A337377.
%Y Sum of prime indices of a(n) is A161511, reverse version A359043.
%Y A048793 lists binary indices, ranked by A019565.
%Y A066099 lists standard comps, partial sums A358134 (ranked by A358170).
%Y Cf. A001222, A029837, A059893, A241916, A242628, A253566, A359042.
%K nonn,easy,nice,tabf,look
%O 1,2
%A _N. J. A. Sloane_ and _J. H. Conway_
%E More terms from _Robert G. Wilson v_, Feb 22 2005
%E Sign in a formula switched and Maple program added by _R. J. Mathar_, Mar 06 2010
%E Binary tree illustration and keyword tabf added by _Antti Karttunen_, Dec 21 2014
| 