A219666 -id:A219666 - cp's OEIS frontend

A230411 a(n) = minimal k for which A219665(k) >= n; a(n) = one more than the factorial base width (A084558) of the (n-1)th term in the infinite trunk of factorial beanstalk (A219666).

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

Offset: 1

Antti Karttunen, Oct 22 2013

nonn

a(1)=1, after which each term n occurs A219661(n-1) times.

Auxiliary function for computing A219666, A230431 and A230432.

Antti Karttunen, Table of n, a(n) for n = 1..5000

Cf. A219661, A219665, A219666, A230431, A230432.

Analogous sequence for binary system: A213711.

a(n) = 1 + A084558(A219666(n-1)) = 1 + A084558(A230416(n-1)). [Each a(n) is one more than the number of digits needed in factorial base to write the (n-1)-th term in the infinite trunk of factorial beanstalk]

A230428 Triangle T(n,k) giving the smallest term in "the infinite trunk of factorial beanstalk" (A219666) whose factorial base representation contains n digits (A084558) and the most significant such digit (A099563) is k.

Original entry on oeis.org

1, 2, 5, 7, 12, 23, 25, 48, 74, 97, 121, 240, 362, 481, 605, 721, 1440, 2162, 2881, 3605, 4326, 5041, 10080, 15122, 20161, 25205, 30246, 35288, 40321, 80640, 120962, 161281, 201605, 241926, 282248, 322568, 362881, 725760, 1088642, 1451521, 1814405, 2177286, 2540168, 2903048, 3265923

Offset: 1

Antti Karttunen, Oct 18 2013

			The first rows of this triangular table are:
1;
2, 5;
7, 12, 23;
25, 48, 74, 97;
121, 240, 362, 481, 605;
...
T(3,1) = 7 as 7 has factorial base representation 101, which is the smallest such three digit term in A219666 beginning with factorial base digit 1 (in other words, for which A084558(x)=3 and A099563(x)=1).
T(3,2) = 12 as 12 has factorial base representation 200, which is the smallest such three digit term in A219666 beginning with factorial base digit 2.
T(3,3) = 23 as 23 has factorial base representation 321, which is the smallest such three digit term in A219666 beginning with factorial base digit 3.

Subset of A219666. Corresponding largest terms: A230429. Cf. also A230420.

(define (A230428 n) (if (< n 3) n (let ((k (A002260 n))) (let loop ((i (A230429 n)) (prev_i 0)) (cond ((not (= (A099563 i) k)) prev_i) (else (loop (A219651 i) i)))))))

A230429 Triangle T(n,k) giving the largest member of "the infinite trunk of factorial beanstalk" (A219666) whose factorial base representation contains n digits (A084558) and the most significant such digit (A099563) is k.

Original entry on oeis.org

1, 2, 5, 10, 17, 23, 46, 70, 92, 119, 238, 358, 476, 597, 719, 1438, 2158, 2876, 3597, 4319, 5039, 10078, 15118, 20156, 25197, 30239, 35279, 40319, 80638, 120958, 161276, 201597, 241919, 282239, 322558, 362879, 725758, 1088638, 1451516, 1814397, 2177279, 2540159, 2903038, 3265912, 3628799

Offset: 1

Antti Karttunen, Oct 18 2013

See A007623 for the factorial number system representation.

			The first rows of this triangular table are:
1;
2, 5;
10, 17, 23;
46, 70, 92, 119;
238, 358, 476, 597, 719;
...
T(3,1) = 10 as 10 has factorial base representation 120, which is the largest such three digit term in A219666 beginning with factorial base digit 1 (in other words, for which A084558(x)=3 and A099563(x)=1).
T(3,2) = 17 as 17 has factorial base representation 221, which is the largest such three digit term in A219666 beginning with factorial base digit 2.
T(3,3) = 23 as 23 has factorial base representation 321, which is the largest such three digit term in A219666 beginning with factorial base digit 3.

Subset of A219666. Corresponding smallest terms: A230428. Can be used to compute A230420. Right edge: A033312.

(define (A230429 n) (if (= (A002024 n) (A002260 n)) (- (A000142 (+ (A002024 n) 1)) 1) (A219651 (A230428 (+ 1 n)))))

A230426 a(n)=0 if n is in the infinite trunk of factorial beanstalk (in A219666), otherwise the number of terminal nodes (leaves) in that finite branch of the beanstalk.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Nov 10 2013

nonn

This sequence relates to the factorial base representation (A007623) in the same way as A213726 relates to the binary system.

			From 11 sprouts the following finite side-tree of "factorial beanstalk":
    18  19
     \  /
  14  15
   \  /
    11
Its leaves are the numbers 14, 18 and 19 (which all occur in A219658), whose factorial base representations (see A007623) are '210', '300' and '301' respectively. The corresponding parent nodes are obtained by subtracting the sum of factorial base digits, thus we get 18-3 = 15 and also 19-4 = 15, thus 15 ('211' in factorial base) is the parent of 18 and 19. For 14 and 15 we get 14-3 = 15-4 = 11, thus 11 is the parent of both 14 and 15, and the common ancestor of all numbers 11, 14, 15, 18 and 19.
For numbers not occurring in A219666 this sequence gives the number of leaves in such subtrees. Thus a(11)=3, a(14)=1 (counting just the leaf 14 itself), a(15)=2 and a(18) = a(19) = 1.

Antti Karttunen, Table of n, a(n) for n = 0..10080

A219658 gives the position of ones in this sequence (which are the leaves of the tree).

Differs from A230425 for the first time at n=34, where a(n)=4, while A230425(34)=3. Cf. also A230427.

If A230412(n)=0, a(n)=1; otherwise, if n is in A219666, a(n)=0; otherwise a(n) = a(A230423(n)) + a(A230424(n)).

A231723 a(n) = the difference between the n-th node of the infinite trunk of the factorial beanstalk (A219666(n)) and the smallest integer (A219653(n)) which is as many A219651-iteration steps distanced from the root (zero); a(n) = A219666(n) - A219653(n).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 0, 1, 3, 1, 2, 0, 1, 2, 4, 0, 0, 1, 3, 4, 2, 1, 1, 2, 1, 0, 1, 3, 1, 2, 0, 1, 2, 4, 0, 0, 1, 3, 4, 2, 1, 1, 2, 1, 0, 0, 1, 3, 2, 4, 0, 0, 1, 3, 4, 2, 1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 1, 3, 4, 2, 1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 1, 3, 5, 7, 8, 1, 0

Offset: 0

Antti Karttunen, Nov 13 2013

nonn

For all n, the following holds: A219653(n) <= A219666(n) <= A219655(n). This sequence gives the distance of the node n in the infinite trunk of factorial beanstalk (A219666(n)) from the left (lesser) edge of the A219654(n) wide window which it at that point must pass through.

This sequence relates to the factorial base representation (A007623) in the same way as A218603 relates to the binary system and similar remarks apply here.

Antti Karttunen, Table of n, a(n) for n = 0..3149

Cf. A231724, A230409, A219662 & A219663.

(define (A231723 n) (- (A219666 n) (A219653 n)))

a(n) = A219666(n) - A219653(n).

A219654(n) = a(n) + A231724(n) + 1.

A231724 a(n) = the difference between the n-th node of the infinite trunk of the factorial beanstalk (A219666(n)) and the greatest integer (A219655(n)) which is as many A219651-iteration steps distanced from the root (zero); a(n) = A219655(n) - A219666(n).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 3, 2, 0, 0, 1, 3, 2, 1, 1, 3, 3, 2, 2, 1, 3, 4, 4, 3, 4, 5, 6, 0, 0, 1, 3, 2, 1, 1, 3, 3, 2, 2, 1, 3, 4, 4, 3, 4, 5, 7, 4, 4, 5, 1, 3, 3, 2, 2, 1, 3, 4, 4, 3, 4, 5, 7, 5, 7, 7, 5, 6, 6, 1, 3, 4, 4, 3, 4, 5, 7, 5, 7, 7, 5, 6, 6, 2, 2, 3, 4, 5

Offset: 0

Antti Karttunen, Nov 13 2013

nonn

For all n, the following holds: A219653(n) <= A219666(n) <= A219655(n). This sequence gives the distance of the node n in the infinite trunk of factorial beanstalk (A219666(n)) from the right (greater) edge of the A219654(n) wide window which it at that point must pass through.

This sequence relates to the factorial base representation (A007623) in the same way as A218604 relates to the binary system and similar remarks apply here.

Antti Karttunen, Table of n, a(n) for n = 0..3149

Cf. A231723, A230409, A219662 & A219663.

(define (A231724 n) (- (A219655 n) (A219666 n)))

a(n) = A219655(n) - A219666(n).

A219654(n) = a(n) + A231723(n) + 1.

A278534 a(n) = A278236(A219666(n)).

Original entry on oeis.org

1, 2, 2, 12, 6, 12, 4, 180, 360, 6, 12, 6, 420, 180, 360, 4, 36, 420, 1260, 1800, 24, 120, 360, 1080, 48, 48, 720, 75600, 6, 12, 6, 420, 180, 360, 6, 60, 2310, 4620, 2520, 60, 420, 1260, 2520, 120, 120, 360, 83160, 5040, 720, 75600, 4, 36, 420, 1260, 1800, 60, 420, 1260, 2520, 180, 180, 900, 12600, 360, 12600, 5400, 720, 277200, 529200, 24, 120, 360, 1080, 120, 120

Offset: 0

Antti Karttunen, Nov 30 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..3149

Cf. A219666, A230406, A278232, A278236, A278262, A278532.

(define (A278534 n) (A278236 (A219666 n)))

a(n) = A278236(A219666(n)).

A179016 The infinite trunk of binary beanstalk: The only infinite sequence such that a(n-1) = a(n) - number of 1's in binary representation of a(n).

Original entry on oeis.org

0, 1, 3, 4, 7, 8, 11, 15, 16, 19, 23, 26, 31, 32, 35, 39, 42, 46, 49, 53, 57, 63, 64, 67, 71, 74, 78, 81, 85, 89, 94, 97, 101, 104, 109, 112, 116, 120, 127, 128, 131, 135, 138, 142, 145, 149, 153, 158, 161, 165, 168, 173, 176, 180, 184, 190, 193, 197, 200, 205, 209

Offset: 0

Carl R. White, Jun 24 2010

a(n) tells in what number we end in n steps, when we start climbing up the infinite trunk of the "binary beanstalk" from its root (zero). The name "beanstalk" is due to Antti Karttunen.

There are many finite sequences such as 0,1,2; 0,1,3,4,7,9; etc. obeying the same condition (see A218254) and as the length increases, so (necessarily) does the similarity to this infinite sequence.

Alois P. Heinz and Antti Karttunen, Table of n, a(n) for n = 0..16405 (first 1000 terms from Alois P. Heinz)
Paul Tek, Illustration of the first terms

Cf. A000120, A010062, A011371, A213710, A213711, A213717, A213730, A213731, A218600, A218616, A218789, A233271, A218602, A054429. First differences: A213712, complement: A213713.
A subsequence of A005187, i.e., a(n) = A005187(A213715(n)). For all n,
A071542(a(n)) = n, and furthermore A213708(n) <= a(n) <= A173601(n). (Cf. A218603, A218604).
Rows of A218254, when reversed, converge towards this sequence.
Cf. A276623, A219648, A219666, A255056, A276573, A276583, A276613 for analogous constructions, and also A259934.

TakeWhile[Reverse@ NestWhileList[# - DigitCount[#, 2, 1] &, 10^3, # > 0 &], # <= 209 &] (* Michael De Vlieger, Sep 12 2016 *)

a(0)=0, a(1)=1, and for n > 1, if n = A218600(A213711(n)) then a(n) = (2^A213711(n)) - 1, and in other cases, a(n) = a(n+1) - A213712(n+1). (This formula is based on Carl White's observation that this iterated/converging path must pass through each (2^n)-1. However, it would be very interesting to know whether the sequence admits more traditional recurrence(s), referring to previous, not to further terms in the sequence in their definition!) - Antti Karttunen, Oct 26 2012
a(n) = A218616(A218602(n)). - Antti Karttunen, Mar 04 2013
a(n) = A054429(A233271(A218602(n))). - Antti Karttunen, Dec 12 2013

Starting offset changed from 1 to 0 by Antti Karttunen, Nov 05 2012

A034968 Minimal number of factorials that add to n.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7

Offset: 0

Erich Friedman

nonn

Equivalently, sum of digits when n is written in factorial base (A007623).
Equivalently, a(0)...a(n!-1) give the total number of inversions of the permutations of n elements in lexicographic order (the factorial numbers in rising base are the inversion tables of the permutations and their sum of digits give the total number of inversions, see example and the Fxtbook link). - Joerg Arndt, Jun 17 2011
Also minimum number of adjacent transpositions needed to produce each permutation in the list A055089, or number of swappings needed to bubble sort each such permutation. (See A055091 for the minimum number of any transpositions.)

			a(205) = a(1!*1 + 3!*2 + 4!*3 + 5!*1) = 1+2+3+1 = 7. [corrected by Shin-Fu Tsai, Mar 23 2021]
From _Joerg Arndt_, Jun 17 2011: (Start)
   n:    permutation   inv. table a(n)  cycles
   0:    [ 0 1 2 3 ]   [ 0 0 0 ]   0    (0) (1) (2) (3)
   1:    [ 0 1 3 2 ]   [ 0 0 1 ]   1    (0) (1) (2, 3)
   2:    [ 0 2 1 3 ]   [ 0 1 0 ]   1    (0) (1, 2) (3)
   3:    [ 0 2 3 1 ]   [ 0 1 1 ]   2    (0) (1, 2, 3)
   4:    [ 0 3 1 2 ]   [ 0 2 0 ]   2    (0) (1, 3, 2)
   5:    [ 0 3 2 1 ]   [ 0 2 1 ]   3    (0) (1, 3) (2)
   6:    [ 1 0 2 3 ]   [ 1 0 0 ]   1    (0, 1) (2) (3)
   7:    [ 1 0 3 2 ]   [ 1 0 1 ]   2    (0, 1) (2, 3)
   8:    [ 1 2 0 3 ]   [ 1 1 0 ]   2    (0, 1, 2) (3)
   9:    [ 1 2 3 0 ]   [ 1 1 1 ]   3    (0, 1, 2, 3)
  10:    [ 1 3 0 2 ]   [ 1 2 0 ]   3    (0, 1, 3, 2)
  11:    [ 1 3 2 0 ]   [ 1 2 1 ]   4    (0, 1, 3) (2)
  12:    [ 2 0 1 3 ]   [ 2 0 0 ]   2    (0, 2, 1) (3)
  13:    [ 2 0 3 1 ]   [ 2 0 1 ]   3    (0, 2, 3, 1)
  14:    [ 2 1 0 3 ]   [ 2 1 0 ]   3    (0, 2) (1) (3)
  15:    [ 2 1 3 0 ]   [ 2 1 1 ]   4    (0, 2, 3) (1)
  16:    [ 2 3 0 1 ]   [ 2 2 0 ]   4    (0, 2) (1, 3)
  17:    [ 2 3 1 0 ]   [ 2 2 1 ]   5    (0, 2, 1, 3)
  18:    [ 3 0 1 2 ]   [ 3 0 0 ]   3    (0, 3, 2, 1)
  19:    [ 3 0 2 1 ]   [ 3 0 1 ]   4    (0, 3, 1) (2)
  20:    [ 3 1 0 2 ]   [ 3 1 0 ]   4    (0, 3, 2) (1)
  21:    [ 3 1 2 0 ]   [ 3 1 1 ]   5    (0, 3) (1) (2)
  22:    [ 3 2 0 1 ]   [ 3 2 0 ]   5    (0, 3, 1, 2)
  23:    [ 3 2 1 0 ]   [ 3 2 1 ]   6    (0, 3) (1, 2)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000
F. T. Adams-Watters and F. Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS 12 (2009) 09.5.6.
Joerg Arndt, Matters Computational (The Fxtbook), fig.10-1.B on p.234.
Tyler Ball, Joanne Beckford, Paul Dalenberg, Tom Edgar, and Tina Rajabi, Some Combinatorics of Factorial Base Representations, J. Int. Seq., Vol. 23 (2020), Article 20.3.3.
FindStat - Combinatorial Statistic Finder, The number of inversions of a permutation
Index entries for sequences related to factorial base representation

Cf. A368342 (partial sums), A001809 (sums of n! terms).
Cf. A000142, A007489, A007623, A033312, A055091, A139365.
Cf. A000217, A001222, A056019, A060130, A099563, A225901, A257684, A257687, A257694, A276076, A276146.
Cf. A227148 (positions of even terms), A227149 (of odd terms).
Cf. also A219650, A219651, A219666, A230423.
Differs from analogous A276150 for the first time at n=24.
Positions of records are A200748.

[seq(convert(fac_base(j),`+`),j=0..119)]; # fac_base and PermRevLexUnrank given in A055089. Perm2InversionVector in A064039
Or alternatively: [seq(convert(Perm2InversionVector(PermRevLexUnrank(j)),`+`),j=0..119)];
# third Maple program:
b:= proc(n, i) local q;
      `if`(n=0, 0, b(irem(n, i!, 'q'), i-1)+q)
    end:
a:= proc(n) local k;
      for k while k!Alois P. Heinz, Nov 15 2012

a[n_] := Module[{s=0, i=2, k=n}, While[k > 0, k = Floor[n/i!]; s = s + (i-1)*k; i++]; n-s]; Table[a[n], {n, 0, 105}] (* Jean-François Alcover, Nov 06 2013, after Benoit Cloitre *)

a(n)=local(k,r);k=2;r=0;while(n>0,r+=n%k;n\=k;k++);r \\ Franklin T. Adams-Watters, May 13 2009

def a(n):
    k=2
    r=0
    while n>0:
        r+=n%k
        n=n//k
        k+=1
    return r
print([a(n) for n in range(201)]) # Indranil Ghosh, Jun 19 2017, after PARI program

def A034968(n, p=2): return n if nA034968(n//p, p+1) + n%p
print([A034968(n) for n in range(106)]) # Michael S. Branicky, Oct 27 2024

(define (A034968 n) (let loop ((n n) (i 2) (s 0)) (cond ((zero? n) s) (else (loop (quotient n i) (+ 1 i) (+ s (remainder n i)))))))
;; Antti Karttunen, Aug 29 2016

a(n) = n - Sum_{i>=2} (i-1)*floor(n/i!). - Benoit Cloitre, Aug 26 2003
G.f.: 1/(1-x)*Sum_{k>0} (Sum_{i=1..k} i*x^(i*k!))/(Sum_{i=0..k} x^(i*k!)). - Franklin T. Adams-Watters, May 13 2009
From Antti Karttunen, Aug 29 2016: (Start)
a(0) = 0; for n >= 1, a(n) = A099563(n) + a(A257687(n)).
a(0) = 0; for n >= 1, a(n) = A060130(n) + a(A257684(n)).
Other identities. For all n >= 0:
a(n) = A001222(A276076(n)).
a(n) = A276146(A225901(n)).
a(A000142(n)) = 1, a(A007489(n)) = n, a(A033312(n+1)) = A000217(n).
a(A056019(n)) = a(n).
A219651(n) = n - a(n).
(End)

Additional comments from Antti Karttunen, Aug 23 2001

A255056 Trunk of number-of-runs beanstalk: The unique infinite sequence such that a(n-1) = a(n) - number of runs in binary representation of a(n).

Original entry on oeis.org

0, 2, 4, 6, 10, 12, 14, 18, 22, 26, 28, 30, 32, 36, 42, 46, 50, 54, 58, 60, 62, 64, 68, 74, 78, 84, 90, 94, 96, 100, 106, 110, 114, 118, 122, 124, 126, 128, 132, 138, 142, 148, 152, 156, 162, 168, 174, 180, 186, 190, 192, 196, 202, 206, 212, 218, 222, 224, 228, 234, 238, 242, 246, 250, 252, 254

Offset: 0

Antti Karttunen, Feb 14 2015

All numbers of the form (2^n)-2 are present, which guarantees the uniqueness and also provides a well-defined method to compute the sequence, for example, via a partially reversed version A255066.

The sequence was inspired by a similar "binary weight beanstalk", A179016, sharing some general properties with it (like its partly self-copying behavior, see A255071), but also differing in some aspects. For example, here the branching degree is not the constant 2, but can vary from 1 to 4. (Cf. A255058.)

Antti Karttunen, Table of n, a(n) for n = 0..16142

First differences: A255336.
Terms halved: A255057.
Cf. A255053 & A255055 (the lower & upper bound for a(n)) and also A255123, A255124 (distances to those limits).
Cf. A255327, A255058 (branching degree for node n), A255330 (number of nodes in the finite subtrees branching from the node n), A255331, A255332
Subsequence: A000918 (except for -1).
Cf. A255061, A255062, A255071, A255072, A255066, A255122.
Cf. A254113, A254114.
Cf. A255063, A255064, A255125, A255126.
Similar "beanstalk's trunk" sequences using some other subtracting map than A236840: A179016, A219648, A219666.

(define (A255056 n) (A255066 (A255122 n)))

a(n) = A255066(A255122(n)).
Other identities and observations. For all n >= 0:
a(n) = 2*A255057(n).
A255072(a(n)) = n.
A255053(n) <= a(n) <= A255055(n).

cp's OEIS Frontend

A230411 a(n) = minimal k for which A219665(k) >= n; a(n) = one more than the factorial base width (A084558) of the (n-1)th term in the infinite trunk of factorial beanstalk (A219666).

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A230428 Triangle T(n,k) giving the smallest term in "the infinite trunk of factorial beanstalk" (A219666) whose factorial base representation contains n digits (A084558) and the most significant such digit (A099563) is k.

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A230429 Triangle T(n,k) giving the largest member of "the infinite trunk of factorial beanstalk" (A219666) whose factorial base representation contains n digits (A084558) and the most significant such digit (A099563) is k.

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A230426 a(n)=0 if n is in the infinite trunk of factorial beanstalk (in A219666), otherwise the number of terminal nodes (leaves) in that finite branch of the beanstalk.

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A231723 a(n) = the difference between the n-th node of the infinite trunk of the factorial beanstalk (A219666(n)) and the smallest integer (A219653(n)) which is as many A219651-iteration steps distanced from the root (zero); a(n) = A219666(n) - A219653(n).

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A231724 a(n) = the difference between the n-th node of the infinite trunk of the factorial beanstalk (A219666(n)) and the greatest integer (A219655(n)) which is as many A219651-iteration steps distanced from the root (zero); a(n) = A219655(n) - A219666(n).

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A278534 a(n) = A278236(A219666(n)).

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A179016 The infinite trunk of binary beanstalk: The only infinite sequence such that a(n-1) = a(n) - number of 1's in binary representation of a(n).

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Mathematica

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A034968 Minimal number of factorials that add to n.

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