A219666 -id:A219666 - cp's OEIS frontend

A230410 After a(0)=0, a(n) = A230415(A219666(n),A219666(n-1)).

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

Offset: 0

Antti Karttunen, Nov 10 2013

After zero, a(n) = number of positions where digits in the factorial base representations of successive nodes A219666(n-1) and A219666(n) in the infinite trunk of the factorial beanstalk differ from each other.

			a(8) = 1, because A219666(8)=23, whose factorial base representation (A007623(23)) is '321', and A219666(7)=17, whose factorial base representation (A007623(17)) is '221', and they differ just in one digit position.
a(9) = 3, because A219666(9)=25, '...01001' in factorial base, which differs from '...0321' in three digit positions.
Note that A226061(4)=8 (A226061(n) tells the position of (n!)-1 in A219666), and 1+2+3 = 6 happens to be both a triangular number (A000217) and a factorial number (A000142).
The next time 1 occurs in this sequence because of this coincidence is at x=A226061(16) (whose value is currently not known), as at that point A219666(x) = 16!-1 = 20922789887999, whose factorial base representation is (15,14,13,12,11,10,9,8,7,6,5,4,3,2,1), and A000217(15) = 120 = A000142(5), which means that A219666(x-1) = A219651(20922789887999) = 20922789887879, whose factorial base representation is (15,14,13,12,11,10,9,8,7,6,4,4,3,2,1), which differs only in one position from the previous.
Of course 1's occur in this sequence for other reasons as well.

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

Cf. A230415, A230406, A231717, A231719, A232094. A230422 gives the positions of ones.

nn = 1200; m = 1; While[m! < nn, m++]; m; f[n_] := IntegerDigits[n, MixedRadix[Reverse@ Range[2, m]]]; Join[{0}, Function[w, Count[Subtract @@ Map[PadLeft[#, Max@ Map[Length, w]] &, w], k_ /; k != 0]]@ Map[f@ # &, {#1, #2}] & @@@ Partition[#, 2, 1] &@ TakeWhile[Reverse@ NestWhileList[# - Total@ f@ # &, nn, # > 0 &], # <= 500 &]] (* Michael De Vlieger, Jun 27 2016, Version 10 *)

(define (A230410 n) (if (zero? n) n (A230415bi (A219666 n) (A219666 (- n 1))))) ;; Where bi-variate function A230415bi has been given in A230415.

a(0)=0, and for n>=1, a(n) = A230415(A219666(n),A219666(n-1)).

For all n, a(A226061(n+1)) = A232094(n).

A230406 a(n) = A034968(A219666(n)); after zero, the differences between successive nodes in the infinite trunk of the factorial beanstalk (A219666).

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 2, 5, 6, 2, 3, 2, 5, 5, 6, 2, 4, 5, 6, 7, 4, 5, 6, 7, 5, 5, 7, 10, 2, 3, 2, 5, 5, 6, 2, 4, 5, 6, 7, 4, 5, 6, 7, 5, 5, 6, 9, 8, 7, 10, 2, 4, 5, 6, 7, 4, 5, 6, 7, 5, 5, 6, 8, 6, 8, 8, 7, 10, 11, 4, 5, 6, 7, 5, 5, 6, 8, 6, 8, 8, 7, 10, 12, 10, 11

Offset: 0

Antti Karttunen, Nov 09 2013

nonn

Also the first differences of A219666, shifted once right and prepended with zero.

This sequence relates to the factorial base representation (A007623) in the same way as A213712 relates to the binary system.

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

Cf. also A230418, A230410.

(define (A230406 n) (A034968 (A219666 n)))
;; Alternative definition:
(define (A230406 n) (if (zero? n) n (- (A219666 n) (A219666 (- n 1)))))

a(n) = A034968(A219666(n)).
a(0) = 0, and for n>=1, a(n) = A219666(n) - A219666(n-1).
a(A226061(n)) = A000217(n-1) for all n.

A230427 a(n)=0 if n is in the infinite trunk of factorial beanstalk (in A219666), otherwise the number of nodes (including leaves and the node n itself) in that finite branch of the beanstalk.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 3, 0, 1, 1, 0, 5, 0, 1, 1, 3, 3, 0, 1, 1, 1, 1, 1, 0, 3, 0, 1, 1, 0, 5, 0, 1, 1, 3, 7, 0, 1, 1, 3, 3, 0, 1, 1, 1, 1, 1, 0, 5, 0, 1, 1, 3, 0, 5, 1, 1, 5, 0, 3, 1, 1, 3, 5, 0, 1, 1, 1, 1, 1, 3, 0, 5, 1, 1, 0, 11, 3, 1, 1, 0, 7, 3, 1, 1, 3, 0, 5

Offset: 0

Antti Karttunen, Nov 10 2013

nonn

This sequence relates to the factorial base representation (A007623) in the same way as A213727 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 1 + the number of all descendant nodes in such subtrees. Thus a(11)=5, a(14)=1 (counting just the leaf 14 itself), a(15)=3 and a(18) = a(19) = 1.

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

A219658 gives the position of ones (leaves). Cf. also A230407, A230423-A230424, A230425-A230426, A230430.

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

Alternatively, if A230426(n)=0, a(n)=0; otherwise (2*A230426(n))-1.

A231719 After zero, a(n) = largest m such that m! divides the difference between successive nodes A219666(n-1) and A219666(n) in the infinite trunk of the factorial beanstalk.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 2, 1, 1, 3, 2, 2, 1, 3, 1, 2, 1, 3, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 2, 2, 1, 3, 1, 2, 1, 3, 1, 1, 1, 3, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 1, 3, 1, 1, 1, 3, 2, 3, 2, 2, 1, 2, 1, 2, 1, 3, 1, 1, 1, 3, 2, 3, 2, 2, 1, 2, 3, 2, 1, 1, 1

Offset: 0

Antti Karttunen, Nov 12 2013

nonn

The first 4 occurs at n=2206. The first 5 occurs at n = 361788001015 = A226061(16).

Antti Karttunen, Table of n, a(n) for n = 0..21622

Cf. A055881, A219666, A226061, A230406, A230410, A231717, A232096.

nn = 1200; m = 1; While[Factorial@ m < nn, m++]; m; t = TakeWhile[
Reverse@ NestList[# - Total@ IntegerDigits[#, MixedRadix[Reverse@ Range[2, m]]] &, nn, 182], # <= 1000 &]; {0}~Join~Table[SelectFirst[Reverse@ Range@ 10, Divisible[t[[n]] - t[[n - 1]], #!] &], {n, 2, 87}] (* Michael De Vlieger, Jun 27 2016, Version 10.2 *)

(define (A231719 n) (if (zero? n) n (A055881 (A230406 n))))

a(0)=0 and for n>=1, a(n) = A055881(A230406(n)).

For all n, a(A226061(n+1)) = A232096(n).

A230420 Triangle T(n,k) giving the number of terms of A219666 which have n digits (A084558) in their factorial base expansion and whose most significant digit (A099563) in that base is k.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 5, 4, 4, 22, 19, 16, 14, 12, 94, 82, 73, 65, 59, 55, 479, 432, 395, 362, 336, 314, 293, 2886, 2667, 2482, 2324, 2189, 2073, 1971, 1881, 20276, 19123, 18124, 17249, 16473, 15775, 15140, 14555, 14011, 164224, 156961, 150389, 144378, 138828, 133664, 128831, 124289, 120010, 115974

Offset: 1

Antti Karttunen, Oct 18 2013

See A007623 for the factorial number system representation.

			The first rows of this triangular table are:
1;
1, 1;
2, 2, 1;
6, 5, 4, 4;
22, 19, 16, 14, 12;
94, 82, 73, 65, 59, 55;
...
T(4,2) = 5 as only the terms 48, 52, 57, 63 and 70 of A219666 (with factorial base representations 2000, 2020, 2111, 2211 and 2320) have four significant digits in the factorial base, with the most significant digit being 2.

Transpose: A230421. Row sums: A219661. Cf. also A230428, A230429, A219652, A219666.

(define (A230420 n) (if (<= n 3) 1 (let loop ((i (A230429 n)) (s 0)) (cond ((not (= (A099563 i) (A002260 n))) s) (else (loop (A219651 i) (+ 1 s)))))))

T(n,k) = 1 + A219652(A230429(n,k)) - A219652(A230428(n,k)).

A231717 After a(0)=0, a(n) = A231713(A219666(n),A219666(n-1)).

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 3, 1, 6, 3, 3, 3, 2, 1, 6, 2, 3, 1, 3, 5, 3, 1, 3, 6, 2, 2, 3, 10, 3, 3, 3, 2, 1, 6, 2, 3, 1, 3, 5, 3, 1, 3, 6, 2, 1, 3, 5, 5, 3, 10, 2, 3, 1, 3, 5, 3, 1, 3, 6, 2, 1, 2, 4, 2, 4, 5, 3, 3, 9, 3, 1, 3, 6, 2, 1, 2, 4, 2, 4, 5, 3, 2, 4, 3, 10

Offset: 0

Antti Karttunen, Nov 12 2013

nonn

For all n, a(A226061(n+1)) = A232095(n). This works because at the positions given by each x=A226061(n+1), it holds that A219666(x) = (n+1)!-1, which has a factorial base representation (A007623) of (n,n-1,n-2,...,3,2,1) whose digit sum (A034968) is the n-th triangular number, A000217(n). This in turn is always a new record as at those points, in each significant digit position so far employed, a maximal digit value (for factorial number system) is used, and thus the preceding term, A219666(x-1) cannot have any larger digits in its factorial base representation, and so the differences between their digits (in matching positions) are all nonnegative.

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

A231718 gives the positions of ones.

Cf. also A230410, A231719, A232095.

(define (A231717 n) (if (zero? n) n (A231713bi (A219666 n) (A219666 (- n 1)))))

a(0)=0, and for n>=1, a(n) = A231713(A219666(n),A219666(n-1)).

A230407 Absolute value of a(n) tells the size of the n-th side-tree ("tendril", A230430(n)) in the factorial beanstalk; the sign tells on which side of the infinite trunk (A219666) it is.

Original entry on oeis.org

0, -1, 1, 3, -5, -1, 3, 1, 3, -5, -1, 7, -1, -5, -1, -5, 5, 5, -5, -11, 1, 3, -11, 1, -3, 3, 1, 3, -5, -1, 7, -1, -5, -1, -5, 5, 5, -5, -11, 1, 3, -21, 1, -3, -3, 9, 1, -1, -5, -1, -5, 5, 5, -5, -11, 1, 3, -21, 1, -3, -3, -11, -1, -9, -3, 5, 5, -5, -11, 1, 3

Offset: 0

Antti Karttunen, Nov 10 2013

sign

Positive and negative terms correspond to the tendrils that sprout respectively at the left and right sides of the infinite trunk, when the factorial beanstalk is drawn with the lesser numbers branching to the left. The absolute values give the sizes of those tendrils, with all nodes included: The leaves, the internal vertices as well as the root itself (which is at A230430(n)).
Here a(0) = 0 is a special case, as the infinite trunk starts to grow from its child 1, while the other child is 0 itself. (For both k=0 or k=1 it is true that A219651(k)=0).
This sequence relates to the factorial base representation (A007623) in the same way as A218618 relates to the binary system.

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

Partial sums: A230408, A230409.

(define (A230407 n) (* (expt -1 (A230430 n)) (A230427 (A230430 n))))

a(n) = ((-1)^A230430(n)) * A230427(A230430(n)).

A230425 a(n)=0 if n is in the infinite trunk of factorial beanstalk (in A219666), otherwise 1 + number of steps to reach the farthest leaf 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, 3, 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, 5, 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 A213725 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 the numbers 11, 14, 15, 18 and 19.
For numbers not occurring in A219666 this sequence gives one more than the maximum number of steps to reach the most distant leaf in such subtrees. In the above case, there is from 11 only one step to 14, but two steps to both 18 and 19. Thus a(11)=2+1=3. For leaves the result is always 1, for example, a(14) = a(18) = a(19) = 1.

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

Differs from A230426 for the first time at n=34, where a(n)=3, while A230426(34)=4. Cf. also A230427.

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

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

A230416 The infinite trunk of factorial beanstalk (A219666) with reversed subsections.

Original entry on oeis.org

0, 1, 5, 2, 23, 17, 12, 10, 7, 119, 109, 102, 97, 92, 85, 79, 74, 70, 63, 57, 52, 48, 46, 40, 35, 30, 28, 25, 719, 704, 693, 680, 670, 658, 648, 641, 630, 623, 612, 605, 597, 584, 574, 562, 552, 545, 534, 527, 516, 509, 501, 492, 486, 481, 476, 465, 455, 443

Offset: 0

Antti Karttunen, Oct 22 2013

Can be viewed also as an irregular table: after the initial zero on row 0, start each row n with (n!)-1 and subtract repeatedly the sum of factorial expansion digits (A034968) to get successive terms, until the number that has already been listed [which is always (n-1)!-1] is encountered, which is not listed second time, but instead, the current row is finished and the next row starts with ((n+1)!-1), with the same process repeated.

Contains the terms in the infinite trunk of factorial beanstalk (A219666) listed in partially reversed manner: after the initial zero each subsequence lists A219661(n) successive terms from A219666, descending from (n!)-1 downwards.

			This irregular table begins as:
0;
1;
5, 2;
23, 17, 12, 10, 7;
119, 109, 102, 97, 92, 85, 79, 74, 70, 63, 57, 52, 48, 46, 40, 35, 30, 28, 25;
...
After the initial zero (on row 0), each row n is A219661(n) elements long.

Antti Karttunen, Rows 0..7, flattened

Cf. A219666, A219661, A000142, A219651.
The rows are the initial portions of every (n!-1)th row in A219659.
Analogous sequence for binary system: A218616.

For n < 3, a(n) = (n+1)!-1, and for n >= 3, a(n) = (k+2)!-1 if A219651(a(n-1)) is of form k!-1, otherwise just A219651(a(n-1)).

a(n) = A219666(A230432(n)). [Consequence of the definitions]

A230430 After zero, gives the numbers where finite side-trees (or "tendrils") of factorial beanstalk branch off from its infinite trunk (A219666).

Original entry on oeis.org

0, 3, 4, 6, 11, 13, 16, 22, 24, 29, 31, 34, 41, 47, 49, 53, 56, 62, 71, 75, 78, 84, 93, 96, 103, 108, 118, 120, 125, 127, 130, 137, 143, 145, 149, 152, 158, 167, 171, 174, 180, 189, 192, 199, 205, 212, 220, 229, 239, 241, 245, 248, 254, 263, 267, 270, 276, 285

Offset: 0

Antti Karttunen, Nov 10 2013

nonn

This sequence relates to the factorial base representation (A007623) in the same way as A213730 relates to the binary system.

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

Cf. A230407.

(define (A230430 n) (+ (A230423 (A219666 n)) (- 1 (modulo (A219666 (1+ n)) 2))))
;; Alternative definition:
(define (A230430 n) (+ (A219666 (+ 1 n)) (expt -1 (modulo (A219666 (+ 1 n)) 2))))

a(n) = A219666(n+1) + (-1)^A219666(n+1).

a(n) = A230423(A219666(n)) + (1 - (A219666(n+1) mod 2)).

cp's OEIS Frontend

A230410 After a(0)=0, a(n) = A230415(A219666(n),A219666(n-1)).

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A230406 a(n) = A034968(A219666(n)); after zero, the differences between successive nodes in the infinite trunk of the factorial beanstalk (A219666).

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A230427 a(n)=0 if n is in the infinite trunk of factorial beanstalk (in A219666), otherwise the number of nodes (including leaves and the node n itself) in that finite branch of the beanstalk.

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A231719 After zero, a(n) = largest m such that m! divides the difference between successive nodes A219666(n-1) and A219666(n) in the infinite trunk of the factorial beanstalk.

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A230420 Triangle T(n,k) giving the number of terms of A219666 which have n digits (A084558) in their factorial base expansion and whose most significant digit (A099563) in that base is k.

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A231717 After a(0)=0, a(n) = A231713(A219666(n),A219666(n-1)).

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A230407 Absolute value of a(n) tells the size of the n-th side-tree ("tendril", A230430(n)) in the factorial beanstalk; the sign tells on which side of the infinite trunk (A219666) it is.

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A230425 a(n)=0 if n is in the infinite trunk of factorial beanstalk (in A219666), otherwise 1 + number of steps to reach the farthest leaf in that finite branch of the beanstalk.

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