A219651 -id:A219651 - cp's OEIS frontend

A219661 Number of steps to go from (n+1)!-1 to n!-1 with map x -> x - (sum of digits in factorial base representation of x).

1, 2, 5, 19, 83, 428, 2611, 18473, 150726, 1377548, 13851248, 152610108, 1835293041, 23925573979, 335859122743, 5049372125352, 80942722123544, 1378487515335424, 24858383452927384, 473228664468684846

Offset: 1

Antti Karttunen, Dec 03 2012

nonn

			(1!)-1 (0) is reached from (2!)-1 (1) with one step by subtracting A034968(1) from 1.
(2!)-1 (1) is reached from (3!)-1 (5) with two steps by first subtracting A034968(5) from 5 -> 2, and then subtracting A034968(2) from 2 -> 1.
(3!)-1 (5) is reached from (4!)-1 (23) with five steps by repeatedly subtracting the sum of digits in factorial expansion as: 23 - 6 = 17, 17 - 5 = 12, 12 - 2 = 10, 10 - 3 = 7, 7 - 2 = 5.
Thus a(1)=1, a(2)=2 and a(3)=5.

Hiroaki Yamanouchi, Fast Python-program for computing terms of this sequence

Row sums of A230420 and A230421.
Cf. A000142, A007623, A219651, A219652, A219662, A219663, A219665, A226061.
Cf. also A213709 (analogous sequence for base-2), A261234 (for base-3).

Table[Length@ NestWhileList[# - Total@ IntegerDigits[#, MixedRadix[Reverse@ Range[2, 120]]] &, (n + 1)! - 1, # > n! - 1 &] - 1, {n, 0, 8}] (* Michael De Vlieger, Jun 27 2016, Version 10.2 *)

(define (A219661 n) (if (zero? n) n (let loop ((i (-1+ (A000142 (1+ n)))) (steps 1)) (cond ((isA000142? (1+ (A219651 i))) steps) (else (loop (A219651 i) (1+ steps)))))))
(define (isA000142? n) (and (> n 0) (let loop ((n n) (i 2)) (cond ((= 1 n) #t) ((not (zero? (modulo n i))) #f) (else (loop (/ n i) (1+ i)))))))
;; Alternative definition:
(define (A219661 n) (- (A219652 (-1+ (A000142 (1+ n)))) (A219652 (-1+ (A000142 n)))))

a(n) = A219652((n+1)!-1) - A219652(n!-1).

a(n) = A219662(n) + A219663(n).

Terms a(16) - a(20) computed with Hiroaki Yamanouchi's Python-program by Antti Karttunen, Jun 27 2016

A230423 a(n) = smallest natural number x such that x=n+A034968(x), or zero if no such number exists.

Original entry on oeis.org

0, 2, 4, 0, 0, 6, 8, 10, 0, 0, 12, 14, 16, 0, 0, 18, 20, 22, 0, 0, 0, 0, 0, 24, 26, 28, 0, 0, 30, 32, 34, 0, 0, 36, 38, 40, 0, 0, 42, 44, 46, 0, 0, 0, 0, 0, 48, 50, 52, 0, 0, 54, 56, 58, 0, 0, 60, 62, 64, 0, 0, 66, 68, 70, 0, 0, 0, 0, 0, 72, 74, 76, 0, 0, 78

Offset: 0

Antti Karttunen, Oct 31 2013

nonn

Also, if n can be partitioned into sum d1*(k1!-1) + d2*(k2!-1) + ... + dj*(kj!-1), where all k's are distinct and greater than one and each di is in range [1,ki] (in other words, if A230412(n)=1), then a(n) = d1*k1! + d2*k2! + ... + dj*kj!. If this is not possible, then n is one of the terms of A219658, and a(n)=0.

Antti Karttunen, Table of n, a(n) for n = 0..10079

a(A219650(n)) = A005843(n) = 2n. Cf. also A230414, A230424.
Can be used to compute A230425-A230427.
This sequence relates to the factorial base representation (A007623) in a similar way as A213723 relates to the binary system.

(define (A230423 n) (let loop ((k n)) (cond ((= (A219651 k) n) k) ((> k (+ n n)) 0) (else (loop (+ 1 k))))))

a(n) = 2*A230414(n).

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

Original entry on oeis.org

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).

A219662 Number of times an even number is encountered, when going from (n+1)!-1 to n!-1 using the iterative process described in A219652.

Original entry on oeis.org

1, 1, 2, 10, 49, 268, 1505, 9667, 81891, 779193, 7726623, 80770479, 921442854, 11621384700, 159894957124

Offset: 1

Antti Karttunen, Dec 03 2012

At least for n=7, 8, 9 and 10, a(n) is equal to a(n+1) when taken modulo n.

			(1!)-1 (0) is reached from (2!)-1 (1) with one step by subtracting A034968(1) from 1. Zero is an even number, so a(1)=1.
(2!)-1 (1) is reached from (3!)-1 (5) with two steps by first subtracting A034968(5) from 5 -> 2, and then subtracting A034968(2) from 2 -> 1. Two is an even number, but one is not, so a(2)=1.
(3!)-1 (5) is reached from (4!)-1 (23) with five steps by repeatedly subtracting the sum of digits in factorial expansion as: 23 - 6 = 17, 17 - 5 = 12, 12 - 2 = 10, 10 - 3 = 7, 7 - 2 = 5. Of these only 12 and 10 are even numbers, so a(3)=2.

Antti Karttunen, Table of n, a(n) for n = 1..15

Cf. A034968, A219666, A218542, A218543, A219652, A219661, A219663.

(definec (A219662 n) (if (< n 2) n (let loop ((i (- (A000142 (1+ n)) (A000217 n) 1)) (s 0)) (cond ((isA000142? (1+ i)) (+ s (- 1 (modulo i 2)))) (else (loop (A219651 i) (+ s (- 1 (modulo i 2)))))))))
(define (isA000142? n) (and (> n 0) (let loop ((n n) (i 2)) (cond ((= 1 n) #t) ((not (zero? (modulo n i))) #f) (else (loop (/ n i) (1+ i)))))))

a(n) = A219661(n) - A219663(n).

A219663 Number of times an odd number is encountered, when going from (n+1)!-1 to n!-1 using the iterative process described in A219652.

Original entry on oeis.org

0, 1, 3, 9, 34, 160, 1106, 8806, 68835, 598355, 6124625, 71839629, 913850187, 12304189279, 175964165619

Offset: 1

Antti Karttunen, Dec 03 2012

Ratio a(n)/A219662(n) develops as follows:
0, 1, 1.5, 0.9, 0.694..., 0.597..., 0.735..., 0.911..., 0.841..., 0.768..., 0.793..., 0.889..., 0.992..., 1.059..., 1.100...
Compare this to how the ratio A218543(n)/A218542(n) develops (ratios listed in entry A218543) and see also the associated graphs plotted by OEIS Server.

			(1!)-1 (0) is reached from (2!)-1 (1) with one step by subtracting A034968(1) from 1. Zero is not an odd number, so a(1)=0.
(2!)-1 (1) is reached from (3!)-1 (5) with two steps by first subtracting A034968(5) from 5 -> 2, and then subtracting A034968(2) from 2 -> 1. Two is not an odd number, but one is, so a(2)=1.
(3!)-1 (5) is reached from (4!)-1 (23) with five steps by repeatedly subtracting the sum of digits in factorial expansion as: 23 - 6 = 17, 17 - 5 = 12, 12 - 2 = 10, 10 - 3 = 7, 7 - 2 = 5. Of these (after 23) only 17, 7 and 5 are odd numbers, so a(3)=3.

Antti Karttunen, Table of n, a(n) for n = 1..15
Antti Karttunen, Sequence plotted together with A219662 showing how their ratio develops.

Cf. A034968, A218542, A218543, A219652, A219661, A219662, A219666.

(definec (A219663 n) (if (< n 2) 0 (let loop ((i (- (A000142 (1+ n)) (A000217 n) 1)) (s 0)) (cond ((isA000142? (1+ i)) (+ s (modulo i 2))) (else (loop (A219651 i) (+ s (modulo i 2))))))))
(define (isA000142? n) (and (> n 0) (let loop ((n n) (i 2)) (cond ((= 1 n) #t) ((not (zero? (modulo n i))) #f) (else (loop (/ n i) (1+ i)))))))

a(n) = A219661(n) - A219662(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)).

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)).

A227191 a(n) = n minus (product of nonzero digits in factorial base representation of n).

Original entry on oeis.org

0, 1, 2, 2, 3, 5, 6, 7, 8, 8, 9, 10, 11, 12, 13, 12, 13, 15, 16, 17, 18, 16, 17, 23, 24, 25, 26, 26, 27, 29, 30, 31, 32, 32, 33, 34, 35, 36, 37, 36, 37, 39, 40, 41, 42, 40, 41, 46, 47, 48, 49, 48, 49, 52, 53, 54, 55, 54, 55, 56, 57, 58, 59, 56, 57, 60, 61, 62

Offset: 1

Antti Karttunen, Jul 04 2013

			22 has factorial expansion A007623(22) = "320", and multiplying the nonzero digits, we get 3*2 = 6, and 22-6 = 16, thus a(22)=16.

Antti Karttunen, Table of n, a(n) for n = 1..5040
Index entries for sequences related to factorial base representation.

Cf. A007623, A227153, A219651, A227190.

a[n_] := Module[{k = n, m = 2, r, p = 1}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, If[r > 0, p *= r]; m++]; n - p]; Array[a, 100] (* Amiram Eldar, Feb 14 2024 *)

Scheme
```
(define (A227191 n) (- n (A227153 n)))
```

a(n) = n - A227153(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]

A237449 a(n) = n - A236855(n).

Original entry on oeis.org

0, 0, 1, 1, 1, 4, 4, 5, 5, 5, 7, 7, 7, 7, 13, 13, 14, 14, 14, 17, 17, 18, 18, 18, 20, 20, 20, 20, 25, 25, 26, 26, 26, 28, 28, 28, 28, 31, 31, 31, 31, 31, 41, 41, 42, 42, 42, 45, 45, 46, 46, 46, 48, 48, 48, 48, 54, 54, 55, 55, 55, 58, 58, 59, 59, 59, 61, 61, 61, 61

Offset: 0

Antti Karttunen, Apr 18 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16796

Cf. A236855, A239903, A236859.

Cf. also A011371, A219641, A219651, A227190, A227191, A236840.

A236855list[m_] := With[{r = 2*Range[2, m]-1}, Reverse[Map[Total[r-#] &, Select[Subsets[Range[2, 2*m-1], {m-1}], Min[r-#] >= 0 &]]]];
With[{m = 6}, Range[0, CatalanNumber[m]-1] - A236855list[m]] (* Generates C(m) terms *) (* Paolo Xausa, Feb 20 2024 *)

Scheme
```
(define (A237449 n) (- n (A236855 n)))
```

a(n) = n - A236855(n).

cp's OEIS Frontend

A219661 Number of steps to go from (n+1)!-1 to n!-1 with map x -> x - (sum of digits in factorial base representation of x).

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A230423 a(n) = smallest natural number x such that x=n+A034968(x), or zero if no such number exists.

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

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A219662 Number of times an even number is encountered, when going from (n+1)!-1 to n!-1 using the iterative process described in A219652.

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A219663 Number of times an odd number is encountered, when going from (n+1)!-1 to n!-1 using the iterative process described in A219652.

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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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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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A227191 a(n) = n minus (product of nonzero digits in factorial base representation of n).

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