A219659 -id:A219659 - cp's OEIS frontend

A219657 Positions of zeros in A219659.

0, 2, 5, 8, 12, 16, 21, 26, 32, 38, 44, 50, 57, 64, 71, 78, 86, 94, 102, 110, 119, 128, 137, 146, 156, 166, 177, 188, 199, 210, 222, 234, 246, 258, 271, 284, 297, 310, 324, 338, 352, 366, 381, 396, 411, 426, 441, 456, 472, 488, 504, 520, 537, 554, 571, 588, 606

Offset: 0

Antti Karttunen, Nov 25 2012

nonn

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

Cf. A219656, A219659. Analogous sequence for binary system: A213707, for Zeckendorf expansion: A219647.

Scheme
```
(define (A219657 n) (+ n (A219656 n)))
```

a(n) = n + A219656(n).

A219666 The infinite trunk of factorial expansion beanstalk. The only infinite sequence such that a(n-1) = a(n) - sum of digits in factorial expansion of a(n).

Original entry on oeis.org

0, 1, 2, 5, 7, 10, 12, 17, 23, 25, 28, 30, 35, 40, 46, 48, 52, 57, 63, 70, 74, 79, 85, 92, 97, 102, 109, 119, 121, 124, 126, 131, 136, 142, 144, 148, 153, 159, 166, 170, 175, 181, 188, 193, 198, 204, 213, 221, 228, 238, 240, 244, 249, 255, 262, 266, 271, 277

Offset: 0

Antti Karttunen, Nov 25 2012

a(n) tells in what number we end in n steps, when we start climbing up the infinite trunk of the "factorial beanstalk" from its root (zero).
There are many finite sequences such as 0,1,2,4; 0,1,2,5,6; etc. obeying the same condition (see A219659) and as the length increases, so (necessarily) does the similarity to this infinite sequence.
See A007623 for the factorial number system representation.

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

Cf. A007623, A034968, A219651, A230411, A226061. For all n, A219652(a(n)) = n and A219653(n) <= a(n) <= A219655(n).
Characteristic function: Χ_A219666(n) = A230418(n+1)-A230418(n).
The first differences: A230406.
Other derived sequences: A230425-A230427, A230430, A230407-A230409, A219662 & A219663, A231723 & A231724, A230420, A230410, A231717, A231719.
Subsets: A230428 & A230429.
Analogous sequence for binary system: A179016, for Fibonacci number system: A219648.

nn = 10^3; m = 1; While[m! < Floor[6 nn/5], m++]; m; t = TakeWhile[Reverse@ NestWhileList[# - Total@ IntegerDigits[#, MixedRadix[Reverse@ Range[2, m]]] &, Floor[6 nn/5], # > 0 &], # <= nn &] (* Michael De Vlieger, Jun 27 2016, Version 10.2 *)

;; Memoizing definec-macro from Antti Karttunen's IntSeq-library
(definec (A219666 n) (cond ((<= n 2) n) ((= (A226061 (A230411 n)) n) (- (A000142 (A230411 n)) 1)) (else (- (A219666 (+ n 1)) (A034968 (A219666 (+ n 1)))))))
;; Another variant, utilizing A230416 (which gives a more convenient way to compute large number of terms of this sequence):
(define (A219666 n) (A230416 (A230432 n)))
;; This function is for checking whether n belongs to this sequence:
(define (inA219666? n) (or (zero? n) (= 1 (- (A230418 (+ 1 n)) (A230418 n)))))

a(0) = 0, a(1) = 1, and for n>1, if A226061(A230411(n)) = n then a(n) = A230411(n)!-1, otherwise a(n) = a(n+1) - A034968(a(n+1)).

a(n) = A230416(A230432(n)).

A219651 a(n) = n minus (sum of digits in factorial base expansion of n).

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 5, 6, 6, 7, 7, 10, 10, 11, 11, 12, 12, 15, 15, 16, 16, 17, 17, 23, 23, 24, 24, 25, 25, 28, 28, 29, 29, 30, 30, 33, 33, 34, 34, 35, 35, 38, 38, 39, 39, 40, 40, 46, 46, 47, 47, 48, 48, 51, 51, 52, 52, 53, 53, 56, 56, 57, 57, 58, 58, 61, 61

Offset: 0

Antti Karttunen, Nov 25 2012

See A007623 for the factorial base number system representation.

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

Bisection: A219650. Analogous sequence for binary system: A011371, for Zeckendorf expansion: A219641.

Cf. A007623, A034968, A219652-A219655, A219656, A219659, A219666.

(* First run program for A007623 to define factBaseIntDs *) Table[n - Plus@@factBaseIntDs[n], {n, 0, 99}] (* Alonso del Arte, Nov 25 2012 *)

from itertools import count
def A219651(n):
    c, f = 0, 1
    for i in count(2):
        f *= i
        if f>n:
            break
        c += (i-1)*(n//f)
    return c # Chai Wah Wu, Oct 11 2024

Scheme
```
(define (A219651 n) (- n (A034968 n)))
```

a(n) = n - A034968(n).

A219652 Number of steps to reach 0 starting with n and using the iterated process: x -> x - (sum of digits in factorial expansion of x).

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19

Offset: 0

Antti Karttunen, Nov 25 2012

See A007623 for the factorial number system representation.

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

Analogous sequence for binary system: A071542, for Zeckendorf expansion: A219642. Cf. A007623, A034968, A219650, A219651, A219653-A219655, A219659, A219661, A219666.

nn = 72; m = 1; While[Factorial@ m < nn, m++]; m; Table[Length@ NestWhileList[# - Total@ IntegerDigits[#, MixedRadix[Reverse@ Range[2, m]]] &, n, # > 0 &] - 1, {n, 0, nn}] (* Michael De Vlieger, Jun 27 2016, Version 10.2 *)

a(0)=0; for n>0, a(n) = 1 + a(A219651(n)).

Erroneous description corrected by Antti Karttunen, Dec 03 2012

A219653 Least inverse of A219652; a(n) = minimal i such that A219652(i) = n.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 12, 16, 20, 24, 26, 30, 34, 38, 42, 48, 52, 56, 60, 66, 72, 78, 84, 90, 96, 102, 108, 116, 120, 122, 126, 130, 134, 138, 144, 148, 152, 156, 162, 168, 174, 180, 186, 192, 198, 204, 212, 218, 226, 234, 240, 244, 248, 252, 258, 264, 270, 276

Offset: 0

Antti Karttunen, Nov 25 2012

nonn

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

Cf. A219655 for the greatest inverse. A219654 gives the first differences.

This sequence is based on Factorial number system: A007623. Analogous sequence for binary system: A213708 and for Zeckendorf expansion: A219643. Cf. A219652, A219659, A219666.

A219654 Run lengths in A219652.

Original entry on oeis.org

1, 1, 2, 2, 2, 4, 4, 4, 4, 2, 4, 4, 4, 4, 6, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 8, 4, 2, 4, 4, 4, 4, 6, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 8, 6, 8, 8, 6, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 8, 6, 8, 8, 6, 8, 10, 6, 6, 6, 6, 6, 6, 6, 8, 6, 8, 8, 6, 8, 10, 8, 10, 12, 6

Offset: 0

Antti Karttunen, Nov 25 2012

nonn

a(n) tells from how many starting values one can end to 0 in n steps, with the iterative process described in A219652 (if going around in 0->0 loop is disallowed).

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

a(n) = 1+(A219655(n)-A219653(n)). This sequence is based on Factorial number system: A007623. Analogous sequence for binary system: A086876, for Zeckendorf expansion: A219644. Cf. A219652, A219659, A219666.

a(n) = A219653(n+1)-A219653(n). (The first differences of A219653).

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]

A219649 Irregular table, where row n (n >= 0) starts with n, the next term is A219641(n), and the successive terms are obtained by repeatedly subtracting the number of 1's in the previous term's Zeckendorf expansion, until zero is reached, after which the next row starts with one larger n.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 2, 1, 0, 5, 4, 2, 1, 0, 6, 4, 2, 1, 0, 7, 5, 4, 2, 1, 0, 8, 7, 5, 4, 2, 1, 0, 9, 7, 5, 4, 2, 1, 0, 10, 8, 7, 5, 4, 2, 1, 0, 11, 9, 7, 5, 4, 2, 1, 0, 12, 9, 7, 5, 4, 2, 1, 0, 13, 12, 9, 7, 5, 4, 2, 1, 0, 14, 12, 9, 7, 5, 4, 2, 1, 0, 15

Offset: 0

Antti Karttunen, Nov 24 2012

Rows converge towards A219648 (reversed).

See A014417 for the Fibonacci number system representation, also known as Zeckendorf expansion.

A. Karttunen, Rows 0..233, flattened

Cf. A007895, A014417, A219641, A219647. Analogous sequence for binary system: A218254, for factorial number system: A219659.

cp's OEIS Frontend

A219657 Positions of zeros in A219659.

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

A219666 The infinite trunk of factorial expansion beanstalk. The only infinite sequence such that a(n-1) = a(n) - sum of digits in factorial expansion of a(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Scheme

Formula

A219651 a(n) = n minus (sum of digits in factorial base expansion of n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Scheme

Formula

A219652 Number of steps to reach 0 starting with n and using the iterated process: x -> x - (sum of digits in factorial expansion of x).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A219653 Least inverse of A219652; a(n) = minimal i such that A219652(i) = n.

Views

Author

Keywords

Links

Crossrefs

A219654 Run lengths in A219652.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A219649 Irregular table, where row n (n >= 0) starts with n, the next term is A219641(n), and the successive terms are obtained by repeatedly subtracting the number of 1's in the previous term's Zeckendorf expansion, until zero is reached, after which the next row starts with one larger n.

Views

Author

Keywords

Comments

Links

Crossrefs