A230428 -id:A230428 - cp's OEIS frontend

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).

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)).

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)).

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)))))

cp's OEIS Frontend

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).

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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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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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