A230406 -id:A230406 - 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)).

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).

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).

A230418 a(n) = the least k such that after having climbed k steps up from the root of the infinite trunk of the factorial beanstalk we have reached integer which is at least n.

Original entry on oeis.org

0, 1, 2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 10, 10, 10, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 19, 20, 20

Offset: 0

Antti Karttunen, Nov 09 2013

nonn

a(n) = the least k such that A219666(k) >= n.
After zero, each n occurs A230406(n) times.
The characteristic function of A219666 is given by Χ_A219666(n) = a(n+1)-a(n).

Antti Karttunen, Table of n, a(n) for n = 0..5039

This sequence relates to the factorial base representation (A007623) in the same way as A213718 relates to the binary system.

A230422 Positions of ones in A230410.

Original entry on oeis.org

1, 8, 14, 16, 18, 22, 33, 35, 37, 41, 45, 51, 53, 57, 61, 71, 75, 82, 87, 96, 106, 116, 118, 120, 124, 128, 134, 136, 140, 144, 154, 158, 165, 170, 179, 189, 198, 200, 206, 208, 212, 216, 226, 230, 237, 242, 251, 261, 270, 272, 280, 289, 293, 300, 305, 314, 324

Offset: 1

Antti Karttunen, Nov 10 2013

nonn

This sequence gives all n at which positions the successive terms A219666(n-1) & A219666(n) in the infinite trunk of the factorial beanstalk differ only in one digit position in their factorial base representations (A007623).

Please see further comments and examples in A230410.

			14 is included, because A219666(13) = 40 = '1220' in factorial base representation, while A219666(14) = 46 = '1320' in factorial base, and they differ only by their third least significant digit.
16 is included, because A219666(15) = 48 = '2000' in factorial base representation, while A219666(16) = 52 = '2020' in factorial base, and they differ only by their second least significant digit.

Antti Karttunen, Table of n, a(n) for n = 1..17517

Subset: A231718. Cf. also A230410 and A258010 (first differences).

nn = 10^4; m = 1; While[m! < Floor[6 nn/5], m++]; m; f[n_] := IntegerDigits[n, MixedRadix[Reverse@ Range[2, m]]]; Position[#, 1] &[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@ # &, Floor[6 nn/5], # > 0 &], # <= nn &]] // Flatten (* Michael De Vlieger, Jun 27 2016, Version 10.2 *)

For all n, A230406(a(n)) is one of the terms of A051683.

A231718 Positions of ones in A231717.

Original entry on oeis.org

1, 8, 14, 18, 22, 33, 37, 41, 45, 53, 57, 61, 71, 75, 87, 116, 120, 124, 128, 136, 140, 144, 154, 158, 170, 208, 212, 216, 226, 230, 242, 289, 293, 305, 362, 544, 548, 552, 556, 564, 568, 572, 582, 586, 598, 636, 640, 644, 654, 658, 670, 717, 721, 733, 790, 1021

Offset: 1

Antti Karttunen, Nov 12 2013

nonn

This sequence gives all n at which positions the successive terms A219666(n-1) & A219666(n) in the infinite trunk of the factorial beanstalk differ only in one digit position in their factorial base representations (A007623) and the difference of those digits is exactly one.

			14 is included, because A219666(13) = 40 = '1220' in factorial base representation, while A219666(14) = 46 = '1320' in factorial base, and they differ only by the third least significant digits, and 3-2 = 1.

Antti Karttunen, Table of n, a(n) for n = 1..11905

Subset of A230422.

For all n, A230406(a(n)) = A000142(A231719(a(n))).

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)).

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

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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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A230418 a(n) = the least k such that after having climbed k steps up from the root of the infinite trunk of the factorial beanstalk we have reached integer which is at least n.

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A230422 Positions of ones in A230410.

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A231718 Positions of ones in A231717.

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

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