A219653 -id:A219653 - cp's OEIS frontend

A231723 a(n) = the difference between the n-th node of the infinite trunk of the factorial beanstalk (A219666(n)) and the smallest integer (A219653(n)) which is as many A219651-iteration steps distanced from the root (zero); a(n) = A219666(n) - A219653(n).

0, 0, 0, 1, 1, 2, 0, 1, 3, 1, 2, 0, 1, 2, 4, 0, 0, 1, 3, 4, 2, 1, 1, 2, 1, 0, 1, 3, 1, 2, 0, 1, 2, 4, 0, 0, 1, 3, 4, 2, 1, 1, 2, 1, 0, 0, 1, 3, 2, 4, 0, 0, 1, 3, 4, 2, 1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 1, 3, 4, 2, 1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 1, 3, 5, 7, 8, 1, 0

Offset: 0

Antti Karttunen, Nov 13 2013

nonn

For all n, the following holds: A219653(n) <= A219666(n) <= A219655(n). This sequence gives the distance of the node n in the infinite trunk of factorial beanstalk (A219666(n)) from the left (lesser) edge of the A219654(n) wide window which it at that point must pass through.

This sequence relates to the factorial base representation (A007623) in the same way as A218603 relates to the binary system and similar remarks apply here.

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

Cf. A231724, A230409, A219662 & A219663.

(define (A231723 n) (- (A219666 n) (A219653 n)))

a(n) = A219666(n) - A219653(n).

A219654(n) = a(n) + A231724(n) + 1.

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)).

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

A219655 Greatest inverse of A219652; a(n) = maximal i such that A219652(i) = n.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 15, 19, 23, 25, 29, 33, 37, 41, 47, 51, 55, 59, 65, 71, 77, 83, 89, 95, 101, 107, 115, 119, 121, 125, 129, 133, 137, 143, 147, 151, 155, 161, 167, 173, 179, 185, 191, 197, 203, 211, 217, 225, 233, 239, 243, 247, 251, 257, 263, 269, 275, 281

Offset: 0

Antti Karttunen, Nov 25 2012

nonn

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

Cf. A219653 for the least inverse. A219654 gives the first differences.

This sequence is based on factorial number system: A007623. Analogous sequence for binary system: A173601, for Zeckendorf expansion: A219645.

a(n) = A219653(n) + A219654(n) - 1.

A219643 Least inverse of A219642; a(n) = minimal i such that A219642(i) = n.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 8, 10, 13, 15, 18, 21, 23, 26, 29, 32, 34, 36, 39, 42, 45, 47, 50, 54, 55, 57, 60, 63, 66, 68, 71, 75, 78, 81, 84, 89, 91, 94, 97, 100, 102, 105, 109, 112, 115, 118, 123, 126, 130, 134, 138, 143, 144, 146, 149, 152, 155, 157, 160, 164, 167

Offset: 0

Antti Karttunen, Nov 24 2012

nonn

A. Karttunen, Table of n, a(n) for n = 0..10000

Cf. A219645 for the greatest inverse. A219644 gives the first differences.

This sequence is based on Fibonacci number system (Zeckendorf expansion): A014417. For analogous sequence in binary system, see: A213708, for factorial number system: A219653.

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).

A255053 Least inverse of A255072; a(n) = smallest k such that A255072(k) = n.

Original entry on oeis.org

0, 1, 3, 6, 7, 11, 14, 15, 19, 23, 27, 30, 31, 35, 39, 44, 47, 51, 55, 59, 62, 63, 67, 71, 76, 79, 86, 91, 95, 99, 103, 108, 111, 115, 119, 123, 126, 127, 131, 135, 140, 143, 150, 155, 159, 166, 172, 176, 182, 187, 191, 195, 199, 204, 207, 214, 219, 223, 227, 231, 236, 239, 243, 247, 251, 254, 255

Offset: 0

Antti Karttunen, Feb 14 2015

nonn

Also positions of records in A255072.

Antti Karttunen, Table of n, a(n) for n = 0..16143

Cf. A255072, A255054, A255055, A255056, A255123.

Analogous sequences: A213708, A219643, A219653.

Other identities. For all n >= 0:
a(0) = 0; for n > 0: a(n) = a(n-1) + A255054(n-1).
a(n) = A255056(n) - A255123(n).

A231724 a(n) = the difference between the n-th node of the infinite trunk of the factorial beanstalk (A219666(n)) and the greatest integer (A219655(n)) which is as many A219651-iteration steps distanced from the root (zero); a(n) = A219655(n) - A219666(n).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 3, 2, 0, 0, 1, 3, 2, 1, 1, 3, 3, 2, 2, 1, 3, 4, 4, 3, 4, 5, 6, 0, 0, 1, 3, 2, 1, 1, 3, 3, 2, 2, 1, 3, 4, 4, 3, 4, 5, 7, 4, 4, 5, 1, 3, 3, 2, 2, 1, 3, 4, 4, 3, 4, 5, 7, 5, 7, 7, 5, 6, 6, 1, 3, 4, 4, 3, 4, 5, 7, 5, 7, 7, 5, 6, 6, 2, 2, 3, 4, 5

Offset: 0

Antti Karttunen, Nov 13 2013

nonn

For all n, the following holds: A219653(n) <= A219666(n) <= A219655(n). This sequence gives the distance of the node n in the infinite trunk of factorial beanstalk (A219666(n)) from the right (greater) edge of the A219654(n) wide window which it at that point must pass through.

This sequence relates to the factorial base representation (A007623) in the same way as A218604 relates to the binary system and similar remarks apply here.

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

Cf. A231723, A230409, A219662 & A219663.

(define (A231724 n) (- (A219655 n) (A219666 n)))

a(n) = A219655(n) - A219666(n).

A219654(n) = a(n) + A231723(n) + 1.

cp's OEIS Frontend

A231723 a(n) = the difference between the n-th node of the infinite trunk of the factorial beanstalk (A219666(n)) and the smallest integer (A219653(n)) which is as many A219651-iteration steps distanced from the root (zero); a(n) = A219666(n) - A219653(n).

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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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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).

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A219655 Greatest inverse of A219652; a(n) = maximal i such that A219652(i) = n.

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A219643 Least inverse of A219642; a(n) = minimal i such that A219642(i) = n.

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A219654 Run lengths in A219652.

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A255053 Least inverse of A255072; a(n) = smallest k such that A255072(k) = n.

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A231724 a(n) = the difference between the n-th node of the infinite trunk of the factorial beanstalk (A219666(n)) and the greatest integer (A219655(n)) which is as many A219651-iteration steps distanced from the root (zero); a(n) = A219655(n) - A219666(n).

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