A219666 -id:A219666 - cp's OEIS frontend

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.

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).

A219665 One more than the partial sums of A219661.

Original entry on oeis.org

1, 2, 4, 9, 28, 111, 539, 3150, 21623, 172349, 1549897, 15401145, 168011253, 2003304294, 25928878273, 361788001016, 5411160126368, 86353882249912, 1464841397585336, 26323224850512720, 499551889319197566

Offset: 1

Antti Karttunen, May 28 2013

nonn

Are there any cases after n>2, for which A219666(a(n)) = n! instead of n!+1 ? (At least for all terms a(3) - a(14) that number is n!+1.)

Compare to the conjecture given at A213710.

One more than A226061.
Cf. A219652, A219666, A226061.
Cf. also A213710 (analogous sequence for base-2).

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

Scheme
```
(define (A219665 n) (+ 1 (A226061 n)))
```

a(n) = A226061(n)+1 = A219652(n!).

Terms a(16) - a(21) computed from the new terms of A219661 by Antti Karttunen, Jun 27 2016

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).

A230409 Partial sums of A230407.

Original entry on oeis.org

0, -1, 0, 3, -2, -3, 0, 1, 4, -1, -2, 5, 4, -1, -2, -7, -2, 3, -2, -13, -12, -9, -20, -19, -22, -19, -18, -15, -20, -21, -14, -15, -20, -21, -26, -21, -16, -21, -32, -31, -28, -49, -48, -51, -54, -45, -44, -45, -50, -51, -56, -51, -46, -51, -62, -61, -58, -79

Offset: 0

Antti Karttunen, Nov 10 2013

sign

The term a(n) indicates approximately the "balance" of the factorial beanstalk (cf. A219666) at n steps up from the root, which in turn correlates with the behavior of such sequences as A219662 and A219663.
This sequence relates to the factorial base representation (A007623) in the same way as A218789 relates to the binary system.
Question: When will a negative term occur next time, after a(251) = -41 ?

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

Cf. A230407 & A230408, A219662 & A219663.

a(0) = 0, a(n) = a(n-1) + A230407(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.

A230432 Simple self-inverse permutation of natural numbers: after zero, list each block of A219661(n) numbers in reverse order, from A226061(n+1) to A219665(n).

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 6, 5, 4, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 110, 109, 108, 107, 106, 105, 104, 103, 102, 101, 100, 99, 98, 97, 96, 95, 94, 93, 92, 91, 90, 89, 88, 87, 86, 85, 84, 83, 82, 81, 80, 79, 78, 77, 76, 75, 74, 73

Offset: 0

Antti Karttunen, Oct 22 2013

nonn

This permutation can be used to map between the sequences A219666 and A230416. E.g. A219666(n) = A230416(a(n)) and vice versa: A230416(n) = A219666(a(n)).

Antti Karttunen, Table of n, a(n) for n = 0..3149
Index entries for sequences that are permutations of the natural numbers

Cf. A219661, A219665, A226061, A230411, A219666, A230416, A230431.

Analogous sequence for binary system: A218602.

(define (A230432 n) (if (zero? n) n (- (A219665 (A230411 (+ 1 n))) (A230431 n) 1)))

a(n) = A219665(A230411(n+1)) - A230431(n) - 1.

A230408 Partial sums of absolute values of A230407.

Original entry on oeis.org

0, 1, 2, 5, 10, 11, 14, 15, 18, 23, 24, 31, 32, 37, 38, 43, 48, 53, 58, 69, 70, 73, 84, 85, 88, 91, 92, 95, 100, 101, 108, 109, 114, 115, 120, 125, 130, 135, 146, 147, 150, 171, 172, 175, 178, 187, 188, 189, 194, 195, 200, 205, 210, 215, 226, 227, 230, 251, 252

Offset: 0

Antti Karttunen, Nov 10 2013

nonn

The term a(n) tells how many nodes there are in total in all side-trees ("tendrils") encountered (see A230430) after we have climbed n steps up along the infinite trunk of the factorial beanstalk, A219666.

This sequence relates to the factorial base representation (A007623) in the same way as A218785 relates to the binary system.

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

Cf. A230407, A230409.

a(0) = 0, a(n) = a(n-1) + |A230407(n)| [Where | | indicates the absolute value].

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))).

cp's OEIS Frontend

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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A219665 One more than the partial sums of A219661.

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A219654 Run lengths in A219652.

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A230409 Partial sums of A230407.

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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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A230432 Simple self-inverse permutation of natural numbers: after zero, list each block of A219661(n) numbers in reverse order, from A226061(n+1) to A219665(n).

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A230408 Partial sums of absolute values of A230407.

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A230422 Positions of ones in A230410.

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A231718 Positions of ones in A231717.

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