A219666 -id:A219666 - cp's OEIS frontend

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

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).

A328316 Iterates of A276086 starting from 0.

Original entry on oeis.org

0, 1, 2, 3, 6, 5, 18, 125, 43218, 258413198822535882125

Offset: 0

Antti Karttunen, Oct 14 2019

nonn

The unique infinite sequence such that a(0) = 0, a(n) = A276085(a(n+1)) for n >= 0, and A129251(a(n)) = 0 for n >= 1, i.e., all nonzero terms must be in A048103.

a(10) is 240 decimal digits long (can be found in b-file), and a(11) is too big to fit even into a b-file as it is 32700 decimal digits long, but it can be found in the given a-file.

Antti Karttunen, Table of n, a(n) for n = 0..10
Antti Karttunen, Terms a(0) .. a(11), without running index
Index entries for sequences related to primorial base

Cf. A002110, A048103, A129251, A276085, A276086, A328317 (the smallest prime not dividing a(n)), A328318, A328319 (digit sum in primorial base), A328322 (max. digit), A328323.
Cf. A153013, and also A109162, A179016, A219666, A259934 for more or less analogous sequences.
Cf. also A328313.

A276086[n0_] := Module[{m = 1, i = 1, n = n0, p}, While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m];
NestList[A276086, 0, 10] (* Jean-François Alcover, Dec 01 2021, after Antti Karttunen in A276086 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328316(n) = if(!n,0,A276086(A328316(n-1)));

a(0) = 0; and for n > 0, a(n) = A276086(a(n-1)).

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

A219648 The infinite trunk of Zeckendorf beanstalk. The only infinite sequence such that a(n-1) = a(n) - number of 1's in Zeckendorf representation of a(n).

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 9, 12, 14, 17, 20, 22, 24, 27, 29, 33, 35, 37, 40, 42, 45, 47, 50, 54, 56, 58, 61, 63, 67, 70, 74, 76, 79, 83, 88, 90, 92, 95, 97, 101, 104, 108, 110, 113, 117, 121, 123, 126, 130, 134, 138, 143, 145, 147, 150, 152, 156, 159, 163, 165, 168

Offset: 0

Antti Karttunen, Nov 24 2012

nonn

a(n) tells in what number we end in n steps, when we start climbing up the infinite trunk of the "Zeckendorf beanstalk" from its root (zero).
There are many finite sequences such as 0,1,2; 0,1,2,4,5; etc. (see A219649) and as the length increases, so (necessarily) does the similarity to this infinite sequence.
There can be only one infinite trunk in "Zeckendorf beanstalk" as all paths downwards from numbers >= A000045(i) must pass through A000045(i)-1 (i.e. A000071(i)). This provides also a well-defined method to compute the sequence, for example, via a partially reversed version A261076.
See A014417 for the Fibonacci number system representation, also known as Zeckendorf expansion.

Antti Karttunen, Table of n, a(n) for n = 0..11817

Cf. A000045, A000071, A007895, A014417, A219641, A219649, A261076, A261102. For all n, A219642(a(n)) = n and A219643(n) <= a(n) <= A219645(n). Cf. also A261083 & A261084.

Other similarly constructed sequences: A179016, A219666, A255056.

(define (A219648 n) (A261076 (A261102 n)))

a(n) = A261076(A261102(n)).

A226061 Partial sums of A219661.

Original entry on oeis.org

0, 1, 3, 8, 27, 110, 538, 3149, 21622, 172348, 1549896, 15401144, 168011252, 2003304293, 25928878272, 361788001015, 5411160126367, 86353882249911, 1464841397585335, 26323224850512719, 499551889319197565

Offset: 1

Antti Karttunen, May 28 2013

nonn

a(n) tells the position of (n!)-1 in A219666.

One less than A219665.
Cf. A219661, A219666.
Analogous sequence for binary system: A218600.
Cf. also A230410, A231719.

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

a(n) = a(n-1) + A219661(n-1) with a(1) = 0.
a(n) = A219652(n!-1).
a(n) = A219665(n) - 1.

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

A219658 Complement of A219650. Natural numbers that do not occur in A219651.

Original entry on oeis.org

3, 4, 8, 9, 13, 14, 18, 19, 20, 21, 22, 26, 27, 31, 32, 36, 37, 41, 42, 43, 44, 45, 49, 50, 54, 55, 59, 60, 64, 65, 66, 67, 68, 72, 73, 77, 78, 82, 83, 87, 88, 89, 90, 91, 95, 96, 100, 101, 105, 106, 110, 111, 112, 113, 114, 115, 116, 117, 118, 122, 123, 127

Offset: 1

Antti Karttunen, Nov 25 2012

nonn

These are positive integers i for which there does not exist any k such that A034968(i+k) = k.

A. Karttunen, Table of n, a(n) for n = 1..10000

A219650, A219651, A219666. Analogous sequence for binary system: A055938, for Zeckendorf expansion: A219638.

A276623 The infinite trunk of ternary beanstalk: The only infinite sequence such that a(n-1) = a(n) - A053735(a(n)), where A053735(n) = base-3 digit sum of n.

Original entry on oeis.org

0, 2, 4, 8, 10, 12, 16, 20, 26, 28, 30, 34, 38, 42, 46, 52, 56, 62, 68, 72, 80, 82, 84, 88, 92, 96, 100, 106, 110, 116, 122, 126, 134, 140, 144, 152, 160, 164, 170, 176, 180, 188, 194, 198, 204, 212, 216, 224, 232, 242, 244, 246, 250, 254, 258, 262, 268, 272, 278, 284, 288, 296, 302, 306, 314, 322, 326, 332, 338, 342, 350, 356, 360

Offset: 0

Antti Karttunen, Sep 11 2016

Antti Karttunen, Table of n, a(n) for n = 0..6250

Cf. A004128, A024023, A053735, A054861, A261231 (left inverse), A261233, A276622, A276624, A276603 (terms divided by 2), A276604 (first differences).
Cf. A179016, A219648, A219666, A255056, A259934, A276573, A276583, A276613 for similar constructions.
Cf. also A263273.

(define (A276623 n) (A276624 (A276622 n)))

a(n) = A276624(A276622(n)).
Other identities. For all n >= 0:
A261231(a(n)) = n.
a(A261233(n)) = A024023(n) = 3^n - 1.

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.

A219659 Irregular table where row n (n >= 0) starts with n, the next term is A219651(n), and the successive terms are obtained by repeatedly subtracting the sum of digits in the previous term's factorial 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, 1, 0, 4, 2, 1, 0, 5, 2, 1, 0, 6, 5, 2, 1, 0, 7, 5, 2, 1, 0, 8, 6, 5, 2, 1, 0, 9, 6, 5, 2, 1, 0, 10, 7, 5, 2, 1, 0, 11, 7, 5, 2, 1, 0, 12, 10, 7, 5, 2, 1, 0, 13, 10, 7, 5, 2, 1, 0, 14, 11, 7, 5, 2, 1, 0, 15, 11, 7, 5, 2, 1, 0, 16, 12, 10, 7, 5, 2, 1, 0

Offset: 0

Antti Karttunen, Nov 25 2012

Rows converge towards A219666 (reversed).

See A007623 for the Factorial number system representation.

A. Karttunen, Rows 0..120, flattened

Cf. A007623, A034968, A219651, A219657. Analogous sequence for binary system: A218254, for Zeckendorf expansion: A219649.

A219662 Number of times an even number is encountered, when going from (n+1)!-1 to n!-1 using the iterative process described in A219652.

Original entry on oeis.org

1, 1, 2, 10, 49, 268, 1505, 9667, 81891, 779193, 7726623, 80770479, 921442854, 11621384700, 159894957124

Offset: 1

Antti Karttunen, Dec 03 2012

At least for n=7, 8, 9 and 10, a(n) is equal to a(n+1) when taken modulo n.

			(1!)-1 (0) is reached from (2!)-1 (1) with one step by subtracting A034968(1) from 1. Zero is an even number, so a(1)=1.
(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 an even number, but one is not, 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 only 12 and 10 are even numbers, so a(3)=2.

Antti Karttunen, Table of n, a(n) for n = 1..15

Cf. A034968, A219666, A218542, A218543, A219652, A219661, A219663.

(definec (A219662 n) (if (< n 2) n (let loop ((i (- (A000142 (1+ n)) (A000217 n) 1)) (s 0)) (cond ((isA000142? (1+ i)) (+ s (- 1 (modulo i 2)))) (else (loop (A219651 i) (+ s (- 1 (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) - A219663(n).

cp's OEIS Frontend

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

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A328316 Iterates of A276086 starting from 0.

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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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A219648 The infinite trunk of Zeckendorf beanstalk. The only infinite sequence such that a(n-1) = a(n) - number of 1's in Zeckendorf representation of a(n).

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A226061 Partial sums of A219661.

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A219658 Complement of A219650. Natural numbers that do not occur in A219651.

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A276623 The infinite trunk of ternary beanstalk: The only infinite sequence such that a(n-1) = a(n) - A053735(a(n)), where A053735(n) = base-3 digit sum of n.

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

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A219659 Irregular table where row n (n >= 0) starts with n, the next term is A219651(n), and the successive terms are obtained by repeatedly subtracting the sum of digits in the previous term's factorial expansion, until zero is reached, after which the next row starts with one larger n.

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