A084558 -id:A084558 - cp's OEIS frontend

A275725 a(n) = A275723(A002110(1+A084558(n)), n); prime factorization encodings of cycle-polynomials computed for finite permutations listed in the order that is used in tables A060117 / A060118.

2, 4, 18, 8, 12, 8, 150, 100, 54, 16, 24, 16, 90, 40, 54, 16, 36, 16, 60, 40, 36, 16, 24, 16, 1470, 980, 882, 392, 588, 392, 750, 500, 162, 32, 48, 32, 270, 80, 162, 32, 108, 32, 120, 80, 72, 32, 48, 32, 1050, 700, 378, 112, 168, 112, 750, 500, 162, 32, 48, 32, 450, 200, 162, 32, 72, 32, 300, 200, 108, 32, 48, 32, 630, 280, 378, 112, 252, 112, 450, 200

Offset: 0

Antti Karttunen, Aug 09 2016

nonn

In this context "cycle-polynomials" are single-variable polynomials where the coefficients (encoded with the exponents of prime factorization of n) are equal to the lengths of cycles in the permutation listed with index n in tables A060117 or A060118. See the examples.

			Consider the first eight permutations (indices 0-7) listed in A060117:
  1 [Only the first 1-cycle explicitly listed thus a(0) = 2^1 = 2]
  2,1 [One transposition (2-cycle) in beginning, thus a(1) = 2^2 = 4]
  1,3,2 [One fixed element in beginning, then transposition, thus a(2) = 2^1 * 3^2 = 18]
  3,1,2 [One 3-cycle, thus a(3) = 2^3 = 8]
  3,2,1 [One transposition jumping over a fixed element, a(4) = 2^2 * 3^1 = 12]
  2,3,1 [One 3-cycle, thus a(5) = 2^3 = 8]
  1,2,4,3 [Two 1-cycles, then a 2-cycle, thus a(6) = 2^1 * 3^1 * 5^2 = 150].
  2,1,4,3 [Two 2-cycles, not crossed, thus a(7) = 2^2 * 5^2 = 100]
and also the seventeenth one at n=16 [A007623(16)=220] where we have:
  3,4,1,2 [Two 2-cycles crossed, thus a(16) = 2^2 * 3^2 = 36].

Antti Karttunen, Table of n, a(n) for n = 0..40319
Indranil Ghosh, Python program for computing this sequence
Index entries for sequences related to factorial base representation

Cf. A000040, A001222, A001221, A002110, A007814, A046660, A048675, A051903, A056169, A056170, A084558, A243054, A257510, A275723, A275803, A275832.
Cf. A275807 (terms divided by 2).
Cf. also A060129, A060128, A060130, A060131, A072411, A275726, A275812, A275851.
Cf. also A275733, A275734, A275735 for other such prime factorization encodings of A060117/A060118-related polynomials.

(define (A275725 n) (A275723bi (A002110 (+ 1 (A084558 n))) n)) ;; Code for A275723bi given in A275723.

a(n) = A275723(A002110(1+A084558(n)), n).
Other identities:
A001221(a(n)) = 1+A257510(n) (for all n >= 1).
A001222(a(n)) = 1+A084558(n).
A007814(a(n)) = A275832(n).
A048675(a(n)) = A275726(n).
A051903(a(n)) = A275803(n).
A056169(a(n)) = A275851(n).
A046660(a(n)) = A060130(n).
A072411(a(n)) = A060131(n).
A056170(a(n)) = A060128(n).
A275812(a(n)) = A060129(n).
a(n!) = 2 * A243054(n) = A000040(n)*A002110(n) for all n >= 1.

A275851 a(n) = number of elements in range [1..(1+A084558(n))] fixed by the permutation with rank n of permutation list A060117 (or A060118).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 3, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 3, 1, 2, 1, 2, 1, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 3, 2, 2, 1, 1, 1, 1, 0, 1, 0, 0, 0, 2, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 3, 2, 1, 1, 2, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 4

Offset: 0

Antti Karttunen, Aug 11 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to factorial base representation

Cf. A060117, A060118.
Cf. A056169, A060129, A084558, A275725.
Cf. A275852 (indices of zeros).

a(n) = A056169(A275725(n)).

a(n) = 1 + A084558(n) - A060129(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)).

A220655 For n with a unique factorial base representation n = duu! + ... + d22! + d11! (each di in range 0..i, cf. A007623), a(n) = (du+1)u! + ... + (d2+1)2! + (d1+1)1!; a(n) = n + A007489(A084558(n)).

Original entry on oeis.org

2, 5, 6, 7, 8, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

Offset: 1

Antti Karttunen, Dec 17 2012

nonn

Used for computing A107346.

Term a(n) can be obtained by adding one to each digit of factorial base representation of n (A007623(n)) and then reinterpreting it as a kind of pseudo-factorial base representation, ignoring the fact that now some of the digits might be over the maximum allowed in that position. Please see the example section. - Antti Karttunen, Nov 29 2013

			1 has a factorial base representation A007623(1) = '1', as 1 = 1*1!. Incrementing the digit 1 with 1, we get 2*1! = 2, thus a(1) = 2. (Note that although '2' is not a valid factorial base representation, it doesn't matter here.)
2 has a factorial base representation '10', as 2 = 1*2! + 0*1!. Incrementing the digits by one, we get 2*2! + 1*1! = 5, thus a(2) = 5.
3 has a factorial base representation '11', as 3 = 1*2! + 1*1!. Incrementing the digits by one, we get 2*2! + 2*1! = 6, thus a(3) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..5039

Complement: A220695.
One less than A220656.
Cf. A000142, A003422, A007489, A007623, A084558, A212598, A153880, A231720.

Block[{nn = 66, m = 1}, While[Factorial@ m < nn, m++]; m = MixedRadix[Reverse@ Range[2, m]]; Array[FromDigits[1 + IntegerDigits[#, m], m] &, nn]] (* Michael De Vlieger, Jan 20 2020 *)

;; Standalone iterative implementation (Nov 29 2013):
(define (A220655 n) (let loop ((n n) (z 0) (i 2) (f 1)) (cond ((zero? n) z) (else (loop (quotient n i) (+ (* f (+ 1 (remainder n i))) z) (+ 1 i) (* f i))))))
;; Alternative implementation:
(define (A220655 n) (+ n (A007489 (A084558 n))))

a(n) = A220656(n)-1 = A003422(A084558(n)+1) + A000142(A084558(n)) + A212598(n) - 1. [The original definition]
a(n) = n + A007489(A084558(n)). [The above formula reduces to this, which proves that the original Dec 17 2012 description and the new main description produce the same sequence. Essentially, we are adding to n a factorial base repunit '1...111' with as many fact.base digits as n has.] - Antti Karttunen, Nov 29 2013
For n >= 1, A231720(n) = a(A153880(n)).

Name changed by Antti Karttunen, Nov 29 2013

A220658 Irregular table, where the n-th row consists of A084558(n)+1 copies of n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Dec 18 2012

Equally, for n>=1, each i in range [n!,(n+1)!-1] occurs n+1 times.

Used for computing A220659, A055089 and A060118: The n-th term a(n) tells which permutation (counted from the start, zero-based) of A055089 or A060117/A060118 the n-th term in those sequence belongs to.

			Rows of this irregular table begin as:
0;
1, 1;
2, 2, 2;
3, 3, 3;
4, 4, 4;
5, 5, 5;
6, 6, 6, 6;
The terms A055089(3), A055089(4) and A055089(5) are 1,3,2. As a(3), a(4) and a(5) are all 2, we see that "132" is the second permutation in A055089-list, after the identity permutation "1", which has the index zero.

A. Karttunen, Rows 0..720 of the irregular table, flattened.

Cf. A220657, A220659.

A220659 Irregular table: row n (n >= 1) consists of numbers 0..A084558(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3

Offset: 0

Antti Karttunen, Dec 18 2012

The term a(n) gives the position (zero-based, starting from the left hand end of permutation) of a corresponding permutation A055089 and A060117/A060118 from which the term A055089(n) or A060118(n) is to be picked.

			Rows of this irregular table begin as:
0;
0, 1;
0, 1, 2;
0, 1, 2;
0, 1, 2;
0, 1, 2;
0, 1, 2, 3;
The term 2 occurs four times in A084558, in positions 2, 3, 4 and 5. Thus rows 2, 3, 4 and 5 (zero-based) of this irregular table are all 0,1,2.

Antti Karttunen, Rows 0..720 of the irregular table, flattened.

(define (A220659 n) (- n (A220657 (A220658 n))))

a(n) = n - A220657(A220658(n)).

A275849 Number of unoccupied slopes in factorial base representation of n: a(n) = A084558(n) - A060502(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 3, 2, 2, 1, 2, 2, 2, 1, 1, 0, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 3, 2, 2, 1, 2, 2, 3, 2, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 3, 2, 3, 2, 2, 2, 2, 1, 2, 1, 1, 1, 3, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 3, 3, 4

Offset: 0

Antti Karttunen, Aug 15 2016

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to factorial base representation

Cf. A060501, A060502, A084558.
Cf. A007489 (the indices of zeros).
Cf. also A225901, A275850, A275851, A275853.

(define (A275849 n) (- (A084558 n) (A060502 n)))

a(n) = A084558(n) - A060502(n).
Other identities. For all n >= 0:
a(n) = A275850(A225901(n)).
a(n) = A060501(n)-1. [To be proved.]

A220657 Partial sums of A084558+1.

Original entry on oeis.org

0, 1, 3, 6, 9, 12, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, 63, 67, 71, 75, 79, 83, 87, 92, 97, 102, 107, 112, 117, 122, 127, 132, 137, 142, 147, 152, 157, 162, 167, 172, 177, 182, 187, 192, 197, 202, 207, 212, 217, 222, 227, 232, 237, 242, 247, 252, 257

Offset: 0

Antti Karttunen, Dec 18 2012

nonn

Auxiliary function for computing A220658 and A220659.

A230411 a(n) = minimal k for which A219665(k) >= n; a(n) = one more than the factorial base width (A084558) of the (n-1)th term in the infinite trunk of factorial beanstalk (A219666).

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 1

Antti Karttunen, Oct 22 2013

nonn

a(1)=1, after which each term n occurs A219661(n-1) times.

Auxiliary function for computing A219666, A230431 and A230432.

Antti Karttunen, Table of n, a(n) for n = 1..5000

Cf. A219661, A219665, A219666, A230431, A230432.

Analogous sequence for binary system: A213711.

a(n) = 1 + A084558(A219666(n-1)) = 1 + A084558(A230416(n-1)). [Each a(n) is one more than the number of digits needed in factorial base to write the (n-1)-th term in the infinite trunk of factorial beanstalk]

A230428 Triangle T(n,k) giving the smallest term in "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, 7, 12, 23, 25, 48, 74, 97, 121, 240, 362, 481, 605, 721, 1440, 2162, 2881, 3605, 4326, 5041, 10080, 15122, 20161, 25205, 30246, 35288, 40321, 80640, 120962, 161281, 201605, 241926, 282248, 322568, 362881, 725760, 1088642, 1451521, 1814405, 2177286, 2540168, 2903048, 3265923

Offset: 1

Antti Karttunen, Oct 18 2013

			The first rows of this triangular table are:
1;
2, 5;
7, 12, 23;
25, 48, 74, 97;
121, 240, 362, 481, 605;
...
T(3,1) = 7 as 7 has factorial base representation 101, which is the smallest 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) = 12 as 12 has factorial base representation 200, which is the smallest 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 smallest such three digit term in A219666 beginning with factorial base digit 3.

Subset of A219666. Corresponding largest terms: A230429. Cf. also A230420.

(define (A230428 n) (if (< n 3) n (let ((k (A002260 n))) (let loop ((i (A230429 n)) (prev_i 0)) (cond ((not (= (A099563 i) k)) prev_i) (else (loop (A219651 i) i)))))))

cp's OEIS Frontend

A275725 a(n) = A275723(A002110(1+A084558(n)), n); prime factorization encodings of cycle-polynomials computed for finite permutations listed in the order that is used in tables A060117 / A060118.

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A275851 a(n) = number of elements in range [1..(1+A084558(n))] fixed by the permutation with rank n of permutation list A060117 (or A060118).

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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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A220655 For n with a unique factorial base representation n = du*u! + ... + d2*2! + d1*1! (each di in range 0..i, cf. A007623), a(n) = (du+1)*u! + ... + (d2+1)*2! + (d1+1)*1!; a(n) = n + A007489(A084558(n)).

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A220658 Irregular table, where the n-th row consists of A084558(n)+1 copies of n.

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A220659 Irregular table: row n (n >= 1) consists of numbers 0..A084558(n).

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A275849 Number of unoccupied slopes in factorial base representation of n: a(n) = A084558(n) - A060502(n).

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A220657 Partial sums of A084558+1.

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A230411 a(n) = minimal k for which A219665(k) >= n; a(n) = one more than the factorial base width (A084558) of the (n-1)th term in the infinite trunk of factorial beanstalk (A219666).

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A220655 For n with a unique factorial base representation n = duu! + ... + d22! + d11! (each di in range 0..i, cf. A007623), a(n) = (du+1)u! + ... + (d2+1)2! + (d1+1)1!; a(n) = n + A007489(A084558(n)).