cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A276005 Numbers with hit-free factorial base representations; positions of zeros in A276004 & A276007.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 12, 14, 16, 18, 19, 20, 22, 23, 24, 25, 26, 28, 29, 48, 49, 54, 55, 60, 66, 67, 72, 74, 76, 78, 84, 86, 88, 90, 92, 94, 96, 97, 98, 100, 101, 102, 103, 108, 110, 112, 114, 115, 116, 118, 119, 120, 121, 122, 124, 125, 126, 127, 132, 134, 136, 138, 139, 140, 142, 143, 240, 241, 242, 244, 245, 264, 265, 266, 268, 269, 288, 289, 312, 314, 316
Offset: 0

Views

Author

Antti Karttunen, Aug 17 2016

Keywords

Comments

We say there is a "hit" in factorial base representation (A007623) of n when there is any such pair of nonzero digits d_i and d_j in positions i > j so that (i - d_i) = j. Here the rightmost (least significant digit) occurs at position 1. This sequence gives all "hit-free" numbers, meaning that for every nonzero digit d_i (in position i) in their factorial base representation the digit at the position (i - d_i) is 0.
Also numbers n for which A060502(n) = A060128(n), in other words, the numbers n for which the number of slopes in their factorial base representation (A007623) is equal to the number of non-singleton cycles of the permutation listed as n-th permutation in the list A060117 (or A060118).
This can be viewed as a factorial base analog of base-2 related A003714.

Examples

			n=14 (factorial base "210") is included because 2 occurs in position 3 and 1 occurs in position 2, thus as (3-2) = 1 <> 2, 2 does not "hit" digit 1.
n=15 ("211") is NOT included because 2 occurring in position 3 hits the rightmost 1 in position 1 (as 3-2 = 1), and moreover, also the middle 1 hits the rightmost 1 as 2-1 = 1.

Links

Crossrefs

Complement: A276006.
Cf. A000110, A000142, A060128, A060502, A276004, A276007.
Cf. A060112 (a subsequence).
Intersection with A275804 gives A261220.
Cf. also A003714, A060117 and A060118.

Formula

Other identities. For all n >= 1:
a(A000110(n)) = n! = A000142(n). [To be proved.]

A276006 Numbers with at least one hit in their factorial base representations (see comments for the definition); positions of nonzeros in A276004 & A276007.

Original entry on oeis.org

3, 8, 9, 10, 11, 13, 15, 17, 21, 27, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 56, 57, 58, 59, 61, 62, 63, 64, 65, 68, 69, 70, 71, 73, 75, 77, 79, 80, 81, 82, 83, 85, 87, 89, 91, 93, 95, 99, 104, 105, 106, 107, 109, 111, 113, 117, 123, 128, 129, 130, 131, 133, 135, 137, 141, 144, 145, 146, 147, 148
Offset: 1

Views

Author

Antti Karttunen, Aug 17 2016

Keywords

Comments

We say there is a "hit" in factorial base representation (A007623) of n when there is any such pair of nonzero digits d_i and d_j in positions i > j so that (i - d_i) = j. Here the rightmost (least significant digit) occurs at position 1. This sequence gives all numbers that contain at least one such hit, meaning that there exists such a nonzero digit d_i in some position i of their factorial base representation that the digit at the position (i - d_i) is nonzero.

Examples

			3 ("11" in factorial base) is included because the most significant 1 at position 2 hits the least significant 1 at position 1 as (2-1) = 1.
17 ("221") is included because the most significant 2 at position 3 hits the 1 at position 1 as (3-2) = 1.

Links

Crossrefs

Complement: A276005.
Indices of nonzeros in A276004 and A276007.

A060502 a(n) = number of occupied digit slopes in the factorial base representation of n (see comments for the definition); number of drops in the n-th permutation of list A060117.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 2, 3, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 3, 2, 2, 2, 3, 3, 4, 3, 3, 2, 3, 2, 3, 3, 3, 2, 2, 3, 3, 2, 2, 1, 2, 2, 3, 2, 2, 1, 2, 2, 3, 2, 2, 2, 3, 2, 3, 3, 3, 2, 2, 3, 3, 2, 2, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1
Offset: 0

Views

Author

Antti Karttunen, Mar 22 2001

Keywords

Comments

From Antti Karttunen, Aug 11-24 2016: (Start)
a(n) gives the number of occupied "digit slopes" in the factorial base representation of n, or more formally, the number of distinct elements in a multiset [(i_x - d_x) | where d_x ranges over each nonzero digit present in factorial base representation of n and i_x is that digit's position from the right]. Here one-based indexing is used, thus the least significant digit is in position 1. Each value {digit's position} - {digit's value} determines on which slope that particular nonzero digit is. The nonzero digits for which (position - digit) = 0, are said to be on the "maximal slope" (see A260736), those with value 1 on "sub-maximal", etc.
The number of occupied digit slopes translates directly to the number of drops in the n-th permutation as given in the list A060117 because only the largest (and thus leftmost) of all nonzero digits on any particular slope adds a (single) drop to the permutation, when constructed by the unranking algorithm employed in A060117.
The original definition of this sequence is (essentially):
a(n) = the average of digits (where "digits" may eventually obtain also any values > 9) in each siteswap pattern A060498(n) constructed from each permutation in list A060117, which is equal to number of balls used in that pattern.
The equivalence of the old and the new definitions is seen from the following (as kindly pointed by Olivier Gérard in personal mail): For any permutation p of [1..n], Sum(i=1..n) p(i)-i = 0 (whether taken modulo n or not), thus Sum(i=1..n) (p(i)-i modulo n) = Sum(i={set of nondrops}) (p(i)-i) + Sum(i={set of drops}) (n + (p(i)-i)) = 0 + n * #{set of drops}, where drops is the set of those i where p[i] < i and nondrops are those i for which p[i] >= 1.
Involution A225901 maps this metric to another metric A275806 which gives the number of distinct nonzero digits in factorial base representation of n. See also A275811.
A007489 (repunits in this context) gives the positions where a(n) = A084558(n) (the length of factorial base representation of n). These are also the positions of records.
(End)

Examples

			For n=23 ("321" in factorial base representation, A007623), all the digits are maximal for their positions (they occur on the "maximal slope"), thus there is only one distinct digit slope present and a(23)=1. Also, for the 23rd permutation in the ordering A060117, [2341], there is just one drop, as p[4] = 1 < 4.
For n=29 ("1021"), there are three nonzero digits, where both 2 and the rightmost 1 are on the maximal slope, while the most significant 1 is on the "sub-sub-sub-maximal", thus there are two occupied slopes in total, and a(29) = 2. In the 29th permutation of A060117, [23154], there are two drops as p[3] = 1 < 3 and p[5] = 4 < 5.
For n=37 ("1201"), there are three nonzero digits, where the rightmost 1 is on the maximal slope, 2 is on the submaximal, and the most significant 1 is on the "sub-sub-sub-maximal", thus there are three occupied slopes in total, and a(37) = 3. In the 37th permutation of A060117, [51324], there are three drops at indices 2, 4 and 5.

Links

Crossrefs

Cf. A000120, A001221, A007489, A007623, A033312, A060117, A060118, A060130, A060498, A225901, A275734, A275806.
Cf. A007489 (positions of records, the first occurrence of each n).
Cf. A276001, A276002, A276003 (positions where a(n) obtains values 1, 2, 3).
Cf. also A060500, A060501, A060125, A084558, A153880, A255411, A260736, A273670, A275804, A275811, A275946, A275947, A275849, A275956, A276004,
A276005, A276010.

Programs

  • Maple
    # The following program follows the original 2001 interpretation of this sequence:
A060502 := n -> avg(Perm2SiteSwap3(PermUnrank3R(n)));
with(group);
permul := (a, b) -> mulperms(b, a);
# factorial_base(n) gives the digits of A007623(n) as a list, uncorrupted even when there are digits > 9:
factorial_base := proc(nn) local n, a, d, j, f; n := nn; if(0 = n) then RETURN([0]); fi; a := []; f := 1; j := 2; while(n > 0) do d := floor(`mod`(n, (j*f))/f); a := [d, op(a)]; n := n - (d*f); f := j*f; j := j+1; od; RETURN(a); end;
# PermUnrank3R(r) gives the permutation with rank r in list A060117:
PermUnrank3R := proc(r) local n; n := nops(factorial_base(r)); convert(PermUnrank3Raux(n+1, r, []), 'permlist', 1+(((r+2) mod (r+1))*n)); end;
PermUnrank3Raux := proc(n, r, p) local s; if(0 = r) then RETURN(p); else s := floor(r/((n-1)!)); RETURN(PermUnrank3Raux(n-1, r-(s*((n-1)!)), permul(p, [[n, n-s]]))); fi; end;
Perm2SiteSwap3 := proc(p) local ip,n,i,a; n := nops(p); ip := convert(invperm(convert(p,'disjcyc')),'permlist',n); a := []; for i from 1 to n do if(0 = ((ip[i]-i) mod n)) then a := [op(a),0]; else a := [op(a), n-((ip[i]-i) mod n)]; fi; od; RETURN(a); end;
avg := a -> (convert(a, `+`)/nops(a));

Formula

From Antti Karttunen, Aug 11-21 2016: (Start)
The following formula reflects the original definition of computing the average, with a few unnecessary steps eliminated:
a(n) = 1/s * Sum_{i=1..s} ((p[i]-i) modulo s), where p is the permutation of rank n as ordered in the list A060117, and s is its size (the number of its elements) computed as s = 1+A084558(n).
a(n) = Sum_{i=1..s} [p[i]
a(n) = 1/s * Sum_{i=1..s} ((i-p[i]) modulo s). [If inverse permutations from list A060118 are used, then we just flip the order of difference that is used in the first formula].
Following formulas do not need intermediate construction of permutation lists:
a(n) = A001221(A275734(n)).
a(n) = A275806(A225901(n)).
a(n) = A000120(A276010(n)).
Other identities and observations. For all n >= 0:
a(n) = A275946(n) + A275947(n).
a(n) = A060500(A060125(n)).
a(n) = A060128(n) + A276004(n).
a(n) = A060129(n) - A060500(n).
a(n) = A084558(n) - A275849(n) = 1 + A084558(n) - A060501(n).
a(A007489(n)) = n. [Particularly, A007489(n) gives the position of the first occurrence of each n.]
A060128(n) <= a(n) <= A060129(n).
a(n!) = 1.
a(A033312(n)) = 1 for all n > 1.
a(A059590(n)) = A000120(n).
a(A060112(n)) = A007895(n).
a(n) = a(A153880(n)) = a(A255411(n)). [The shift-operations do not change the number of distinct slopes.]
a(A275804(n)) = A060130(A275804(n)). [A275804 gives all the positions where this coincides with A060130.]
(End)

Extensions

Entry revised, with a new interpretation and formulas. Maple-code cleaned up. - Antti Karttunen, Aug 11 2016
Another new interpretation added and the original definition moved to the comments - Antti Karttunen, Aug 24 2016

A060128 a(n) is the number of disjoint cycles (excluding 1-cycles, i.e., fixed elements) in the n-th permutation of A060117 and A060118.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2
Offset: 0

Views

Author

Antti Karttunen, Mar 05 2001

Keywords

Links

Crossrefs

Cf. A055090, A056170, A060118, A060120, A060129, A060130, A275725, A278225, A276004.
Cf. A276005 (positions where coincides with A060502).

Programs

  • Maple
    A060128(n) = nops(convert(PermUnrank3L(n), 'disjcyc')); # Code for function PermUnrank3L given in A060118.

Formula

a(n) = A060129(n) - A060130(n).
From Antti Karttunen, Aug 07 2017: (Start)
a(n) = A056170(A275725(n)).
a(n) = A055090(A060120(n)).
a(n) = A060502(n) - A276004(n).
(End)

A276007 a(n) = number of nonzero digits in factorial base representation of n that hit less significant nonzero digits to the right. See comments for exact definition.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 2, 3, 2, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 0, 0, 1, 2, 1, 1, 0, 0, 2, 3, 2, 2, 0, 1, 1, 3, 1, 2, 0, 0, 1, 2, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 3, 1, 2, 0, 2, 0, 3, 0, 2, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0
Offset: 0

Views

Author

Antti Karttunen, Aug 17 2016

Keywords

Comments

a(n) = Number of times a nonzero digit d_i appears in such position i of factorial base representation of n for which there is another nonzero digit in position i - d_i. Here one-based indexing is used for digits, thus the least significant digit is in position 1.

Examples

			For n=15 ("211" in factorial base) both 2 at position 3 and 1 at position 2 hit the least significant 1 at position 1 as (2-1) = (3-2) = 1, the position where the least significant 1 itself is. These both cases are included in the count, because this sequence counts the total number of hitting digits, thus a(15)=2.

Links

Crossrefs

Cf. A276005 (indices of zeros), A276006 (of nonzeros).
Differs from A276004 for the first time at n=15, where a(15)=2, while A276004(15)=1.

Programs

  • Scheme
    (define (A276007 n) (let ((fv (list->vector (cons 0 (reverse (n->factbase n)))))) (let loop ((i 1) (c 0)) (if (>= i (vector-length fv)) c (let ((d (vector-ref fv i))) (if (zero? d) (loop (+ 1 i) c) (loop (+ 1 i) (+ c (if (not (zero? (vector-ref fv (- i d)))) 1 0)))))))))
(define (n->factbase n) (let loop ((n n) (fex (if (zero? n) (list 0) (list))) (i 2)) (cond ((zero? n) fex) (else (loop (floor->exact (/ n i)) (cons (modulo n i) fex) (+ 1 i))))))
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