A060502 -id:A060502 - cp's OEIS frontend

A060498 Each permutation in the list A060117 converted to Site Swap notation, with digits reversed and inverted. "Zero throws" (fixed elements) indicated with 0's.

0, 11, 120, 222, 201, 111, 1300, 1313, 2330, 3333, 3302, 2312, 2020, 3023, 1120, 1223, 2222, 3122, 3001, 2011, 3131, 2231, 1201, 1111, 14000, 14014, 14140, 14244, 14203, 14113, 24400, 24414, 34440, 44444, 44403, 34413, 34030, 44034, 24130

Offset: 0

Antti Karttunen, Mar 22 2001

nonn

This sequence is not well-defined for n >= 3628800 because the Site Swap notation can contain values exceeding 9, for example, the Site Swap notation for a(3628800) is [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 10]. - Sean A. Irvine, Nov 25 2022

Cf. A060495, A060496, A060499. Average of digits gives number of balls: A060502.

SiteSwap3ToDec := proc(s) local i,z,n; n := nops(s); z := 0; for i from n by -1 to 1 do z := 10*z; if(s[i] > 0) then z := z + (n-s[i]); fi; od; RETURN(z); end;

a(n) = SiteSwap3ToDec(Perm2SiteSwap1(PermUnrank3R(n))).

A275804 Numbers with at most one nonzero digit on each digit slope of the factorial base representation of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 16, 18, 20, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 36, 37, 40, 42, 44, 48, 49, 50, 51, 52, 60, 61, 64, 66, 68, 72, 73, 76, 78, 79, 82, 90, 96, 98, 102, 104, 108, 120, 121, 122, 123, 124, 126, 127, 128, 129, 130, 132, 133, 136, 138, 140, 144, 145, 146, 147, 148, 150, 151, 152, 153, 154, 156, 157, 160

Offset: 0

Antti Karttunen, Aug 10 2016

Indexing starts from zero, because a(0) = 0 is a special case in this sequence.
Numbers n for which A275947(n) = 0 or equally, for which A275811(n) <= 1.
Numbers n for which A008683(A275734(n)) <> 0, that is, indices of squarefree terms in A275734.
Numbers n for which A060130(n) = A060502(n).
Numbers with at most one nonzero digit on each digit slope of the factorial base representation of n (see A275811 and A060502 for the definition of slopes in this context). More exactly: numbers n in whose factorial base representation (A007623) there does not exist a pair of digit positions i_1 and i_2 with nonzero digits d_1 and d_2, such that (i_1 - d_1) = (i_2 - d_2).

Antti Karttunen, Table of n, a(n) for n = 0..10000
Index entries for sequences related to factorial base representation

Complement: A275805.
Indices of zeros in A275947 and A275962.
Intersection with A276005 gives A261220.
Cf. A059590 (a subsequence).
Cf. A007623, A008683, A060130, A060502, A275734, A275811.

from operator import mul
from sympy import prime, factorial as f
from sympy.ntheory.factor_ import core
def a007623(n, p=2): return n if n0 else '0' for i in x)[::-1]
    return 0 if n==1 else sum([int(y[i])*f(i + 1) for i in range(len(y))])
def a(n): return 1 if n==0 else a275732(n)*a(a257684(n))
def ok(n): return 1 if n==0 else core(a(n))==a(n)
print([n for n in range(201) if ok(n)]) # Indranil Ghosh, Jun 19 2017

A275806 a(n) = number of distinct nonzero digits in factorial base representation of n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Aug 11 2016

			For n=0, with factorial base representation (A007623) also 0, there are no nonzero digits, thus a(0) = 0.
For n=2, with factorial base representation "10", there is one distinct nonzero digit, thus a(2) = 1.
For n=3, with factorial base representation "11", there is just one distinct nonzero digit, thus a(3) = 1.
For n=44, with factorial base representation "1310", there are two distinct nonzero digits ("1" and "3"), thus a(44) = 2.

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

Cf. A000142, A001221, A007623, A033312, A060130, A060502, A225901, A265349, A275735.

Cf. also A153880, A255411.

a[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; Length[Union[Select[s, # > 0 &]]]]; Array[a, 100, 0] (* Amiram Eldar, Feb 07 2024 *)

from sympy import prime, primefactors
from operator import mul
import collections
def a007623(n, p=2): return n if nIndranil Ghosh, Jun 20 2017

(define (A275806 n) (A001221 (A275735 n)))

a(n) = A001221(A275735(n)).
a(n) = A060502(A225901(n)).
Other identities. For all n >= 0:
a(n) = a(A153880(n)) = a(A255411(n)). [The shift-operations do not change the number of distinct nonzero digits.]
a(A265349(n)) = A060130(A265349(n)).
a(A000142(n)) = 1.
a(A033312(n)) = n-1, for all n >= 1.

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

Antti Karttunen, Mar 05 2001

nonn

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

Cf. A055090, A056170, A060118, A060120, A060129, A060130, A275725, A278225, A276004.

Cf. A276005 (positions where coincides with A060502).

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

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)

A275946 Number of nonzero digits that are the sole occupants of their slope in factorial base representation: a(n) = A056169(A275734(n)). (See comments for more exact definition.)

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Aug 15 2016

Total number of such nonzero digits d_x in the factorial base representation (A007623) of n for which it holds that for all other nonzero digits d_y present (i_x - d_x) <> (i_y - d_y), where i_x and i_y are the indices of the digits d_x and d_y respectively.

Equally: Number of digit slopes occupied by just one nonzero digit in the factorial base representation of n. In other words, a(n) is the number of elements with multiplicity one in multiset [(i_x - d_x) | where d_x ranges over each nonzero digit present and i_x is its position from the right].

			For n=2, in factorial base "10", there is only one slope occupied by a single nonzero digit (1 is on the sub-maximal slope as 2-1 = 1), thus a(2) = 1.
For n=3, in factorial base "11", there are two occupied slopes, each having just one digit present, thus a(3) = 2.
For n=5, in factorial base "21", there is just one distinct occupied slope, but it contains two nonzero digits (2 and 1 both occupy the maximal slope as 2-2 = 1-1 = 0), thus there are no slopes with just one nonzero digit and a(5) = 0.
For n=525, in factorial base "41311", there are three occupied slopes. The maximal slope contains the nonzero digits "3.1", the sub-maximal digits "4..1.", and the sub-sub-sub-maximal just "1..." (the 1 in the position 4 from right is the sole occupier of its own slope). Thus only one of the slopes is occupied by a lonely occupant and a(525) = 1.

Antti Karttunen, Table of n, a(n) for n = 0..40320
Indranil Ghosh, Python program for generating this sequence
Index entries for sequences related to factorial base representation

Cf. A056169, A275734.

Cf. also A060502, A060130, A225901, A275947, A275948, A275962.

(define (A275946 n) (A056169 (A275734 n)))

a(n) = A056169(A275734(n)).
Other identities. For all n >= 0.
a(n) = A275948(A225901(n)).
A060502(n) = a(n) + A275947(n).
A060130(n) = a(n) + A275962(n).

A275947 Number of distinct slopes with multiple nonzero digits in factorial base representation of n: a(n) = A056170(A275734(n)). (See comments for more exact definition).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Aug 15 2016

a(n) gives the number of distinct elements that have multiplicity > 1 in a multiset [(i_x - d_x) | where d_x ranges over each nonzero digit present and i_x is its position from the right].

			For n=525, in factorial base "41311", there are three occupied slopes. The maximal slope contains the nonzero digits "3.1", the sub-maximal digits "4..1.", and the sub-sub-sub-maximal just "1..." (the 1 in the position 4 from right is the sole occupier of its own slope). Thus there are two slopes with more than one nonzero digit, and a(525) = 2.
Equally, when we form a multiset of (digit-position - digit-value) differences for all nonzero digits present in "41311", we obtain a multiset [0, 0, 1, 1, 3], in which the distinct elements that occur multiple times are 0 and 1, thus a(525) = 2.

Antti Karttunen, Table of n, a(n) for n = 0..40320
Indranil Ghosh, Python program for computing this sequence
Index entries for sequences related to factorial base representation

Cf. A056170, A275734.
Cf. A275804 (indices of zeros), A275805 (of nonzeros).
Cf. also A060502, A225901, A275946, A275949, A275962.

(define (A275947 n) (A056170 (A275734 n)))

a(n) = A056170(A275734(n)).
Other identities and observations. For all n >= 0.
a(n) = A275949(A225901(n)).
A060502(n) = A275946(n) + a(n).
a(n) <= A275962(n).

A275811 Number of nonzero digits on a maximally occupied slope of factorial base representation of n: a(n) = A051903(A275734(n)). See comments for the definition.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Aug 10 2016

Digit slopes are called "maximal", "sub-maximal", "sub-sub-maximal", etc. For digit-positions we employ one-based indexing, thus we say that the least significant digit of factorial base expansion of n is in position 1, etc. The maximal digit slope is occupied when there is at least one digit-position k that contains digit k (maximal digit allowed in that position), so that A260736(n) > 0, and n is thus a term of A273670. The sub-maximal digit slope is occupied when there is at least one nonzero digit k in a digit-position k+1. The sub-sub-maximal slope is occupied when there is at least one nonzero digit k in a digit-position k+2, etc. This sequence gives the number of nonzero digits on a slope (of possibly several) for which there exists no other slopes with more nonzero digits. See the examples.
In other words: a(n) gives the number of occurrences of a most common element in the 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].
Involution A225901 maps this metric to another metric A264990 which gives the maximal number of equal nonzero digits occurring in factorial base representation (A007623) of n. See also A060502.

			For n=23 ("321" in factorial base representation, A007623), all three nonzero digits are maximal for their positions (they all occur on "maximal slope"), thus the "maximal slope" is also the "maximally occupied slope" (as there are no other occupied slopes present), and a(23) = 3.
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 here the "maximal slope" is also the "maximally occupied slope" (with 2 nonzero digits present), and a(29) = 2.
For n=37 ("1201"), there are three nonzero digits, where the rightmost 1 is on the maximal slope, 2 is on the sub-maximal, and the most significant 1 is on the "sub-sub-sub-maximal", thus there are three occupied slopes in total, all with just one nonzero digit present, and a(37) = 1.
For n=55 ("2101"), the least significant 1 is on the maximal slope, and the digits "21" at the beginning are together on the sub-sub-maximal slope (as they are both two less than the maximal digit values 4 and 3 allowed in those positions), thus here the sub-sub-maximal slope is the "maximally occupied slope" with its two occupiers, and a(55) = 2.

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

Cf. A051903, A060502, A225901, A264990, A275734.

Cf. A275804 (gives the indices of 0 and 1's), A275805 (gives the indices of terms > 1).

from sympy import prime, factorint
from operator import mul
from functools import reduce
from sympy import factorial as f
def a051903(n): return 0 if n==1 else max(factorint(n).values())
def a007623(n, p=2): return n if n0 else '0' for i in x])[::-1]
    return 0 if n==1 else sum([int(y[i])*f(i + 1) for i in range(len(y))])
def a275734(n): return 1 if n==0 else a275732(n)*a275734(a257684(n))
def a(n): return 0 if n==0 else a051903(a275734(n))
print([a(n) for n in range(201)]) # Indranil Ghosh, Jun 20 2017

a(n) = A051903(A275734(n)).

a(n) = A264990(A225901(n)).

Signs in comment corrected and clarification added by Antti Karttunen, Aug 16 2016

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

Antti Karttunen, Aug 17 2016

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.

			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.

Antti Karttunen, Table of n, a(n) for n = 0..4140
Index entries for sequences related to factorial base representation

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

Other identities. For all n >= 1:

a(A000110(n)) = n! = A000142(n). [To be proved.]

A278234 Filter-sequence for factorial base (digit slopes): Least number with the same prime signature as A275734(n).

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 2, 6, 6, 30, 6, 12, 2, 6, 4, 12, 6, 12, 2, 4, 6, 12, 4, 8, 2, 6, 6, 30, 6, 12, 6, 30, 30, 210, 30, 60, 6, 30, 12, 60, 30, 60, 6, 12, 30, 60, 12, 24, 2, 6, 6, 30, 6, 12, 4, 12, 12, 60, 12, 36, 6, 30, 12, 60, 30, 60, 6, 12, 30, 60, 12, 24, 2, 6, 4, 12, 6, 12, 6, 30, 12, 60, 30, 60, 4, 12, 8, 24, 12, 36, 6, 12, 12, 36, 12, 24, 2, 4, 6, 12, 4, 8, 6

Offset: 0

Antti Karttunen, Nov 16 2016

nonn

This sequence can be used for filtering certain factorial base (A007623) related sequences, because it matches only with any such sequence b that can be computed as b(n) = f(A275734(n)), where f(n) is any function that depends only on the prime signature of n (some of these are listed under the index entry for "sequences computed from exponents in ...").

Matching in this context means that the sequence a matches with the sequence b iff for all i, j: a(i) = a(j) => b(i) = b(j). In other words, iff the sequence b partitions the natural numbers to the same or coarser equivalence classes (as/than the sequence a) by the distinct values it obtains.

Antti Karttunen, Table of n, a(n) for n = 0..40320
Indranil Ghosh, Python program for computing this sequence
Index entries for sequences computed from exponents in factorization of n
Index entries for sequences related to factorial base representation

Cf. A007623, A046523, A225901, A275734.
Other filter-sequences related to factorial base: A278225, A278235, A278236.
Sequences that partition N into same or coarser equivalence classes: A060130, A060502, A275811, A275946, A275962.

(define (A278234 n) (A046523 (A275734 n)))

a(n) = A046523(A275734(n)).

a(n) = A278235(A225901(n)).

A060501 Average of digits of each term in A060495, number of balls in each such siteswap juggling pattern.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Mar 22 2001

nonn

Cf. A060500 (for avg), A060502.

Perm2SiteSwap2 := 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), n]; else a := [op(a),((ip[i]-i) mod n)]; fi; od; RETURN(a); end;

a(n) = avg(Perm2SiteSwap2(PermUnrank3R(n)))

cp's OEIS Frontend

A060498 Each permutation in the list A060117 converted to Site Swap notation, with digits reversed and inverted. "Zero throws" (fixed elements) indicated with 0's.

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A275804 Numbers with at most one nonzero digit on each digit slope of the factorial base representation of n.

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A275806 a(n) = number of distinct nonzero digits in factorial base representation of n.

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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.

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A275946 Number of nonzero digits that are the sole occupants of their slope in factorial base representation: a(n) = A056169(A275734(n)). (See comments for more exact definition.)

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A275947 Number of distinct slopes with multiple nonzero digits in factorial base representation of n: a(n) = A056170(A275734(n)). (See comments for more exact definition).

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A275811 Number of nonzero digits on a maximally occupied slope of factorial base representation of n: a(n) = A051903(A275734(n)). See comments for the definition.

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

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