A275804 -id:A275804 - cp's OEIS frontend

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.

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

Antti Karttunen, Mar 22 2001

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)

			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.

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

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.

# 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));

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)

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

A275734 Prime-factorization representations of "factorial base slope polynomials": a(0) = 1; for n >= 1, a(n) = A275732(n) * a(A257684(n)).

Original entry on oeis.org

1, 2, 3, 6, 2, 4, 5, 10, 15, 30, 10, 20, 3, 6, 9, 18, 6, 12, 2, 4, 6, 12, 4, 8, 7, 14, 21, 42, 14, 28, 35, 70, 105, 210, 70, 140, 21, 42, 63, 126, 42, 84, 14, 28, 42, 84, 28, 56, 5, 10, 15, 30, 10, 20, 25, 50, 75, 150, 50, 100, 15, 30, 45, 90, 30, 60, 10, 20, 30, 60, 20, 40, 3, 6, 9, 18, 6, 12, 15, 30, 45, 90, 30, 60, 9, 18, 27

Offset: 0

Antti Karttunen, Aug 08 2016

These are prime-factorization representations of single-variable polynomials where the coefficient of term x^(k-1) (encoded as the exponent of prime(k) in the factorization of n) is equal to the number of nonzero digits that occur on the slope (k-1) levels below the "maximal slope" in the factorial base representation of n. See A275811 for the definition of the "digit slopes" in this context.

			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 a(23) = prime(1)^3 = 2^3 = 8.
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 a(29) = prime(1)^2 * prime(4)^1 = 2*7 = 28.
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 a(37) = prime(1) * prime(2) * prime(4) = 2*3*7 = 42.
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 a(55) = prime(1)^1 * prime(3)^2 = 2*25 = 50.

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

Cf. A001221, A001222, A002110, A007489, A007814, A048675, A051903, A056169, A056170, A060130, A060502, A225901.
Cf. A257684, A275732.
Cf. A275811.
Cf. A260736, A275728, A275808, A275812, A275946, A275947, A275962.
Cf. A275804 (indices of squarefree terms), A275805 (of terms not squarefree).
Cf. also A275725, A275733, A275735, A276076 for other such prime factorization encodings of A060117/A060118-related polynomials.

from operator import mul
from sympy import prime, factorial as f
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))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 19 2017

a(0) = 1; for n >= 1, a(n) = A275732(n) * a(A257684(n)).
Other identities and observations. For all n >= 0:
a(n) = A275735(A225901(n)).
a(A007489(n)) = A002110(n).
A001221(a(n)) = A060502(n).
A001222(a(n)) = A060130(n).
A007814(a(n)) = A260736(n).
A051903(a(n)) = A275811(n).
A048675(a(n)) = A275728(n).
A248663(a(n)) = A275808(n).
A056169(a(n)) = A275946(n).
A056170(a(n)) = A275947(n).
A275812(a(n)) = A275962(n).

A261220 Ranks of involutions in permutation orderings A060117 and A060118.

Original entry on oeis.org

0, 1, 2, 4, 6, 7, 12, 16, 18, 20, 24, 25, 26, 28, 48, 49, 60, 66, 72, 76, 78, 90, 96, 98, 102, 108, 120, 121, 122, 124, 126, 127, 132, 136, 138, 140, 240, 241, 242, 244, 288, 289, 312, 316, 336, 338, 360, 361, 372, 378, 384, 385, 432, 450, 456, 468, 480, 484, 486, 498, 504, 508, 528, 546, 576, 582, 600, 602, 606, 612, 624, 626, 648, 660, 672, 678, 720, 721

Offset: 0

Antti Karttunen, Aug 26 2015

From Antti Karttunen, Aug 17 2016: (Start)
Intersection of A275804 and A276005. In other words, these are numbers in whose factorial base representation (A007623, see A260743) there does not exist any such pair of nonzero digits d_i and d_j in positions i and j that either (i - d_i) = j or (i - d_i) = (j - d_j) would hold. Here one-based indexing is used so that the least significant digit at right is in position 1.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..1000
Index entries for sequences related to factorial base representation

Intersection of A275804 and A276005.
Same sequence shown in factorial base: A260743.
Cf. A060117, A060118, A275947, A276007.
Positions of zeros in A261219.
Positions of 1 and 2's in A060131 and A275803.
Subsequence: A060112.
Cf. also A014489.

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).

A275962 Total number of nonzero digits that occur on the multiply occupied slopes of the factorial base representation of n: a(n) = A275812(A275734(n)). (See comments for more exact definition).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Aug 15 2016

a(n) gives the total number of elements (counted with multiplicity) 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 the 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). There are two slopes with more than one nonzero digit, each having two such digits, and thus a(525) = 2+2 = 4.
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 elements that occur multiple times are [0, 0, 1, 1], thus a(525) = 4.

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. A275734, A275812.
Cf. A275804 (indices of zeros), A275805 (of nonzeros).
Cf. also A060130, A225901, A275946, A275947, A275964.

(define (A275962 n) (A275812 (A275734 n)))

a(n) = A275812(A275734(n)).
Other identities and observations. For all n >= 0.
a(n) = A275964(A225901(n)).
a(n) = A060130(n) - A275946(n).
a(n) >= A275947(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.]

A275805 Indices of nonsquarefree terms in A275734; numbers with at least one digit slope (in their factorial base representation) with multiple nonzero digits. (See comments for the exact definition).

Original entry on oeis.org

5, 11, 14, 15, 17, 19, 21, 22, 23, 29, 35, 38, 39, 41, 43, 45, 46, 47, 53, 54, 55, 56, 57, 58, 59, 62, 63, 65, 67, 69, 70, 71, 74, 75, 77, 80, 81, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99, 100, 101, 103, 105, 106, 107, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 125, 131, 134, 135, 137, 139, 141, 142, 143, 149, 155

Offset: 1

Antti Karttunen, Aug 10 2016

Numbers n for which A008683(A275734(n)) = 0.
Numbers n for which A275811(n) > 1.
Numbers n in whose factorial base representation (A007623) there exists at least one 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).

			For n=5, "21" in factorial base (A007623), the pair 2 (in position 2) and 1 (in position 1) satisfies the condition, as (2-2) = (1-1), thus 5 is included.
For n=55, "2101" in factorial base, the pair 2 (in position 4) and 1 (in position 3) satisfies the condition, as (4-2) = (3-1), thus 55 is included.
For n=67, "2301" in factorial base, the pair 3 (in position 3) and 1 (in position 1) satisfies the condition, as (3-3) = (1-1), thus 67 is included in the sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to factorial base representation

Complement: A275804.
Cf. A007623, A008683, A275734, A275811.
Cf. A275809 (a subsequence apart from its initial 0-term).
Subsequence of A115945.

from operator import mul
from sympy import prime, factorial as f, mobius
from functools import reduce
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))
print([n for n in range(201) if mobius(a(n))==0]) # Indranil Ghosh, Jun 19 2017

Showing 1-10 of 10 results.

cp's OEIS Frontend

A059590 Numbers obtained by reinterpreting base-2 representation of n in the factorial base: a(n) = Sum_{k>=0} A030308(n,k)*A000142(k+1).

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A060130 Number of nonzero digits in factorial base representation (A007623) of n; minimum number of transpositions needed to compose each permutation in the lists A060117 & A060118.

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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.

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A275734 Prime-factorization representations of "factorial base slope polynomials": a(0) = 1; for n >= 1, a(n) = A275732(n) * a(A257684(n)).

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A261220 Ranks of involutions in permutation orderings A060117 and A060118.

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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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A275962 Total number of nonzero digits that occur on the multiply occupied slopes of the factorial base representation of n: a(n) = A275812(A275734(n)). (See comments for more exact definition).

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