A275727 -id:A275727 - cp's OEIS frontend

A276008 Substitute ones for all nonzero digits in factorial base representation of n: a(n) = A059590(A275727(n)).

0, 1, 2, 3, 2, 3, 6, 7, 8, 9, 8, 9, 6, 7, 8, 9, 8, 9, 6, 7, 8, 9, 8, 9, 24, 25, 26, 27, 26, 27, 30, 31, 32, 33, 32, 33, 30, 31, 32, 33, 32, 33, 30, 31, 32, 33, 32, 33, 24, 25, 26, 27, 26, 27, 30, 31, 32, 33, 32, 33, 30, 31, 32, 33, 32, 33, 30, 31, 32, 33, 32, 33, 24, 25, 26, 27, 26, 27, 30, 31, 32, 33, 32, 33, 30, 31, 32, 33, 32, 33, 30, 31, 32, 33, 32, 33, 24

Offset: 0

Antti Karttunen, Aug 18 2016

			For n=23 whose factorial base representation is "321", when we replace each nonzero digit with 1, we get "111", the factorial base representation of 9, thus a(23) = 9.
From n=37 ("1201") we get "1101", thus a(37) = 31 as A007623(31) = 1101.

Antti Karttunen, Table of n, a(n) for n = 0..40319
Index entries for sequences related to factorial base representation.

Cf. A059590, A275727.

Cf. also A276009.

a[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; s = Min[#, 1]& /@ s; Total[s*Range[Length[s]]!]]; Array[a, 100, 0] (* Amiram Eldar, Feb 14 2024 *)

from sympy import factorial as f
def a007623(n, p=2): return n if n0 else '0' for i in x)[::-1]
    return sum([int(y[i])*f(i + 1) for i in range(len(y))])
print([a(n) for n in range(201)]) # Indranil Ghosh, Jun 21 2017

(define (A276008 n) (A059590 (A275727 n)))
;; Standalone program:
(define (A276008 n) (let loop ((n n) (s 0) (f 1) (i 2)) (if (zero? n) s (let ((d (modulo n i))) (if (zero? d) (loop (/ n i) s (* i f) (+ 1 i)) (loop (/ (- n d) i) (+ s f) (* i f) (+ 1 i)))))))

a(n) = A059590(A275727(n)).

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).

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 8, 9, 24, 25, 26, 27, 30, 31, 32, 33, 120, 121, 122, 123, 126, 127, 128, 129, 144, 145, 146, 147, 150, 151, 152, 153, 720, 721, 722, 723, 726, 727, 728, 729, 744, 745, 746, 747, 750, 751, 752, 753, 840, 841, 842, 843, 846, 847, 848, 849, 864, 865

Offset: 0

Henry Bottomley, Jan 24 2001

Numbers that are sums of distinct factorials (0! and 1! not treated as distinct).
Complement of A115945; A115944(a(n)) > 0; A115647 is a subsequence. - Reinhard Zumkeller, Feb 02 2006
A115944(a(n)) = 1. - Reinhard Zumkeller, Dec 04 2011
From Tilman Piesk, Jun 04 2012: (Start)
The inversion vector (compare A007623) of finite permutation a(n) (compare A055089, A195663) has only zeros and ones. Interpreted as a binary number it is 2*n (or n when the inversion vector is defined without the leading 0).
The inversion set of finite permutation a(n) interpreted as a binary number (compare A211362) is A211364(n).
(End)

			128 is in the sequence since 5! + 3! + 2! = 128.
a(22) = 128. a(22) = a(6) + (1 + floor(log(16) / log(2)))! = 8 + 5! = 128. Also, 22 = 10110_2. Therefore, a(22) = 1 * 5! + 0 * 4! + 1 * 3! + 1 + 2! + 0 * 0! = 128. - _David A. Corneth_, Aug 21 2016

Reinhard Zumkeller (terms 0..500) & Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to factorial base representation
Index entries for sequences related to factorial numbers

Cf. A007088, A007623, A014597, A051760, A051761, A059589.
Indices of zeros in A257684.
Cf. A275736 (left inverse).
Cf. A025494, A060112 (subsequences).
Cf. A153880, A225901.
Subsequence of A060132, A256450 and A275804.
Other sequences that are built by replacing 2^k in the binary representation with other numbers: A029931 (naturals), A089625 (primes), A022290 (Fibonacci), A197433 (Catalans), A276091 (n*n!), A275959 ((2n)!/2). Cf. also A276082 & A276083.

import Data.List (elemIndices)
a059590 n = a059590_list !! n
a059590_list = elemIndices 1 $ map a115944 [0..]
-- Reinhard Zumkeller, Dec 04 2011

[seq(bin2facbase(j),j=0..64)]; bin2facbase := proc(n) local i; add((floor(n/(2^i)) mod 2)*((i+1)!),i=0..floor_log_2(n)); end;
floor_log_2 := proc(n) local nn,i; nn := n; for i from -1 to n do if(0 = nn) then RETURN(i); fi; nn := floor(nn/2); od; end;
# next Maple program:
a:= n-> (l-> add(l[j]*j!, j=1..nops(l)))(Bits[Split](n)):
seq(a(n), n=0..57);  # Alois P. Heinz, Aug 12 2025

a[n_] :=  Reverse[id = IntegerDigits[n, 2]].Range[Length[id]]!; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Jun 19 2012, after Philippe Deléham *)

a(n) = if(n>0, a(n-msb(n)) + (1+logint(n,2))!, 0)
msb(n) = 2^#binary(n)>>1
{my(b = binary(n)); sum(i=1,#b,b[i]*(#b+1-i)!)} \\ David A. Corneth, Aug 21 2016

def facbase(k, f):
    return sum(f[i] for i, bi in enumerate(bin(k)[2:][::-1]) if bi == "1")
def auptoN(N): # terms up to N factorial-base digits; 13 generates b-file
    f = [factorial(i) for i in range(1, N+1)]
    return list(facbase(k, f) for k in range(2**N))
print(auptoN(5)) # Michael S. Branicky, Oct 15 2022

G.f. 1/(1-x) * Sum_{k>=0} (k+1)!*x^2^k/(1+x^2^k). - Ralf Stephan, Jun 24 2003
a(n) = Sum_{k>=0} A030308(n,k)*A000142(k+1). - Philippe Deléham, Oct 15 2011
From Antti Karttunen, Aug 19 2016: (Start)
a(0) = 0, a(2n) = A153880(a(n)), a(2n+1) = 1+A153880(a(n)).
a(n) = A225901(A276091(n)).
a(n) = A276075(A019565(n)).
a(A275727(n)) = A276008(n).
A275736(a(n)) = n.
A276076(a(n)) = A019565(n).
A007623(a(n)) = A007088(n).
(End)
a(n) = a(n - mbs(n)) + (1 + floor(log(n) / log(2)))!. - David A. Corneth, Aug 21 2016

Name changed (to emphasize the functional nature of the sequence) with the old definition moved to the comments by Antti Karttunen, Aug 21 2016

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.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Mar 02 2001

nonn

			19 = 3*(3!) + 0*(2!) + 1*(1!), thus it is written as "301" in factorial base (A007623). The count of nonzero digits in that representation is 2, so a(19) = 2.

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

Cf. A007623, A034968, A055091, A060117, A060118, A060128, A060129, A060131, A060502, A257687, A275734, A275735, A276076.
Cf. A227130 (positions of even terms), A227132 (of odd terms).
Cf. also A225901, A232094, A257694, A257695.
The topmost row and the leftmost column in array A230415, the left edge of triangle A230417.
Differs from similar A267263 for the first time at n=30.

A060130(n) = count_nonfixed(convert(PermUnrank3R(n), 'disjcyc'))-nops(convert(PermUnrank3R(n), 'disjcyc')) or nops(fac_base(n))-nops(positions(0, fac_base(n)))
fac_base := n -> fac_base_aux(n, 2); fac_base_aux := proc(n, i) if(0 = n) then RETURN([]); else RETURN([op(fac_base_aux(floor(n/i), i+1)), (n mod i)]); fi; end;
count_nonfixed := l -> convert(map(nops, l), `+`);
positions := proc(e, ll) local a, k, l, m; l := ll; m := 1; a := []; while(member(e, l[m..nops(l)], 'k')) do a := [op(a), (k+m-1)]; m := k+m; od; RETURN(a); end;
# For procedure PermUnrank3R see A060117

Block[{nn = 105, r}, r = MixedRadix[Reverse@ Range[2, -1 + SelectFirst[Range@ 12, #! > nn &]]]; Array[Count[IntegerDigits[#, r], k_ /; k > 0] &, nn, 0]] (* Michael De Vlieger, Dec 30 2017 *)

(define (A060130 n) (let loop ((n n) (i 2) (s 0)) (cond ((zero? n) s) (else (loop (quotient n i) (+ 1 i) (+ s (if (zero? (remainder n i)) 0 1)))))))
;; Two other implementations, that use memoization-macro definec:
(definec (A060130 n) (if (zero? n) n (+ 1 (A060130 (A257687 n)))))
(definec (A060130 n) (if (zero? n) n (+ (A257511 n) (A060130 (A257684 n)))))
;; Antti Karttunen, Dec 30 2017

a(0) = 0; for n > 0, a(n) = 1 + a(A257687(n)).
a(0) = 0; for n > 0, a(n) = A257511(n) + a(A257684(n)).
a(n) = A060129(n) - A060128(n).
a(n) = A084558(n) - A257510(n).
a(n) = A275946(n) + A275962(n).
a(n) = A275948(n) + A275964(n).
a(n) = A055091(A060119(n)).
a(n) = A069010(A277012(n)) = A000120(A275727(n)).
a(n) = A001221(A275733(n)) = A001222(A275733(n)).
a(n) = A001222(A275734(n)) = A001222(A275735(n)) = A001221(A276076(n)).
a(n) = A046660(A275725(n)).
a(A225901(n)) = a(n).
A257511(n) <= a(n) <= A034968(n).
A275806(n) <= a(n).
a(A275804(n)) = A060502(A275804(n)). [A275804 gives all the positions where this coincides with A060502.]
a(A276091(n)) = A260736(A276091(n)). [A276091 gives all the positions where this coincides with A260736.]

Example-section added, name edited, the old Maple-code moved away from the formula-section, and replaced with all the new formulas by Antti Karttunen, Dec 30 2017

A275736 a(n) has base-2 representation with ones in those digit-positions where n contains ones in its factorial base representation, and zeros in all the other positions.

Original entry on oeis.org

0, 1, 2, 3, 0, 1, 4, 5, 6, 7, 4, 5, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 0, 1, 8, 9, 10, 11, 8, 9, 12, 13, 14, 15, 12, 13, 8, 9, 10, 11, 8, 9, 8, 9, 10, 11, 8, 9, 0, 1, 2, 3, 0, 1, 4, 5, 6, 7, 4, 5, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 0, 1, 4, 5, 6, 7, 4, 5, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 0, 1, 4, 5, 6, 7, 4, 5, 0

Offset: 0

Antti Karttunen, Aug 09 2016

Each natural numbers occurs an infinite number of times.

Can be used when computing A275727.

			22 has factorial base representation "320" (= A007623(22)), which does not contain any "1". Thus a(22) = 0, as the empty sum is 0.
35 has factorial base representation "1121" (= A007623(35)). Here 1's occur in the following positions, when counted from right (starting with 0 for the least significant position): 0, 2 and 3. Thus a(35) = 2^0 + 2^2 + 2^3 = 1*4*8 = 13.

Cf. A000079, A000120, A000225, A007489, A048675, A257261, A257511, A275727, A275730.
Left inverse of A059590.
Cf. A255411 (indices of zeros).
Cf. also A275732.

nn = 120; m = 1; While[Factorial@ m < nn, m++]; m; Map[FromDigits[#, 2] &[IntegerDigits[#, MixedRadix[Reverse@ Range[2, m]]] /. k_ /; k != 1 -> 0] &, Range[0, nn]] (* Michael De Vlieger, Aug 11 2016, Version 10.2 *)

If A257261(n) = 0, then a(n) = 0, otherwise a(n) = A000079(A257261(n)-1) + a(A275730(n, A257261(n)-1)).  [Here A275730(n,p) is a bivariate function that "clears" the digit at zero-based position p in the factorial base representation of n].
Other identities and observations. For all n >= 0:
a(n) = A048675(A275732(n)).
A000120(a(n)) = A257511(n).
a(A007489(n)) = A000225(n).
a(A059590(n)) = n.
a(A255411(n)) = 0.

A275733 a(0) = 1; for n >= 1, a(n) = A275732(n) * A003961(a(A257684(n))).

Original entry on oeis.org

1, 2, 3, 6, 3, 6, 5, 10, 15, 30, 15, 30, 5, 10, 15, 30, 15, 30, 5, 10, 15, 30, 15, 30, 7, 14, 21, 42, 21, 42, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 7, 14, 21, 42, 21, 42, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 7, 14, 21, 42, 21, 42

Offset: 0

Antti Karttunen, Aug 08 2016

a(n) = product of primes whose indices are positions of nonzero-digits in factorial base representation of n (see A007623). Here positions are one-based, so that the least significant digit is the position 1, the next least significant the position 2, etc.

			For n=19, A007623(19) = 301, thus a(19) = prime(3)*prime(1) = 5*2 = 10.
For n=52, A007623(52) = 2020, thus a(52) = prime(2)*prime(4) = 3*7 = 21.

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. A001221, A002110, A003961, A005117, A007489, A007623, A048675, A060130, A061395, A084558, A257684, A275732.
Subsequence of A005117.
Cf. A275727.
Cf. also A275725, A275734, A275735 for other such prime factorization encodings of A060117/A060118-related polynomials.

a(0) = 1; for n >= 1, a(n) = A275732(n) * A003961(a(A257684(n))).
Other identities and observations. For all n >= 0:
a(A007489(n)) = A002110(n).
A001221(a(n)) = A001222(a(n)) = A060130(n).
A048675(a(n)) = A275727(n).
A061395(a(n)) = A084558(n).

A275728 a(0) = 0, for n >= 1, a(n) = A275736(n) + a(A257684(n)); a(n) = A048675(A275734(n)).

Original entry on oeis.org

0, 1, 2, 3, 1, 2, 4, 5, 6, 7, 5, 6, 2, 3, 4, 5, 3, 4, 1, 2, 3, 4, 2, 3, 8, 9, 10, 11, 9, 10, 12, 13, 14, 15, 13, 14, 10, 11, 12, 13, 11, 12, 9, 10, 11, 12, 10, 11, 4, 5, 6, 7, 5, 6, 8, 9, 10, 11, 9, 10, 6, 7, 8, 9, 7, 8, 5, 6, 7, 8, 6, 7, 2, 3, 4, 5, 3, 4, 6, 7, 8, 9, 7, 8, 4, 5, 6, 7, 5, 6, 3, 4, 5, 6, 4, 5, 1, 2, 3, 4

Offset: 0

Antti Karttunen, Aug 09 2016

See graph: pine trees on a snowy mountain. - N. J. A. Sloane, Aug 12 2016

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

Cf. A048675, A257684, A275734, A275736.

Cf. also A275727, A275808.

a(0) = 0, for n >= 1, a(n) = A275736(n) + a(A257684(n)).

a(n) = A048675(A275734(n)).

A276004 a(n) is the number of nonzero digits in the factorial-base representation of n that are matched by more significant digits from left; a(n) = A060502(n) - A060128(n).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Aug 17 2016

a(n) is the number of times a nonzero digit d_i appears in position i of the factorial-base representation of n (where the least significant digit is in position 1) such that there is another nonzero digit d_j in such position j > i that j - d_j = i.

			For n=15 ("211" in factorial base) the least significant 1 at position 1 is matched by its immediate left neighbor 1 and also by 2 at position 3, as (2-1) = (3-2) = 1, the position where the least significant 1 itself is. However, this is counted just as one match, because this sequence gives the number of digits that are matched, instead of the number of digits that match, thus a(15)=1.

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

Cf. A000120, A004198, A060128, A060502, A275727, A276010.
Cf. A276005 (indices of zeros), A276006 (of nonzeros).
Differs from A276007 for the first time at n=15, where a(15)=1, while A276004(15)=2.

a(n) = A060502(n) - A060128(n).

a(n) = A000120(2*A275727(n) AND A276010(n)), where AND is a bitwise-and given in A004198.

cp's OEIS Frontend

A276008 Substitute ones for all nonzero digits in factorial base representation of n: a(n) = A059590(A275727(n)).

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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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A275736 a(n) has base-2 representation with ones in those digit-positions where n contains ones in its factorial base representation, and zeros in all the other positions.

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A275733 a(0) = 1; for n >= 1, a(n) = A275732(n) * A003961(a(A257684(n))).

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A275728 a(0) = 0, for n >= 1, a(n) = A275736(n) + a(A257684(n)); a(n) = A048675(A275734(n)).

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A276004 a(n) is the number of nonzero digits in the factorial-base representation of n that are matched by more significant digits from left; a(n) = A060502(n) - A060128(n).

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