A034968 -id:A034968 - cp's OEIS frontend

A358976 Numbers that are coprime to the sum of their factorial base digits (A034968).

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 25, 28, 29, 31, 32, 33, 37, 39, 41, 43, 44, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 76, 77, 79, 83, 84, 85, 87, 88, 89, 92, 93, 95, 97, 98, 101, 102, 103, 106, 107, 109, 110

Offset: 1

Amiram Eldar, Dec 07 2022

Numbers k such that gcd(k, A034968(k)) = 1.
The factorial numbers (A000142) are terms. These are also the only factorial base Niven numbers (A118363) in this sequence.
Includes all the prime numbers.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 7, 59, 601, 6064, 60729, 607567, 6083420, 60827602, 607643918, 6079478119, ... . Conjecture: The asymptotic density of this sequence exists and equals 6/Pi^2 = 0.607927... (A059956), the same as the density of A094387.

			3 is a term since A034968(3) = 2, and gcd(3, 2) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A034968, A059956, A118363.
Subsequences: A000040, A000142.
Similar sequences: A094387, A339076, A358975, A358977, A358978.

q[n_] := Module[{k = 2, s = 0, m = n, r}, While[m > 0, r=Mod[m,k]; s+=r; m=(m-r)/k; k++]; CoprimeQ[n, s]]; Select[Range[120], q]

is(n)={my(k=2, s=0, m=n); while(m>0, s+=m%k; m\=k; k++); gcd(s,n)==1;}

A377384 a(n) is the number of iterations that n requires to reach a noninteger or a factorial number under the map x -> x / f(x), where f(k) = A034968(k) is the sum of digits in the factorial-base representation of k; a(n) = 0 if n is a factorial number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 27 2024

The factorial numbers are fixed points of the map, since f(k!) = 1 for all k >= 0. Therefore they are arbitrarily assigned the value a(k!) = 0.

Each number n starts a chain of a(n) integers: n, n/f(n), (n/f(n))/f(n/f(n)), ..., of them the first a(n)-1 integers are factorial-base Niven numbers (A118363).

			a(8) = 2 since 8/f(8) = 4 and 4/f(4) = 2 is a factorial number that is reached after 2 iterations.
a(27) = 3 since 27/f(27) = 9, 9/f(9) = 3 and 3/f(3) = 3/2 is a noninteger that is reached after 3 iterations.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000142, A034968, A118363, A286607, A377385, A377386.

Analogous sequences: A376615 (binary), A377208 (Zeckendorf).

fdigsum[n_] := Module[{k = n, m = 2, r, s = 0}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, s += r; m++]; s]; a[n_] := a[n] = Module[{s = fdigsum[n]}, If[s == 1, 0, If[!Divisible[n, s], 1, 1 + a[n/s]]]]; Array[a, 100]

fdigsum(n) = {my(k = n, m = 2, r, s = 0); while([k, r] = divrem(k, m); k != 0 || r != 0, s += r; m++); s;}
a(n) = {my(f = fdigsum(n)); if(f == 1, 0, if(n % f, 1, 1 + a(n/f)));}

def f(n, p=2): return n if nMichael S. Branicky, Oct 27 2024

a(n) = 0 if and only if n is in A000142 (by definition).
a(n) = 1 if and only if n is in A286607.
a(n) >= 2 if and only if n is in A118363 \ A000142 (i.e., n is a factorial-base Niven number that is not a factorial number).
a(n) >= 3 if and only if n is in A377385 \ A000142.
a(n) >= 4 if and only if n is in A377386 \ A000142.
a(n) < A000005(n).

A377386 Factorial-base Niven numbers (A118363) k such that m = k/f(k) and m/f(m) are also factorial-base Niven numbers, where f(k) = A034968(k) is the sum of digits in the factorial-base representation of k.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 24, 36, 40, 48, 54, 72, 80, 96, 108, 120, 135, 144, 180, 192, 240, 280, 288, 360, 384, 432, 480, 576, 594, 600, 720, 840, 864, 1200, 1215, 1225, 1296, 1344, 1440, 1680, 1728, 1800, 2160, 2240, 2352, 2400, 2520, 2592, 2704, 2730, 2880, 3000

Offset: 1

Amiram Eldar, Oct 27 2024

			16 is a term since 16/f(16) = 4 is an integer, 4/f(4) = 2 is an integer, and 2/f(2) = 2 is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A118363 and A377385.
A000142 is a subsequence.
Cf. A034968, A377384.
Analogous sequences: A376617 (binary), A377210 (Zeckendorf).

fdigsum[n_] := Module[{k = n, m = 2, r, s = 0}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, s += r; m++]; s]; q[k_] := Module[{f = fdigsum[k], f2, m, n}, IntegerQ[m = k/f] && Divisible[m, f2 = fdigsum[m]] && Divisible[n = m/f2, fdigsum[n]]]; Select[Range[3000], q]

fdigsum(n) = {my(k = n, m = 2, r, s = 0); while([k, r] = divrem(k, m); k != 0 || r != 0, s += r; m++); s;}
is(k) = {my(f = fdigsum(k), f2, m); if(k % f, return(0)); m = k/f; f2 = fdigsum(m); !(m % f2) && !((m/f2) % fdigsum(m/f2)); }

A230419 Square array A(n,k) = difference of digit sums in factorial base representations (A007623) of n and k, n>=0, k>=0, read by antidiagonals; A(n,k) = A034968(n)-A034968(k).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Nov 10 2013

Equivalently, A(n,k) = the sum of differences of digits in matching positions of the factorial base representations (A007623) of n and k.

			The top left corner array is:
   0,  1,  1,  2,  2,  3,  1,  2,  2,  3,  3, ...
  -1,  0,  0,  1,  1,  2,  0,  1,  1,  2,  2, ...
  -1,  0,  0,  1,  1,  2,  0,  1,  1,  2,  2, ...
  -2, -1, -1,  0,  0,  1, -1,  0,  0,  1,  1, ...
  -2, -1, -1,  0,  0,  1, -1,  0,  0,  1,  1, ...
  -3, -2, -2, -1, -1,  0, -2, -1, -1,  0,  0, ...
  -1,  0,  0,  1,  1,  2,  0,  1,  1,  2,  2, ...
  -2, -1, -1,  0,  0,  1, -1,  0,  0,  1,  1, ...
  -2, -1, -1,  0,  0,  1, -1,  0,  0,  1,  1, ...
  -3, -2, -2, -1, -1,  0, -2, -1, -1,  0,  0, ...
  -3, -2, -2, -1, -1,  0, -2, -1, -1,  0,  0, ...
  ...

Antti Karttunen, The first 121 antidiagonals of the table, flattened

The topmost row: A034968 (and also the leftmost column negated).
Cf. A230415 (similar array which gives the number of differing digits).
Cf. A231713 (similar array which gives the sum of absolute differences).

A(col,row) = A034968(col)-A034968(row). [Where col is the column and row the row index of entry A(col,row)]
Equally, as a sequence, a(n) = A034968(A025581(n)) - A034968(A002262(n)).
For each entry, A(j,i) = -A(i,j), or as a sequence, a(A061579(n)) = -a(n). [The array is symmetric up to the sign of entries]
Also, for each entry A(i,j), abs(A(i,j)) <= A231713(i,j).

A319711 Sum of A034968(d) over proper divisors d of n, where A034968 gives the sum of digits in factorial base.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 4, 3, 5, 1, 7, 1, 4, 6, 6, 1, 8, 1, 10, 5, 6, 1, 11, 4, 5, 6, 9, 1, 15, 1, 10, 7, 7, 6, 15, 1, 6, 6, 16, 1, 15, 1, 13, 13, 8, 1, 16, 3, 10, 8, 9, 1, 14, 8, 14, 7, 6, 1, 25, 1, 5, 13, 13, 7, 18, 1, 13, 9, 18, 1, 21, 1, 6, 12, 12, 7, 15, 1, 25, 9, 8, 1, 26, 9, 7, 7, 20, 1, 29, 6, 16, 6, 9, 8, 21, 1, 10, 14, 19, 1, 18, 1, 15, 22

Offset: 1

Antti Karttunen, Oct 02 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to factorial base representation.

Cf. A034968, A319712, A319713.

d[n_] := Module[{k = n, m = 2, s = 0, r}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, s += r; m++]; s]; a[n_] := DivisorSum[n, d[#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Mar 05 2024 *)

A034968(n) = { my(s=0, b=2, d); while(n, d = (n%b); s += d; n = (n-d)/b; b++); (s); };
A319711(n) = sumdiv(n,d,(dA034968(d));

a(n) = Sum_{d|n, dA034968(d).

a(n) = A319712(n) - A034968(n).

A276146 a(n) = A034968(A225901(n)).

Original entry on oeis.org

0, 1, 2, 3, 1, 2, 3, 4, 5, 6, 4, 5, 2, 3, 4, 5, 3, 4, 1, 2, 3, 4, 2, 3, 4, 5, 6, 7, 5, 6, 7, 8, 9, 10, 8, 9, 6, 7, 8, 9, 7, 8, 5, 6, 7, 8, 6, 7, 3, 4, 5, 6, 4, 5, 6, 7, 8, 9, 7, 8, 5, 6, 7, 8, 6, 7, 4, 5, 6, 7, 5, 6, 2, 3, 4, 5, 3, 4, 5, 6, 7, 8, 6, 7, 4, 5, 6, 7, 5, 6, 3, 4, 5, 6, 4, 5, 1, 2, 3, 4, 2, 3, 4, 5, 6, 7, 5, 6, 3, 4, 5, 6, 4, 5, 2, 3, 4, 5, 3, 4, 5

Offset: 0

Antti Karttunen, Aug 29 2016

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

Cf. A034968, A225901.

b = MixedRadix[Reverse@ Range[2, 12]]; h[n_] := Module[{s = 0, i = 2, k = n}, While[k > 0, k = Floor[n/i!]; s = s + (i - 1) k; i++]; n - s]; Table[h@ FromDigits[Map[Boole[# > 0] &, #] (Reverse@ Range[2, Length@ # + 1] - #), b] &@ IntegerDigits[n, b], {n, 0, 120}] (* Version 10.2, or *)
f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Range[# + 1] <= n &]; Most@ Rest[a][[All, 1]] /. {} -> {0}]; g[w_List] := Total[Times @@@ Transpose@{Map[Times @@ # &, Range@ Range[0, Length@ w]], Reverse@ Append[w, 0]}]; h[n_] := Module[{s = 0, i = 2, k = n}, While[k > 0, k = Floor[n/i!]; s = s + (i - 1) k; i++]; n - s]; Table[h@ g[Map[Boole[# > 0] &, #] (Reverse@ Range[2, Length@ # + 1] - #)] &@ f@ n, {n, 0, 120}] (* Michael De Vlieger, Aug 29 2016, function h after Jean-François Alcover at A034968 *)

(define (A276146 n) (A034968 (A225901 n)))

a(n) = A034968(A225901(n)).

A200747 Number of iterations of A034968 required to reach 1.

Original entry on oeis.org

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

Offset: 1

Franklin T. Adams-Watters, Nov 27 2011

nonn

a(n) = 1 when n is a factorial > 1.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A034968.

a(n)=local(r);while(n>1,n=A034968(n);r++);r

A007623 Integers written in factorial base.

Original entry on oeis.org

0, 1, 10, 11, 20, 21, 100, 101, 110, 111, 120, 121, 200, 201, 210, 211, 220, 221, 300, 301, 310, 311, 320, 321, 1000, 1001, 1010, 1011, 1020, 1021, 1100, 1101, 1110, 1111, 1120, 1121, 1200, 1201, 1210, 1211, 1220, 1221, 1300, 1301, 1310, 1311, 1320, 1321, 2000, 2001, 2010

Offset: 0

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

Places reading from right have values (1, 2, 6, 24, 120, ...) = factorials.
Also the reversed inversion vectors for the list of all finite permutations in reversed lexicographic order: A055089.
This concatenated representation is unsatisfactory for large n (above 36287999), when coefficients of 10 or greater start to appear. For these large numbers the representation given in A108731 is better. - N. J. A. Sloane, Jun 04 2012
For n < 10*10!-1, a(n) = concatenation of n-th row of triangle in A108731. - Reinhard Zumkeller, Jun 04 2012
a(n) = A049345(n) for n=0..23. - Reinhard Zumkeller, Jan 05 2014
For n = 36288000 = 10 * 10!, the digits in factorial base are {10, 0, 0, 0, 0, 0, 0, 0, 0, 0}. - Michael De Vlieger, Oct 11 2015, corrected and edited by M. F. Hasler, Nov 27 2018
The alt text in xkcd comic #2835 describes "Numbers larger than about 3.6 million" to be illegal. See links. - David Cleaver, Sep 30 2023

			a(47) = 1321 because 47 = 1*4! + 3*3! + 2*2! + 1*1!

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 192.
F. Smarandache, Definitions solved and unsolved problems, conjectures and theorems in number theory and geometry, edited by M. Perez, Xiquan Publishing House, 2000.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. F. Hasler (terms 0 .. 1000) & Antti Karttunen, Table of n, a(n) for n = 0..40320
Italo J. Dejter, A numeral system for the middle levels, arXiv:1012.0995 [math.CO], 2010-2015.
Italo J. Dejter, Dihedral-symmetry middle-levels problem via a Catalan system of numeration, preprint, 2015.
Italo J. Dejter, Ordering the Levels Lk and Lk+1 of B2k+1, preprint, 2015-2017.
P. Hecht, Post-Quantum Cryptography: S_381 Cyclic Subgroup of High Order, International Journal of Advanced Engineering Research and Science (IJAERS, 2017) Vol. 4, Issue 6, 78-86.
C. A. Laisant, Sur la numération factorielle, application aux permutations, Bulletin de la Société Mathématique de France, 16 (1888), p. 176-183.
Randall Munroe, Factorial Numbers, xkcd Web Comic #2835, Sep 29 2023.
Wikipedia, Factorial base
Index entries for sequences related to factorial base representation

Cf. A000142, A034968 (sum of digits), A060130 (number of nonzero digits), A099563 (the most significant digit).
Cf. also A055089, A055881, A060112, A060495. Permutation of A064039.
See index entry "factorial base representation" for many more related sequences.
See also primorial base A049345.

a007623 n | n <= 36287999 = read $ concatMap show (a108731_row n) :: Int
          | otherwise     = error "representation would be ambiguous"
-- Reinhard Zumkeller, Jun 04 2012
(Scheme, R6RS standard) (define (A007623 n) (let loop ((n n) (s 0) (p 1) (i 2)) (if (zero? n) s (let ((d (mod n i))) (loop (/ (- n d) i) (+ (* p d) s) (* 10 p) (+ 1 i)))))) ;; In older Schemes use modulo instead of mod. - Antti Karttunen, Feb 13 2016

a := n -> if nargs<2 then a(n,2) elif n

factBaseIntDs[n_] := Module[{m, i, len, dList, currDigit}, i = 1; While[n > i!, i++ ]; m = n; len = i; dList = Table[0, {len}]; Do[ currDigit = 0; While[m >= j!, m = m - j!; currDigit++ ]; dList[[len - j + 1]] = currDigit, {j, i, 1, -1}]; If[dList[[1]] == 0, dList = Drop[dList, 1]]; dList]; Table[FromDigits[factBaseIntDs[n]], {n, 0, 50}] (* Alonso del Arte, May 03 2006 *)
lim = 50; m = 1; While[Factorial@ m < lim, m++]; m; IntegerDigits[#, MixedRadix[Reverse@ Range[2, m]]] & /@ Range@ lim (* Michael De Vlieger, Oct 11 2015, Version 10.2 *)

apply( a(n,p=2)=if(nM. F. Hasler, Mar 27 2007; minor edit Nov 26 2018

def a(n, p=2): return n if nIndranil Ghosh, Jun 19 2017, after PARI program
(APL, Dyalog dialect) A007623 ← {⍺←2 ⋄ ⍵<⍺: ⍵ ⋄ (⍺|⍵) + ((⍺+1) ∇ ⍺(⌊÷⍨)⍵)×10} ⍝ (code from Dani Adham) - Antti Karttunen, Feb 20 2024

More terms from R. K. Guy

A055089 List of all finite permutations in reversed colexicographic ordering.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Apr 18 2000

			In this table, each row consists of A001563(n) permutations of n+1 terms; i.e., we have (1/) 2,1/ 1,3,2; 3,1,2; 2,3,1; 3,2,1/ 1,2,4,3; 2,1,4,3; ... .
Append to each an infinite number of fixed terms and we get a list of rearrangements of the natural numbers, but with only a finite number of terms permuted:
1/2,3,4,5,6,7,8,9,...
2,1/3,4,5,6,7,8,9,...
1,3,2/4,5,6,7,8,9,...
3,1,2/4,5,6,7,8,9,...
2,3,1/4,5,6,7,8,9,...
3,2,1/4,5,6,7,8,9,...
1,2,4,3/5,6,7,8,9,...
2,1,4,3/5,6,7,8,9,...
Alternatively, if we take only the first n terms of each such infinite row, then the first n! rows give all permutations of the elements 1,2,...,n.

Antti Karttunen, Rows 0..719 of the irregular table, flattened.
Daniel Forgues, Tilman Piesk, et al., Orderings, OEIS Wiki.
Antti Karttunen, Ranking and unranking functions, OEIS Wiki.
Tilman Piesk, Permutations and partitions in the OEIS, Wikiversity.
Lorenzo Sauras-Altuzarra, Some arithmetical problems that are obtained by analyzing proofs and infinite graphs, arXiv:2002.03075 [math.NT], 2020.
Index entries for sequences related to factorial base representation
Index entries for sequences related to permutations

Inversion vectors: A007623, cycle counts: A055090, minimum number of transpositions: A055091, minimum number of adjacent transpositions: A034968, order of each permutation: A055092, number of non-fixed elements: A055093, positions of inverses: A056019, positions after Foata transform: A065181; positions of fixed-point-free involutions: A064640.
Cf. A057112, A060112, A060117, A060118, A060132, A060133.
Cf. A195663, array of the infinite rows.
This permutation list gives essentially the same information as A030298/A030299, but in a more compact way, by skipping those permutations of A030298 that start with a fixed element.
A220658(n) gives the rank r of the permutation of which the term at a(n) is an element.
A220659(n) gives the zero-based position (from the left) of that a(n) in that permutation of rank r.
A084558(r)+1 gives the size of the finite subsequence (of the r-th infinite, but finitary permutation) which has been included in this list.

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;
fexlist2permlist := proc(a) local n,b,j; n := nops(a); if(0 = n) then RETURN([1]); fi; b := fexlist2permlist(cdr(a)); for j from 1 to n do if(b[j] >= ((n+1)-a[1])) then b[j] := b[j]+1; fi; od; RETURN([op(b),(n+1)-a[1]]); end;
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;
PermRevLexUnrank := n -> `if`((0 = n),[1],fexlist2permlist(fac_base(n)));
cdr := proc(l) if 0 = nops(l) then ([]) else (l[2..nops(l)]); fi; end; # "the tail of the list"
# Same algorithm in different guise, showing how permutations are composed of adjacent transpositions (compare to algorithm PermUnrank3R at A060117):
PermRevLexUnrankAMSDaux := proc(n,r, pp) local s,p,k; p := pp; if(0 = r) then RETURN(p); else s := floor(r/((n-1)!)); for k from n-s to n-1 do p := permul(p,[[k,k+1]]); od; RETURN(PermRevLexUnrankAMSDaux(n-1, r-(s*((n-1)!)), p)); fi; end;
PermRevLexUnrankAMSD := proc(r) local n; n := nops(factorial_base(r)); convert(PermRevLexUnrankAMSDaux(n+1,r,[]),'permlist',1+(((r+2) mod (r+1))*n)); end;

A055089L[n_] := Reverse@SortBy[DeleteCases[Permutations@Range@n, {, n}], Reverse]; Flatten@Array[A055089L, 4] (* JungHwan Min, Aug 28 2016 *)

[seq(op(PermRevLexUnrank(j)), j=0..)]; (see Maple code given below).

Name changed by Tilman Piesk, Feb 01 2012

A276150 Sum of digits when n is written in primorial base (A049345); minimal number of primorials (A002110) that add to n.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 7, 8, 8, 9, 9, 10, 4

Offset: 0

Antti Karttunen, Aug 22 2016

The sum of digits of n in primorial base is odd if n is 1 or 2 (mod 4) and even if n is 0 or 3 (mod 4). Proof: primorials are 1 or 2 (mod 4) and a(n) can be constructed via the greedy algorithm. So if n = 4k + r where 0 <= r < 4, 4k needs an even number of primorials and r needs hammingweight(r) = A000120(r) primorials. Q.E.D. - David A. Corneth, Feb 27 2019

			For n=24, which is "400" in primorial base (as 24 = 4*(3*2*1) + 0*(2*1) + 0*1, see A049345), the sum of digits is 4, thus a(24) = 4.

Antti Karttunen, Table of n, a(n) for n = 0..30030
Index entries for sequences related to primorial base

Cf. A000120, A001222, A002110, A049345, A053589, A235168, A260188, A267263, A276084, A276086, A276151, A277022, A278226, A283477, A319713, A319715 (inverse Möbius transform), A321683, A324342, A324382, A324383, A324386, A324387, A371091, A373605, A373606, A373607.
Cf. A333426 [k such that a(k)|k], A339215 [numbers not of the form x+a(x) for any x], A358977 [k such that gcd(k, a(k)) = 1].
Cf. A338835, A366693.
Cf. A014601, A042963 (positions of even and odd terms), A343048 (positions of records).
Differs from analogous A034968 for the first time at n=24.

nn = 120; b = MixedRadix[Reverse@ Prime@ NestWhileList[# + 1 &, 1, Times @@ Prime@ Range[# + 1] <= nn &]]; Table[Total@ IntegerDigits[n, b], {n, 0, nn}] (* Version 10.2, or *)
nn = 120; f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Prime@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Prime@ Range[# + 1] <= n &]; Rest[a][[All, 1]]]; Table[Total@ f@ n, {n, 0, 120}] (* Michael De Vlieger, Aug 26 2016 *)

A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); }; \\ Antti Karttunen, Feb 27 2019

from sympy import prime, primefactors
def Omega(n): return 0 if n==1 else Omega(n//primefactors(n)[0]) + 1
def a276086(n):
    i=0
    m=pr=1
    while n>0:
        i+=1
        N=prime(i)*pr
        if n%N!=0:
            m*=(prime(i)**((n%N)/pr))
            n-=n%N
        pr=N
    return m
def a(n): return Omega(a276086(n))
print([a(n) for n in range(201)]) # Indranil Ghosh, Jun 23 2017

a(n) = 1 + a(A276151(n)) = 1 + a(n-A002110(A276084(n))), a(0) = 0.
or for n >= 1: a(n) = 1 + a(n-A260188(n)).
Other identities and observations. For all n >= 0:
a(n) = A001222(A276086(n)) = A001222(A278226(n)).
a(n) >= A371091(n) >= A267263(n).
From Antti Karttunen, Feb 27 2019: (Start)
a(n) = A000120(A277022(n)).
a(A283477(n)) = A324342(n).
(End)
a(n) = A373606(n) + A373607(n). - Antti Karttunen, Jun 19 2024

cp's OEIS Frontend

A358976 Numbers that are coprime to the sum of their factorial base digits (A034968).

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A377384 a(n) is the number of iterations that n requires to reach a noninteger or a factorial number under the map x -> x / f(x), where f(k) = A034968(k) is the sum of digits in the factorial-base representation of k; a(n) = 0 if n is a factorial number.

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A377386 Factorial-base Niven numbers (A118363) k such that m = k/f(k) and m/f(m) are also factorial-base Niven numbers, where f(k) = A034968(k) is the sum of digits in the factorial-base representation of k.

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A230419 Square array A(n,k) = difference of digit sums in factorial base representations (A007623) of n and k, n>=0, k>=0, read by antidiagonals; A(n,k) = A034968(n)-A034968(k).

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A319711 Sum of A034968(d) over proper divisors d of n, where A034968 gives the sum of digits in factorial base.

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A276146 a(n) = A034968(A225901(n)).

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A200747 Number of iterations of A034968 required to reach 1.

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A007623 Integers written in factorial base.

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