A258199 -id:A258199 - cp's OEIS frontend

A276333 The most significant digit in greedy A001563-base (A276326): a(n) = floor(n/A258199(n)), a(0) = 0.

0, 1, 2, 3, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 1

Offset: 0

Antti Karttunen, Aug 30 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..4320

Cf. A001563, A276326, A258199, A276334, A276335.

Table[Floor[n/(# #!)] &@ NestWhile[# + 1 &, 0, # #! <= n &[# + 1] &], {n, 96}] (* or *)
f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@#, Last@#} &@ QuotientRemainder[a[[-1, -1]], (# #!) &[# - i]]], {i, 0, # - 1}] &@ NestWhile[# + 1 &, 0, (# #!) &[# + 1] <= n &]; Rest[a][[All, 1]]]; {0}~Join~Table[First@ f@ n, {n, 96}] (* Michael De Vlieger, Aug 31 2016 *)

(define (A276333 n) (if (zero? n) n (floor->exact (/ n (A258199 n)))))

a(0) = 0; for n >= 1, a(n) = floor(n/A258199(n)).

A276334 a(n) = A258199(n) * A276333(n).

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 4, 4, 8, 8, 8, 8, 12, 12, 12, 12, 16, 16, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 72, 72, 72, 72, 72, 72, 72, 72

Offset: 0

Antti Karttunen, Aug 30 2016

a(n) is obtained by first replacing with zeros all other digits except the leftmost (the most significant) in the greedy A001563-base representation of n (A276326), then converting back to decimal. Used to compute A276335.

Antti Karttunen, Table of n, a(n) for n = 0..4320

Cf. A001563, A258199, A276332, A276333, A276335.

Table[# Floor[n/#] &@ (# #!) &@ NestWhile[# + 1 &, 0, # #! <= n &[# + 1] &], {n, 84}] (* Michael De Vlieger, Aug 31 2016 *)

(define (A276334 n) (* (A258199 n) (A276333 n)))

a(n) = A258199(n) * A276333(n).

A276335(n) = n - a(n).

A256450 Numbers that have at least one 1-digit in their factorial base representation (A007623).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 92, 93, 95, 97, 98, 99, 101

Offset: 0

Antti Karttunen, Apr 27 2015

Numbers n for which A257679(n) = 1, i.e., numbers n such that the least nonzero digit in their factorial base representation (A007623) is 1.
Involution A225901 maps each term of this sequence to a unique term of A273670, and vice versa.
Starting offset is zero (with a(0) = 1) because it is the most natural offset for the given fast recurrence.

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

Cf. A007623, A257679.
Complement of A255411.
Cf. A257680 (characteristic function), A273662 (left inverse).
First row of A257503, first column of A257505.
Subsequences: A059590 (apart from its zero-term), A255341, A255342, A255343, A257262, A257263, A258198, A258199.
Cf. also A227187 (numbers with at least one nonleading zero) and A273670, A225901.

Select[Range@ 101, MemberQ[IntegerDigits[#, MixedRadix[Reverse@ Range@ 12]], 1] &] (* Michael De Vlieger, May 30 2016, Version 10.2 *)
r = MixedRadix[Reverse@ Range[2, 12]]; Select[Range@ 101, Min[IntegerDigits[#, r] /. 0 -> Nothing] == 1 &]  (* Michael De Vlieger, Aug 14 2016, Version 10.2 *)

def A(n, p=2): return n if n=1]) # Indranil Ghosh, Jun 19 2017

a(0) = 1, and for n >= 1, if A257511(1+a(n-1)) > 0, then a(n) = a(n-1) + 1, otherwise a(n-1) + 2. [In particular, if the previous term is 2k, then the next term is 2k+1, because all odd numbers are members.]
Other identities:
For all n >= 0, A273662(a(n)) = n. [A273662 works as the left inverse for this sequence.]
From Antti Karttunen, May 26 2015: (Start)
Alternative recurrence for the same sequence:
Set k = A258198(n), d = n - A258199(n) and f = A000142(k+1) = (k+1)! If d < f then b(n) = f+d, otherwise b(n) = ((2+floor((d-f)/A258199(n))) * f) + b((d-f) mod A258199(n)). For offset=1 sequence, define a(n) = b(n-1).
(End)

Starting offset changed from 1 to 0 by Antti Karttunen, May 30 2016

A258198 a(n) = largest k for which A001563(k) = k*k! <= n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 23 2015

nonn

Number of nonzero terms of A001563 <= n.

Each n occurs A001564(n) times.

Antti Karttunen, Table of n, a(n) for n = 0..4320

Cf. A001563, A001564, A084556, A084557, A256450, A258199.

(define (A258198 n) (let loop ((k 1) (f 1)) (if (> (* k f) n) (- k 1) (loop (+ k 1) (* (+ k 1) f)))))

A276328 Digit sum when n is expressed in greedy A001563-base (A276326).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Aug 30 2016

a(n) is the number of terms of A001563 needed to sum to n using the greedy algorithm.

This seems to give also the minimal number of terms of A001563 that sum to n (checked empirically up to n=3265920), but it would be nice to know for sure whether this holds for all n.

			For n=1, the largest term of A001563 <= 1 is A001563(1) = 1, thus a(1) = 1.
For n=2, the largest term of A001563 <= 2 is A001563(1) = 1, thus a(2) = 1 + a(2-1) = 2.
For n=18, the largest term of A001563 <= 18 is A001563(3) = 18, thus a(18) = 1.
For n=20, the largest term of A001563 <= 20 is A001563(3) = 18, thus a(20) = 1 + a(20-18) = 3.
For n=36, the largest term of A001563 <= 36 is A001563(3) = 18, thus a(36) = 1 + a(18) = 2.

Antti Karttunen, Table of n, a(n) for n = 0..4320

Cf. A001563, A258199.
Cf. A000120, A276329, A276330, A276332, A276333, A276335.
Cf. A276091 (gives all n for which a(n) = A276337(n)).
Cf. also A007895, A034968, A265744, A265745 for similar sequences.

f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], (# #!) &[# - i]]], {i, 0, # - 1}] &@NestWhile[# + 1 &, 0, (# #!) &[# + 1] <= n &]; Rest[a][[All, 1]]]; {0}~Join~Table[Total@ f@ n, {n, 120}] (* Michael De Vlieger, Aug 31 2016 *)

a(0) = 0; for n >= 1, a(n) = 1 + a(n-A258199(n)).
a(0) = 0; for n >= 1, a(n) = A276333(n) + a(A276335(n)).
Other identities and observations. For all n >= 0:
a(A276091(n)) = A000120(n).
a(n) >= A276337(n).
It also seems that a(n) <= A276332(n) for all n.

A276330 a(n) = largest term of A001563 that divides n, a(0) = 0.

Original entry on oeis.org

0, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 18, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 18, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 18, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 18, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 18, 1, 4, 1, 1, 1, 96

Offset: 0

Antti Karttunen, Aug 30 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..4319

Cf. A001563, A258199, A276328, A276329, A276331.

f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], (# #!) &[# - i]]], {i, 0, # - 1}] &@ NestWhile[# + 1 &, 0, (# #!) &[# + 1] <= n &]; Rest[a][[All, 1]]]; {0}~Join~Table[# #! &[Length@ TakeWhile[Reverse@ f@ n, # == 0 &] + 1], {n, 120}] (* Michael De Vlieger, Aug 31 2016 *)

(define (A276330 n) (if (zero? n) n (A001563 (A276329 n))))

a(0) = 0; for n >= 1, a(n) = A001563(A276329(n)).
Other identities. For all n >= 0:
A276331(n) = n - a(n).

cp's OEIS Frontend

A276333 The most significant digit in greedy A001563-base (A276326): a(n) = floor(n/A258199(n)), a(0) = 0.

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A276334 a(n) = A258199(n) * A276333(n).

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A256450 Numbers that have at least one 1-digit in their factorial base representation (A007623).

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A258198 a(n) = largest k for which A001563(k) = k*k! <= n.

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A276328 Digit sum when n is expressed in greedy A001563-base (A276326).

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A276330 a(n) = largest term of A001563 that divides n, a(0) = 0.

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