A276332 -id:A276332 - cp's OEIS frontend

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

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.

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).

A276331 a(0) = 0 and for n >= 1, a(n) = n minus the largest term of A001563 that divides n.

Original entry on oeis.org

0, 0, 1, 2, 0, 4, 5, 6, 4, 8, 9, 10, 8, 12, 13, 14, 12, 16, 0, 18, 16, 20, 21, 22, 20, 24, 25, 26, 24, 28, 29, 30, 28, 32, 33, 34, 18, 36, 37, 38, 36, 40, 41, 42, 40, 44, 45, 46, 44, 48, 49, 50, 48, 52, 36, 54, 52, 56, 57, 58, 56, 60, 61, 62, 60, 64, 65, 66, 64, 68, 69, 70, 54, 72, 73, 74, 72, 76, 77, 78, 76, 80, 81, 82, 80

Offset: 0

Antti Karttunen, Aug 30 2016

nonn

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

Cf. A001563, A276330, A276332.

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[n - # #! &[Length@ TakeWhile[Reverse@ f@ n, # == 0 &] + 1], {n, 84}] (* Michael De Vlieger, Aug 31 2016 *)

Scheme
```
(define (A276331 n) (- n (A276330 n)))
```

a(n) = n - A276330(n).

cp's OEIS Frontend

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

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

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A276331 a(0) = 0 and for n >= 1, a(n) = n minus the largest term of A001563 that divides n.

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