A338835 -id:A338835 - cp's OEIS frontend

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

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

A338829 a(n) is the greatest number not yet in the sequence with the same number of digits and the same sum of digits as n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 91, 11, 21, 31, 41, 51, 61, 71, 81, 82, 92, 12, 22, 32, 42, 52, 62, 72, 73, 83, 93, 13, 23, 33, 43, 53, 63, 64, 74, 84, 94, 14, 24, 34, 44, 54, 55, 65, 75, 85, 95, 15, 25, 35, 45, 46, 56, 66, 76

Offset: 0

Rémy Sigrist, Nov 11 2020

This sequence is a self-inverse permutation of the nonnegative integers.

We have a fixed point with m digits and sum of digits k whenever A289410(m, k) is odd.

			For n = 23:
- the numbers with 2 digits and sum of digits 5 are: 14, 23, 32, 41 and 50,
- so  a(14) = 50,
      a(23) = 41,
      a(32) = 32,
      a(41) = 23,
      a(50) = 14.

Rémy Sigrist, Table of n, a(n) for n = 0..9999
Rémy Sigrist, Colored scatterplot of the first 10000 terms (where the color is function of the sum of digits of n)
Rémy Sigrist, PARI program for A338829
Index entries for sequences that are permutations of the natural numbers

Cf. A055642, A289410, A331274 (binary analog), A333659, A338834 (factorial base analog), A338835 (primorial base analog).

Block[{a = {}, f, k}, f[x_] := Total@ IntegerDigits@ x; Do[k = f[i]; AppendTo[a, SelectFirst[Range[10^# - 1, 10^(# - 1), -1] &@ Floor[1 + Log10[i]], And[f[#] == k, FreeQ[a, #]] &]], {i, 67}]; a] (* Michael De Vlieger, Nov 13 2020 *)

PARI
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See Links section.
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A055642(a(n)) = A055642(n).

A007953(a(n)) = A007953(n).

A338834 a(n) is the greatest number not yet in the sequence with the same number of digits and the same sum of digits as n in factorial base.

Original entry on oeis.org

0, 1, 2, 4, 3, 5, 6, 12, 8, 18, 14, 20, 7, 13, 10, 19, 16, 22, 9, 15, 11, 21, 17, 23, 24, 48, 30, 72, 54, 96, 26, 50, 49, 78, 74, 102, 36, 73, 60, 98, 97, 108, 56, 84, 80, 104, 103, 114, 25, 32, 31, 55, 52, 79, 28, 51, 42, 76, 75, 100, 38, 66, 62, 99, 90, 110

Offset: 0

Rémy Sigrist, Nov 11 2020

This sequence is a self-inverse permutation of the nonnegative integers.

			For n = 26:
- the numbers with 4 digits and sum of digits 2 in factorial base are: 25 ("1001"), 26 ("1010"), 30 ("1100") and 48 ("2000"),
- so a(25) = 40,
     a(26) = 30,
     a(30) = 26,
     a(40) = 25.

Rémy Sigrist, Table of n, a(n) for n = 0..5039
Rémy Sigrist, Colored scatterplot of the first 7! terms (where the color is function of A034968(n))
Rémy Sigrist, PARI program for A338834
Index entries for sequences that are permutations of the natural numbers

Cf. A034968, A084558, A337598, A338829 (decimal analog), A338835 (primorial base analog).

PARI
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See Links section.
```

A084558(a(n)) = A084558(n).

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

cp's OEIS Frontend

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

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A338829 a(n) is the greatest number not yet in the sequence with the same number of digits and the same sum of digits as n.

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Mathematica

PARI

Formula

A338834 a(n) is the greatest number not yet in the sequence with the same number of digits and the same sum of digits as n in factorial base.

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