%I #64 Jun 19 2024 16:17:42

%S 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,

%T 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,

%U 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

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

%C 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

%H Antti Karttunen, <a href="/A276150/b276150.txt">Table of n, a(n) for n = 0..30030</a>

%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>

%F a(n) = 1 + a(A276151(n)) = 1 + a(n-A002110(A276084(n))), a(0) = 0.

%F or for n >= 1: a(n) = 1 + a(n-A260188(n)).

%F Other identities and observations. For all n >= 0:

%F a(n) = A001222(A276086(n)) = A001222(A278226(n)).

%F a(n) >= A371091(n) >= A267263(n).

%F From _Antti Karttunen_, Feb 27 2019: (Start)

%F a(n) = A000120(A277022(n)).

%F a(A283477(n)) = A324342(n).

%F (End)

%F a(n) = A373606(n) + A373607(n). - _Antti Karttunen_, Jun 19 2024

%e 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.

%t 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 *)

%t 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 *)

%o (Scheme, two versions)

%o (definec (A276150 n) (if (zero? n) 0 (+ 1 (A276150 (- n (A002110 (A276084 n)))))))

%o (define (A276150 n) (A001222 (A276086 n)))

%o (Python)

%o from sympy import prime, primefactors

%o def Omega(n): return 0 if n==1 else Omega(n//primefactors(n)[0]) + 1

%o def a276086(n):

%o i=0

%o m=pr=1

%o while n>0:

%o i+=1

%o N=prime(i)*pr

%o if n%N!=0:

%o m*=(prime(i)**((n%N)/pr))

%o n-=n%N

%o pr=N

%o return m

%o def a(n): return Omega(a276086(n))

%o print([a(n) for n in range(201)]) # _Indranil Ghosh_, Jun 23 2017

%o (PARI) 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

%Y 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.

%Y 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].

%Y Cf. A338835, A366693.

%Y Cf. A014601, A042963 (positions of even and odd terms), A343048 (positions of records).

%Y Differs from analogous A034968 for the first time at n=24.

%K nonn,look,base

%O 0,4

%A _Antti Karttunen_, Aug 22 2016