A328483 - OEIS

%I #26 Nov 23 2019 14:54:41

%S 0,1,1,1,2,2,1,1,1,1,2,2,1,1,1,1,2,2,1,1,1,1,2,2,2,2,2,2,2,2,1,1,1,1,

%T 2,2,1,1,1,1,2,2,1,1,1,1,2,2,1,1,1,1,2,2,2,2,2,2,2,2,1,1,1,1,2,2,1,1,

%U 1,1,2,2,1,1,1,1,2,2,1,1,1,1,2,2,2,2,2,2,2,2,1,1,1,1,2,2,1,1,1,1,2,2,1,1,1,1

%N Maximum number of times any term appears when n is expressed as a greedy sum of terms of A129912 (maximal digit when n is expressed in greedy A129912-base).

%C Apparently no term is larger than 2.

%C In the initial prefix of 30031 terms, the longest run of 1's is 4 and 2's occur only in runs of lengths 2, 8, 38, 68, 218 and 428. - _Bill McEachen_, Nov 21 2019, clarified by _Antti Karttunen_, Nov 23 2019

%H Antti Karttunen, <a href="/A328483/b328483.txt">Table of n, a(n) for n = 0..30030</a>

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

%F a(A129912(n)) = a(A002110(n)) = 1.

%e Terms of A129912 (numbers that are products of distinct primorial numbers) begin as: 1, 2, 6, 12, 30, 60, 180, 210, 360, 420, 1260, ...

%e Number 5 is expressed as 5 = 2 + 2 + 1 when always choosing the largest term which is <= {what is remaining of the original number}. Thus a(5) = 2 as the most frequent term (2) occurs twice.

%e Number 21 is expressed as 21 = 12 + 6 + 2 + 1, thus a(21) = 1 as no term occurs more than once.

%e Number 720 is expressed as 720 = 420 + 210 + 60 + 30, thus a(720) = 1 as no term occurs twice. Note that 720 = 2*360, so an algorithm which would search for an optimal result would yield a different value at n=720.

%o (PARI)

%o isA129912(n) = { my(o=valuation(n, 2), t); if(o<1||n<2, return(n==1)); n>>=o; forprime(p=3, , t=valuation(n, p); n/=p^t; if(t>o || t<o-1, return(0)); if(t==0, return(n==1)); o=t); }; \\ From A129912

%o prepare_A129912_upto(n) = { my(xs=List([]), k=0); while(k<n, k++; if(isA129912(k), listput(xs,k))); List(Vecrev(xs)); };

%o max_factor_of_terms_in_greedy_sum(n,terms) = { my(m=0); while(n,if(terms[1] > n, listpop(terms,1), m = max(m,(n\terms[1])); n %= terms[1])); (m); };

%o A328483(n) = max_factor_of_terms_in_greedy_sum(n,prepare_A129912_upto(n));

%Y Cf. A002110, A129912, A328481, A328482.

%K nonn

%O 0,5

%A _Antti Karttunen_, Oct 19 2019