A328481 Number of terms required when n is expressed as a greedy sum of terms of A129912 (digital sum of n when written in greedy A129912-base).

0, 1, 1, 2, 2, 3, 1, 2, 2, 3, 3, 4, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 2, 3, 3, 4, 4, 5, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 3, 4, 4, 5, 5, 6, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 3, 4, 4, 5, 5, 6, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 3, 4, 4, 5

Offset

0,4

Links

Antti Karttunen, Table of n, a(n) for n = 0..30030

Index entries for sequences related to primorial numbers

Formula

a(0) = 0; and for n > 0, a(n) = 1 + a(A328480(n)).

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

For all n, a(n) >= A328482(n).

Example

Terms of A129912 (numbers that are products of distinct primorial numbers) begin as: 1, 2, 6, 12, 30, 60, 180, 210, 360, 420, 1260, ...
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) = 3.
Number 21 is expressed as 21 = 12 + 6 + 2 + 1, thus a(21) = 4.
Number 720 is expressed as 720 = 420 + 210 + 60 + 30, thus a(720) = 4. Note that 720 = 2*360, so in this case the greedy algorithm does not produce an optimal result.

Prog

(PARI)
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
prepare_A129912_upto(n) = { my(xs=List([]), k=0); while(k<n, k++; if(isA129912(k), listput(xs, k))); List(Vecrev(xs)); };
number_of_terms_in_greedy_sum(n, terms) = { my(c=0); while(n, if(terms[1] > n, listpop(terms, 1), c += (n\terms[1]); n %= terms[1])); (c); };
number_of_terms_in_greedy_sum_v1(n, terms) = { my(c=0); while(n, if(terms[1] > n, listpop(terms, 1), n -= terms[1]; c++)); (c); }; \\ (Simpler variant)
A328481(n) = number_of_terms_in_greedy_sum(n, prepare_A129912_upto(n));

Crossrefs

Cf. A002110, A129912, A328480, A328482, A328483.

Cf. also A276150.

Sequence in context: A092331 A299029 A200747 * A089293 A034968 A341513

Adjacent sequences: A328478 A328479 A328480 * A328482 A328483 A328484

Keyword

nonn

Author

Antti Karttunen, Oct 19 2019

Status

approved