A372557 -id:A372557 - cp's OEIS frontend

A265745 a(n) is the number of Jacobsthal numbers (A001045) needed to sum to n using the greedy algorithm.

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

Offset: 0

Antti Karttunen, Dec 17 2015

Sum of digits in "Jacobsthal greedy base", A265747.
It would be nice to know for sure whether this sequence gives also the least number of Jacobsthal numbers that add to n, i.e., that there cannot be even better nongreedy solutions.
The integer 63=21+21+21 has 3 for its 'non-greedy' solution, and a(63) = 5 for its greedy solution 63=43+11+5+3+1. - Yuriko Suwa, Jul 11 2021
Positions where a(n) is different from A372555(n) are n=63, 84, 148, 169, 191, 212, 234, 255, etc. See A372557. - Antti Karttunen, May 07 2024

			a(0) = 0, because no numbers are needed to form an empty sum, which is zero.
For n=1 we need just A001045(2) = 1, thus a(1) = 1.
For n=2 we need A001045(2) + A001045(2) = 1 + 1, thus a(2) = 2.
For n=4 we need A001045(3) + A001045(2) = 3 + 1, thus a(4) = 2.
For n=6 we form the greedy sum as A001045(4) + A001045(2) = 5 + 1, thus a(6) = 2. Alternatively, we could form the sum as A001045(3) + A001045(3) = 3 + 3, but the number of summands in that case is no less.
For n=7 we need A001045(4) + A001045(2) + A001045(2) = 5 + 1 + 1, thus a(7) = 3.
For n=8 we need A001045(4) + A001045(3) = 5 + 3, thus a(8) = 2.
For n=10 we need A001045(4) + A001045(4) = 5 + 5, thus a(10) = 2.

Antti Karttunen, Table of n, a(n) for n = 0..10923

Cf. A001045, A130249, A197911, A003158, A265746, A265747, A364377, A364379, A364380, A372288, A372555, A372557.
Cf. A054111 (apparently the positions of the first occurrence of each n > 0).
Cf. also A007895, A014420, A053610, A265404, A265743, A265744.

jacob[n_] := (2^n - (-1)^n)/3; maxInd[n_] := Floor[Log2[3*n + 1]]; A265745[n_] := A265745[n] = 1 + A265745[n - jacob[maxInd[n]]]; A265745[0] = 0; Array[A265745, 100, 0] (* Amiram Eldar, Jul 21 2023 *)

A130249(n) = floor(log(3*n + 1)/log(2));
A001045(n) = (2^n - (-1)^n) / 3;
A265745(n) = {if(n == 0, 0, my(d = n - A001045(A130249(n))); if(d == 0, 1, 1 + A265745(d)));} \\ Amiram Eldar, Jul 21 2023

def greedyJ(n): n1 = (3*n+1).bit_length() - 1; return (2**n1 - (-1)**n1)//3
def a(n): return 0 if n == 0 else 1 + a(n - greedyJ(n))
print([a(n) for n in range(107)]) # Michael S. Branicky, Jul 11 2021

a(0) = 0; for n >= 1, a(n) = 1 + a(n - A001045(A130249(n))). [This formula uses a simple greedy algorithm.]

A372555 Least number of Jacobsthal numbers that add up to n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 07 2024

nonn

Differs from A265745 for the first time at n=63, where a(63) = 3, while A265745(63) = 5. The next differences occur at n=84, 148, 169, 191, 212, 234, 255, etc. See A372557.

See conjecture in A372556, and also in A372561.

			a(5) = 1, because 5 is itself in A001045.
a(7) = 3, because 7 can be expressed as a sum of three Jacobsthal numbers, either as 5+1+1 or 3+3+1, but not as a sum of two Jacobsthal numbers, and neither 7 is itself in A001045.
a(63) = 3, because the least number of Jacobsthal numbers that add up to 63 is obtained when we use A001045(6) = 21 three times, as 21+21+21 = 63. This is the first time this sequence differs from A265745.

Antti Karttunen, Table of n, a(n) for n = 0..87381

Cf. A001045, A130249, A147612, A372556, A372557, A372561.

Cf. also A265745.

up_to = 87381; \\ = A001045(18).
A001045(n) = (2^n - (-1)^n) / 3;
A130249(n) = (#binary(3*n+1)-1);
A372555_or_556list(up_to_n,return_556_instead) = { my(v372555 = vector(up_to_n), v372556 = vector(up_to_n)); v372555[1] = 1; v372556[1] = 2; for(n=2,#v372556, my(m=-1,mk=-1,s=A130249(n)); if(A001045(s)==n, v372555[n] = 1; v372556[n] = s, forstep(k=s, 1, -1, my(c=v372555[n-A001045(k)]); if(m<0 || cA001045(mk)])); if(return_556_instead,v372556,v372555); };
v372555 = A372555_or_556list(up_to,0);
A372555(n) = if(!n,n,v372555[n]);

;; An implementation of memoization-macro definec can be found for example in: http://oeis.org/wiki/Memoization
(definec (A001045 n) (if (<= n 1) n (+ (A001045 (- n 1)) (* 2 (A001045 (- n 2))))))
(define (A130249 n) (floor->exact (/ (log (+ 1 (* 3 n))) (log 2))))
(define (A147612 n) (if (<= n 1) 1 (if (= (A001045 (A130249 n)) n) 1 0)))
(definec (A372555 n) (if (<= n 1) n (+ 1 (A372555 (- n (A001045 (A372556 n)))))))
(definec (A372556 n) (let ((k (A130249 n))) (if (= 1 (A147612 n)) k (let loop ((k k) (m #f) (mk #f)) (cond ((zero? k) mk) (else (let* ((c (A372555 (- n (A001045 k))))) (if (or (not m) (< c m)) (loop (- k 1) c k) (loop (- k 1) m mk)))))))))

a(0) = 0, a(1) = 1; for n > 1, a(n) = 1 + a(n-A001045(A372556(n))).

A372558 Numbers k such that A130249(k) differs from A372556(k).

Original entry on oeis.org

63, 84, 191, 212, 255, 276, 297, 340, 703, 724, 767, 788, 809, 852, 937, 1022, 1023, 1044, 1065, 1108, 1129, 1150, 1193, 1278, 1363, 1364, 2751, 2772, 2815, 2836, 2857, 2900, 2985, 3070, 3071, 3092, 3113, 3156, 3177, 3198, 3241, 3326, 3411, 3412, 3497, 3582, 3667, 3752, 3753, 3838, 3923, 4008, 4093, 4094, 4095

Offset: 1

Antti Karttunen, May 07 2024

nonn

			From the terms of A372557 both 63 and 84 are included here, because the largest summand in their non-greedy solutions is different from the largest summand in their greedy solutions, while 169 is NOT included because the largest summand in both cases is 85. See examples in A372557.

Antti Karttunen, Table of n, a(n) for n = 1..24118

Cf. A001045, A130249, A372555, A372556.

Subsequence of A372557.

A130249(n) = (#binary(3*n+1)-1);
isA372558(k) = (A372556(k)!=A130249(k)); \\ Uses also the program from A372556.

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A265745 a(n) is the number of Jacobsthal numbers (A001045) needed to sum to n using the greedy algorithm.

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A372555 Least number of Jacobsthal numbers that add up to n.

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A372558 Numbers k such that A130249(k) differs from A372556(k).

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