A265745 -id:A265745 - cp's OEIS frontend

A265404 a(n) = number of Spironacci numbers (A078510) needed to sum to n using the greedy algorithm.

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2

Offset: 0

Antti Karttunen, Dec 16 2015

nonn

a(0) = 0, because no numbers are needed to form an empty sum, which is zero.

First 2 occurs as a(17), first 3 at a(234), first 4 at a(3266).

			For n=17, the largest Spironacci number <= 17 is 16 (= A078510(22)). 17 - 16 = 1, which is A078510(1), thus 17 = A078510(22) + A078510(1), requiring only two such numbers for its sum, thus a(17) = 2.
For n=234, the largest Spironacci number <= 234 is 217 (= A078510(45)). 234-217 = 17 (whose decomposition is shown above), so 234 = A078510(45) + A078510(22) + A078510(1), thus a(234) = 3.

Antti Karttunen, Table of n, a(n) for n = 0..10132

Cf. A078510 (from its term a(7) onward gives also the positions of ones here).

Cf. also A007895, A053610, A265743, A265744, A265745.

A364381 Starts of runs of 3 consecutive integers that are greedy Jacobsthal-Niven numbers (A364379).

Original entry on oeis.org

1, 2, 3, 4, 8, 9, 10, 14, 20, 26, 42, 43, 44, 84, 85, 86, 104, 115, 170, 182, 304, 344, 362, 414, 544, 682, 686, 692, 784, 854, 1014, 1370, 1384, 1504, 1673, 1685, 1706, 2224, 2315, 2358, 2730, 2731, 2732, 2763, 2774, 3243, 3594, 3702, 4144, 4688, 4864, 5046, 5408

Offset: 1

Amiram Eldar, Jul 21 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A265745, A265747.
Subsequence of A364379 and A364380.
Subsequences: A364382, A364383.
Similar sequences: A154701, A328206, A328210, A328214, A330932, A331087, A333428, A334310, A331822, A342428, A344343, A351716, A351721, A352091, A352109, A352322, A352344, A352510, A364218.

consecGreedyJN[5500, 3] (* using the function consecGreedyJN from A364380 *)

lista(5500, 3) \\ using the function lista from A364380

A364382 Starts of runs of 4 consecutive integers that are greedy Jacobsthal-Niven numbers (A364379).

Original entry on oeis.org

1, 2, 3, 8, 9, 42, 43, 84, 85, 2730, 2731, 5460, 5461, 21864, 21865, 59477, 60073, 66303, 75048, 112509, 156607, 174762, 174763, 283327, 312190, 320768, 349524, 349525, 351570, 354429, 374589, 384039, 479037, 504510, 527103, 624040, 625470, 656829, 688830, 711423

Offset: 1

Amiram Eldar, Jul 21 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A265745, A265747.
Subsequence of A364379, A364380 and A364381.
A364383 is a subsequence.
Similar sequences: A141769, A328211, A328207, A328215, A330933, A331824, A334311, A342429, A344344, A352092, A352110, A352345, A352511, A364219.

consecGreedyJN[72000, 4] (* using the function consecGreedyJN from A364380 *)

lista(10^5, 4) \\ using the function lista from A364380

A265743 a(n) = number of terms of A005187 needed to sum to n using the greedy algorithm.

Original entry on oeis.org

0, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 2, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 2, 2, 3, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 2, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 2, 2, 3, 4, 1, 1

Offset: 0

Antti Karttunen, Dec 17 2015

nonn

a(0) = 0, because no numbers are needed to form an empty sum, which is zero.

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A005187, A055938.

Cf. also A000120, A007895, A053610, A265404, A265744, A265745.

Other identities. For all n >= 1:

a(A005187(n)) = 1 and a(A055938(n)) > 1.

A265746 Jacobsthal greedy base (A265747) interpreted as base-3 numbers, then shown in decimal.

Original entry on oeis.org

0, 1, 2, 3, 4, 9, 10, 11, 12, 13, 18, 27, 28, 29, 30, 31, 36, 37, 38, 39, 40, 81, 82, 83, 84, 85, 90, 91, 92, 93, 94, 99, 108, 109, 110, 111, 112, 117, 118, 119, 120, 121, 162, 243, 244, 245, 246, 247, 252, 253, 254, 255, 256, 261, 270, 271, 272, 273

Offset: 0

Antti Karttunen, Dec 17 2015

Analogously to "Fibbinary numbers" (A003714) and "Catquaternary numbers" (A244161), this sequence could be called "Jacoternary numbers".

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

Cf. A000244, A001045, A130249.
Cf. A265745, A265747.
Cf. also A003714, A244161.

a[n_] := FromDigits[IntegerDigits[A265747[n]], 3]; Array[a, 100, 0] (* Amiram Eldar, Jul 21 2023 using A265747[n] *)

a(n) = fromdigits(digits(A265747(n)), 3); \\ Amiram Eldar, Jul 21 2023, using A265747(n)

a(0) = 0; for n >= 1, a(n) = 3^(A130249(n)-2) + a(n - A001045(A130249(n))).

A372556 a(n) = largest number k <= A130249(n) for which A372555(n-A001045(k)) = A372555(n)-1, where A372555(n) is the least number of Jacobsthal numbers that add up to n.

Original entry on oeis.org

0, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 0

Antti Karttunen, May 07 2024

nonn

An auxiliary sequence for computing A372555 with a mutually recursive algorithm.
Differs from A130249 for the first time at n=63, 84, 191, 212, 255, etc. See A372558.
Conjecture: For all n, either a(n) = A130249(n) or a(n) = A130249(n)-1. In other words, there is always a minimal solution (in number of summands) for representing n as a sum of Jacobsthal numbers that its largest summand is either A001045(A130249(n)) [same as obtained with a greedy algorithm A265745], or the next smaller Jacobsthal number. - Antti Karttunen, May 10 2024

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

Cf. A001045, A130249, A147612, A372556, A372558.

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); };
v372556 = A372555_or_556list(up_to,1);
A372556(n) = if(!n,n,v372556[n]);

Scheme
```
;; Use the program given in A372555.
```

cp's OEIS Frontend

A265404 a(n) = number of Spironacci numbers (A078510) needed to sum to n using the greedy algorithm.

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A364381 Starts of runs of 3 consecutive integers that are greedy Jacobsthal-Niven numbers (A364379).

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A364382 Starts of runs of 4 consecutive integers that are greedy Jacobsthal-Niven numbers (A364379).

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A265743 a(n) = number of terms of A005187 needed to sum to n using the greedy algorithm.

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A265746 Jacobsthal greedy base (A265747) interpreted as base-3 numbers, then shown in decimal.

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A372556 a(n) = largest number k <= A130249(n) for which A372555(n-A001045(k)) = A372555(n)-1, where A372555(n) is the least number of Jacobsthal numbers that add up to n.

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