A061920 -id:A061920 - cp's OEIS frontend

A344444 Completely additive with a(2) = 12, a(3) = 19; for prime p > 3, a(p) = ceiling((a(p-1) + a(p+1))/2).

0, 12, 19, 24, 28, 31, 34, 36, 38, 40, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 55, 56, 57, 57, 58, 59, 59, 60, 60, 61, 61, 62, 62, 63, 63, 64, 64, 65, 65, 66, 66, 66, 67, 67, 67, 68, 68, 68, 69, 69, 69, 70, 70, 70, 71, 71, 71, 72, 72, 72, 72, 73, 73, 73, 73, 74, 74

Offset: 1

Peter Munn, May 20 2021

Monotonic until a(143) = 87 > 86 = a(144).
The only infinite monotonic completely additive integer sequence is the all 0's sequence (cf. A000004). The challenge taken up here is to specify one that is monotonic for a modestly long number of terms, using a comparatively short prescriptive definition.
To start we specify values for a(2) and a(3) so that a(3)/a(2) approximates log(3)/log(2). 19/12 is a good approximation relative to the size of denominator. This reflects 2^19 = 524288 having a similar magnitude to 3^12 = 531441. Equivalently, we can say 3 is approximately the 19th power of the 12th root of 2. This approximation is used to construct musical scales. (See the Enevoldsen link, also A143800.) There is no better approximation with a denominator smaller than 29. [Revised by Peter Munn, Jun 14 2022]
To find a good specification to use for a(p) for larger primes, p, we are guided by knowing that if 2*a(n) < a(n-1) + a(n+1) then a completely additive sequence is not monotonic after a(n^2-1) because a(n^2) < a((n-1)*(n+1)) = a(n^2-1). Considering n = p, we see we want a(p) >= (a(p-1) + a(p+1))/2; but the same consideration for n = p-1 shows we don't want a(p) larger than necessary. These considerations lead towards the choice of "a(p) = ceiling((a(p-1) + a(p+1))/2)" for use in the definition.

			a(4) = a(2*2) = a(2) + a(2) from the definition of completely additive. So a(4) = 12 + 12 = 24. Similarly, a(6) = a(2*3) = a(2) + a(3) = 12 + 19 = 31.
5 is a prime number greater than 3, so a(5) = ceiling((a(5-1) + a(5+1))/2). Using the values a(4) = 24 and a(6) = 31 that we calculated earlier, we get a(5) = ceiling((24 + 31)/2) = ceiling(27.5) = 28.
The sequence is defined as completely additive, so a(1) = 0, the identity element for addition. (To see this, note that "completely additive" implies a(2) = a(2*1) = a(2)+a(1), and solve the equation for a(1).)

Encyclopedia of Mathematics, Additive arithmetic function
Keith Enevoldsen, Twelve-Tone Musical Scale
Index entries for sequences related to music

Equivalent sequence with a(2)=5, a(3)=8: A344443.
First 10 terms match A143800.
Cf. row 23 of A352957.
Cf. A000004, A020857, A060528, A061920.
For other completely additive sequences see the references in A104244.

a[1] = 0; a[n_] := a[n] = Plus @@ ((Last[#] * Which[First[#] == 2, 12, First[#] == 3, 19, First[#] > 3, Ceiling[(a[First[#] - 1] + a[First[#] + 1])/2]]) & /@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jun 27 2021 *)

a(n*k) = a(n) + a(k).

A061921 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the 11 pairs of target ratios needed to express the 22 steps of the theoretical Hindu scale known as the 22 Srutis: 45/32 and 64/45, 27/20 and 40/27, 4/3 and 3/2, 81/64 and 128/81, 5/4 and 8/5, 6/5 and 5/3, 32/27 and 27/16, 9/8 and 16/9, 10/9 and 9/5, 16/15 and 15/8, 256/243 and 243/128.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 28, 29, 30, 32, 34, 37, 39, 40, 41, 53, 118, 171, 323, 335, 376, 388, 441, 494, 506, 559, 612, 1171, 1783, 2513, 3072, 3125, 3684, 4296, 12276, 16572, 20868, 40565, 44861, 48545, 52841, 57137, 61433, 69413, 73709

Offset: 1

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), May 15 2001

nonn

The sequence was found by a computer search of all the equal divisions of the octave from 1 to 73709. The numerical value of each term represents a musical scale based on an equal division of the octave. The term 32, for example, signifies the scale which is formed by dividing the octave into 32 equal parts.

			118 = 53 + [34 + 31]; Again, 229719 = 78005 + [73709 + 69413 + 4296 + 3684 + 612].

A054540, A060525, A060526, A060527, A060528, A060529, A060233, A061416, A061918, A061919, A061920.

Recurrence rule: The next term equals the current term plus one or more previous terms: a(n+1) = a(n) + a(n-x)... + a(n-y)... + a(n-z), etc.

cp's OEIS Frontend

A344444 Completely additive with a(2) = 12, a(3) = 19; for prime p > 3, a(p) = ceiling((a(p-1) + a(p+1))/2).

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