%I #21 Aug 14 2022 10:15:42

%S 0,0,1,0,1,2,0,2,3,4,0,2,3,4,5,0,3,5,6,7,8,0,3,5,6,7,8,9,0,4,6,8,9,10,

%T 11,12,0,5,8,10,11,13,14,15,16,0,5,8,10,12,13,14,15,16,17,0,5,8,10,12,

%U 13,14,15,16,17,18,0,7,11,14,16,18,19,21,22,23,24,25

%N Triangle read by rows: Row n is the lexicographically earliest strictly monotonic completely additive sequence of length n.

%C Each sequence consists of nonnegative integers indexed from 1.

%C Note in particular in the formula section, the lower bound, floor(n/k), for first differences between terms in a row. This follows (using the additive property) from the strict monotonicity of floor(n/k)+1 consecutive terms near the end of the row.

%C For any k, with increasing length n >= k, the first k terms of the sequences approach similarity with a real-valued logarithmic function defined on the integers. For example, the asymptote of T(n,3)/T(n,2) is log(3)/log(2), A020857.

%H Peter Munn, <a href="/A352957/b352957.txt">Rows n = 1..141, flattened</a>

%H Encyclopedia of Mathematics, <a href="https://encyclopediaofmath.org/wiki/Additive_arithmetic_function">Additive arithmetic function</a>.

%H Peter Munn, <a href="/A352957/a352957.txt">PARI program</a>

%F The definition specifies: T(n,j*k) = T(n,j) + T(n,k); for k > 1, T(n,k) > T(n,k-1).

%F T(n,1) = 0, otherwise T(n,k) >= T(n,k-1) + floor(n/k).

%F For prime p, T(p,p) = T(p-1,p-1) + 1, otherwise T(p,k) = T(p-1,k).

%F T(n,2) >= 2*floor(n/4) + floor(n/9).

%F T(n,3) >= ceiling( (3*T(n,2) + floor(n/9)) / 2).

%F T(11,k) = A344443(k).

%F For k <> 13, T(23,k) = A344444(k).

%e (For row 4.) A completely additive sequence requires T(4,1) = 0. Strict monotonicity requires T(4,4) > T(4,3) > T(4,2). So T(4,4) >= T(4,2) + 2. Using the additivity this becomes T(4,2) + T(4,2) >= T(4,2) + T(4,1) + 2. Subtracting T(4,2) and substituting 0 for T(4,1) we get T(4,2) >= 2. So from T(4,4) > T(4,3) > T(4,2), we see T(4,3) >= 3, T(4,4) >= 4. So row 4 = (0, 2, 3, 4) as it is strictly monotonic and completely additive and from the preceding arguments is seen to be the lexicographically earliest such.

%e Triangle starts:

%e 0;

%e 0, 1;

%e 0, 1, 2;

%e 0, 2, 3, 4;

%e 0, 2, 3, 4, 5;

%e 0, 3, 5, 6, 7, 8;

%e 0, 3, 5, 6, 7, 8, 9;

%e 0, 4, 6, 8, 9, 10, 11, 12;

%e 0, 5, 8, 10, 11, 13, 14, 15, 16;

%e 0, 5, 8, 10, 12, 13, 14, 15, 16, 17;

%e 0, 5, 8, 10, 12, 13, 14, 15, 16, 17, 18;

%e 0, 7, 11, 14, 16, 18, 19, 21, 22, 23, 24, 25;

%e 0, 7, 11, 14, 16, 18, 19, 21, 22, 23, 24, 25, 26;

%e 0, 7, 11, 14, 16, 18, 20, 21, 22, 23, 24, 25, 26, 27;

%e 0, 8, 13, 16, 19, 21, 23, 24, 26, 27, 28, 29, 30, 31, 32;

%e 0, 9, 14, 18, 21, 23, 25, 27, 28, 30, 31, 32, 33, 34, 35, 36;

%Y Cf. A020857.

%Y Completely additive sequences, s, with primes p mapped to a function of s(p-1) and maybe s(p+1): A064097, A344443, A344444; and for functions of earlier terms, see A334200.

%Y For completely additive sequences with primes p mapped to a function of p, see A001414.

%Y For completely additive sequences with prime(k) mapped to a function of k, see A104244.

%Y For completely additive sequences where some primes are mapped to 1, the rest to 0 (notably, some ruler functions) see the cross-references in A249344.

%K nonn,tabl

%O 1,6

%A _Peter Munn_, Apr 11 2022