A352957 Triangle read by rows: Row n is the lexicographically earliest strictly monotonic completely additive sequence of length n.

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

Offset: 1

Author

Peter Munn, Apr 11 2022

Keywords

• nonn
• tabl

Comments

Each sequence consists of nonnegative integers indexed from 1.

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.

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.

Examples

			(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.
Triangle starts:
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, 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, 13, 14, 15, 16, 17, 18;
0, 7, 11, 14, 16, 18, 19, 21, 22, 23, 24, 25;
0, 7, 11, 14, 16, 18, 19, 21, 22, 23, 24, 25, 26;
0, 7, 11, 14, 16, 18, 20, 21, 22, 23, 24, 25, 26, 27;
0, 8, 13, 16, 19, 21, 23, 24, 26, 27, 28, 29, 30, 31, 32;
0, 9, 14, 18, 21, 23, 25, 27, 28, 30, 31, 32, 33, 34, 35, 36;
		

Links

• Peter Munn, Rows n = 1..141, flattened
• Encyclopedia of Mathematics, Additive arithmetic function.
• Peter Munn, PARI program

Crossrefs

Cf. A020857.

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.

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

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

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.

Formula

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

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

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

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

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

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

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