# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a352957 Showing 1-1 of 1 %I A352957 #21 Aug 14 2022 10:15:42 %S A352957 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 A352957 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 A352957 13,14,15,16,17,18,0,7,11,14,16,18,19,21,22,23,24,25 %N A352957 Triangle read by rows: Row n is the lexicographically earliest strictly monotonic completely additive sequence of length n. %C A352957 Each sequence consists of nonnegative integers indexed from 1. %C A352957 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 A352957 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 A352957 Peter Munn, Rows n = 1..141, flattened %H A352957 Encyclopedia of Mathematics, Additive arithmetic function. %H A352957 Peter Munn, PARI program %F A352957 The definition specifies: T(n,j*k) = T(n,j) + T(n,k); for k > 1, T(n,k) > T(n,k-1). %F A352957 T(n,1) = 0, otherwise T(n,k) >= T(n,k-1) + floor(n/k). %F A352957 For prime p, T(p,p) = T(p-1,p-1) + 1, otherwise T(p,k) = T(p-1,k). %F A352957 T(n,2) >= 2*floor(n/4) + floor(n/9). %F A352957 T(n,3) >= ceiling( (3*T(n,2) + floor(n/9)) / 2). %F A352957 T(11,k) = A344443(k). %F A352957 For k <> 13, T(23,k) = A344444(k). %e A352957 (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 A352957 Triangle starts: %e A352957 0; %e A352957 0, 1; %e A352957 0, 1, 2; %e A352957 0, 2, 3, 4; %e A352957 0, 2, 3, 4, 5; %e A352957 0, 3, 5, 6, 7, 8; %e A352957 0, 3, 5, 6, 7, 8, 9; %e A352957 0, 4, 6, 8, 9, 10, 11, 12; %e A352957 0, 5, 8, 10, 11, 13, 14, 15, 16; %e A352957 0, 5, 8, 10, 12, 13, 14, 15, 16, 17; %e A352957 0, 5, 8, 10, 12, 13, 14, 15, 16, 17, 18; %e A352957 0, 7, 11, 14, 16, 18, 19, 21, 22, 23, 24, 25; %e A352957 0, 7, 11, 14, 16, 18, 19, 21, 22, 23, 24, 25, 26; %e A352957 0, 7, 11, 14, 16, 18, 20, 21, 22, 23, 24, 25, 26, 27; %e A352957 0, 8, 13, 16, 19, 21, 23, 24, 26, 27, 28, 29, 30, 31, 32; %e A352957 0, 9, 14, 18, 21, 23, 25, 27, 28, 30, 31, 32, 33, 34, 35, 36; %Y A352957 Cf. A020857. %Y A352957 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 A352957 For completely additive sequences with primes p mapped to a function of p, see A001414. %Y A352957 For completely additive sequences with prime(k) mapped to a function of k, see A104244. %Y A352957 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 A352957 nonn,tabl %O A352957 1,6 %A A352957 _Peter Munn_, Apr 11 2022 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE