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A039952 Maximum cardinality of finite D0L sequence over an alphabet with n symbols.

%I A039952 #23 Jan 05 2018 13:00:54
%S A039952 1,2,3,4,5,6,7,12,15,20,30,31,60,61,84,105,140,210,211,420,421,422,
%T A039952 423,840,841,1260,1261,1540,2310,2520,4620,4621,5460,5461,9240,9241,
%U A039952 13860,13861,16380,16381,27720,30030,32760,60060,60061,60062,60063,120120,120121
%N A039952 Maximum cardinality of finite D0L sequence over an alphabet with n symbols.
%D A039952 P. M. B. Vitanyi, Lindenmayer Systems: Structure, Languages and Growth Functions, Mathematisch Centrum, Math. Centre Tracts #96, 1980, p. 25.
%H A039952 Andrew Howroyd, <a href="/A039952/b039952.txt">Table of n, a(n) for n = 0..1000</a>
%H A039952 O. Osterby, <a href="http://dx.doi.org/10.7146/dpb.v2i19.6438">Prime decompositions with minimum sum</a>, Matematisk Institut, Aarhus Universitet, Technical Report DAIMI PB-19, November 1973; see the third column of Table 5 on page 18.
%H A039952 O. Osterby, <a href="https://doi.org/10.1007/BF01932728">Prime decompositions with minimum sum</a>, Nordisk Tidskr. Informationsbehandling (BIT) 16 (1976), 451-458.
%F A039952 Max { Prod p^a + d : Sum p^a + d = n }, p prime.
%F A039952 a(n) = max(a(n-1)+1, A051703(n)). - _Andrew Howroyd_, Jan 05 2018
%e A039952 a(11) = 31 because we can write 11 = 1 + 2 + 3 + 5 and 31 = 1+2*3*5.
%o A039952 (PARI) \\ here s is A051703 as a vector
%o A039952 s(n)={my(v=vector(n+1)); v[1]=1; forprime(p=2, n, forstep(i=#v, 1, -1, my(q=1); while(q*p<i, q*=p; v[i]=max(v[i], q*v[i-q])))); v}
%o A039952 seq(n)={my(v=s(n)); for(i=2, #v, v[i]=max(v[i], v[i-1]+1)); v} \\ _Andrew Howroyd_, Jan 05 2018
%Y A039952 Cf. A051703.
%K A039952 nonn
%O A039952 0,2
%A A039952 _Jeffrey Shallit_
%E A039952 First 4 values appear incorrectly in cited references; corrected by JOS
%E A039952 a(0)=1 and terms a(35) and beyond from _Andrew Howroyd_, Jan 05 2018