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A202618 a(n) is the smallest integer that is the sum of n distinct terms of A075058.

%I A202618 #32 Jun 11 2023 23:29:55
%S A202618 0,1,4,6,19,42,89,96,289,672,1441,2972,6039,12172,24441,48974,98043,
%T A202618 196172,392419,784922,1569939,3139946,6279987,12560054,25120201
%N A202618 a(n) is the smallest integer that is the sum of n distinct terms of A075058.
%C A202618 Any nonnegative integer can be written as a sum of distinct terms of A075058. a(n) is the smallest integer that is the sum of n distinct terms of A075058 in the same way that A066352 gives a Pillai sequence for the sequence comprising 1 followed by all the primes.
%H A202618 Wikipedia, <a href="http://en.wikipedia.org/wiki/Complete_sequence">"Complete" sequence</a>. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided. - _N. J. A. Sloane_, May 20 2023]
%F A202618 Find the smallest m such that binary(A201997(m)) x {1,1,1,...} = n, where x is the inner product, {1,1,1,1,...} is an infinite binary vector of 1's and binary(A201997(m)) a binary vector with infinite trailing zeros both in ascending powers of 2. Then a(n) = binary(m) x A075058, where x is the inner product and the binary vector is in ascending powers of 2 with infinite trailing zeros.
%e A202618 For n=5, the binary vector at A201997(54) is the smallest binary vector containing 5 1's and when applied to A075058 selects the integer 42. Consequently because 42=23+13+3+2+1 and 1,2,3,13,23 are all terms of the complete sequence A075058, a(5)=42.
%t A202618 prevprime[n_Integer] := (j=n;If[n==1, 1, While[!PrimeQ[j], j--]; j]);aprime[n_Integer] := (aprime[n]=prevprime[Sum[aprime[m], {m, 0, n - 1}]+1]);gentable[n_Integer] := (m=n;ptable={0};While[m!=0, (i=0;While[aprime[i]<=m && ptable[[i+1]]!=1, (AppendTo[ptable, 0];i++)];ptable[[i]] = 1;m=m-aprime[i - 1])];ptable);decimal[n_Integer] := (gentable[n];Sum[2^(k-1)*ptable[[k]], {k, 1, Length[ptable]}]);ones[n_Integer] :=(gentable[n];Sum[ptable[[k]], {k, 1, Length[ptable]}]);changeones[n_Integer] := (p = 0; While[ones[p] < n, p++]; p);aprime[0]=1;Table[changeones[r], {r, 0, 20}]
%Y A202618 Cf. A007924, A066352, A200947, A075058, A201997.
%K A202618 nonn,more
%O A202618 0,3
%A A202618 _Frank M Jackson_, Dec 21 2011
%E A202618 Edited by _N. J. A. Sloane_, May 20 2023