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

0, 1, 4, 6, 19, 42, 89, 96, 289, 672, 1441, 2972, 6039, 12172, 24441, 48974, 98043, 196172, 392419, 784922, 1569939, 3139946, 6279987, 12560054, 25120201

Offset

0,3

Comments

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.

Links

Table of n, a(n) for n=0..24.

Wikipedia, "Complete" sequence. [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]

Formula

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.

Example

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.

Mathematica

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}]

Crossrefs

Cf. A007924, A066352, A200947, A075058, A201997.

Sequence in context: A026521 A343677 A222379 * A006534 A064035 A010364

Adjacent sequences: A202615 A202616 A202617 * A202619 A202620 A202621

Keyword

nonn,more

Author

Frank M Jackson, Dec 21 2011

Extensions

Edited by N. J. A. Sloane, May 20 2023

Status

approved