A279664 Constant whose Engel Expansion is A007775.

1, 1, 5, 6, 9, 0, 5, 1, 5, 3, 7, 5, 4, 0, 2, 8, 9, 5, 4, 5, 0, 1, 3, 4, 5, 8, 1, 5, 5, 7, 2, 3, 2, 1, 4, 6, 5, 3, 5, 2, 5, 5, 4, 0, 2, 8, 9, 4, 8, 7, 9, 5, 3, 6, 4, 7, 0, 0, 3, 9, 9, 3, 8, 9, 5, 9

Offset: 1

Author

Benedict W. J. Irwin, Dec 16 2016

Keywords

• nonn
• cons

Comments

This one constant is enough information to uniquely reconstruct A007775.

There appears to be a general expression for higher sets of k-rough numbers.

Examples

			1.15690515375402895450134581557232146535255402894879536470039938959...
		

Links

• B. W. J. Irwin, Constants Whose Engel Expansions are the k-rough Numbers

Crossrefs

Cf. A007775.

Programs

Mathematica

Prime7[n_] := If[n < 16, Prime[n], If[n == 16, 7^2, Prime[n - 1]]];
RealDigits[N[Pi^4*Sum[Sum[2^(4-n-8*k)*15^(-n-8*k)/Product[Gamma[ Prime7[2+m+n]/30+k], {m,1,8}],{n,1,8}],{k,0,Infinity}], 100]][[1]]

Formula

Define an indexing function over the primes and 7^2.

P(n) = prime(n) for n<16, 49 for n=16, prime(n-1) for n>16.

a = Pi^4*Sum_{k>=0}Sum_{n=1..8} 2^(4-n-8*k)*15^(-n-8*k)/(Prod_{m=1..8} Gamma( P(2+m+n)/30 + k)). - Benedict W. J. Irwin, Dec 16 2016