A295572 First differences of A081881.

1, 2, 6, 16, 43, 117, 318, 865, 2351, 6391, 17372, 47222, 128363, 348927, 948482, 2578241, 7008386, 19050768, 51785356, 140767193, 382644902, 1040136684, 2827384648, 7685628310, 20891703776, 56789538739, 154369971201, 419621087576, 1140648377196, 3100603756393

Offset

1,2

Comments

See A081881 and A295571 for discussion.

If the harmonic series is divided into the longest possible consecutive groups so that the sum of each group is <= 1, then a(n) is the number of terms in the n-th group. - Pablo Hueso Merino, Feb 16 2020

Links

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Formula

a(1) = 1, a(n) = (max(m) : Sum_{s=r..m} 1/s <= 1)-r+1, r = Sum_{k=1..n-1} a(k). - Pablo Hueso Merino, Feb 16 2020

a(n) ~ c * exp(n), where c = (exp(1)-1) * A300897 = 0.290142809280953235916025... - Vaclav Kotesovec, Apr 05 2020

Example

From Pablo Hueso Merino, Feb 16 2020: (Start)
a(1) = 1 because 1 <= 1, 1 is one term (if you added 1/2 the sum would be greater than 1).
a(2) = 2 because 1/2 + 1/3 = 0.8333... <= 1, 1/2 and 1/3 are two terms (if you added 1/4 the sum would be greater than one).
a(3) = 6 because 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 = 0.9956... <= 1, it is a sum of six terms. (End)

Mathematica

a[1]=1;
a[n_]:= a[n]= Module[{sum = 0}, r = 1 + Sum[a[k], {k, n-1}];
   x = r;
   While[sum <= 1, sum += 1/x++];
   p = x-2;
   p -r +1];
Table[a[n], {n, 10}] (* Pablo Hueso Merino, Feb 16 2020 *)

Crossrefs

Cf. A081881, A295571, A300897, A331028, A331030.

Sequence in context: A291142 A027994 A319503 * A027068 A003142 A335686

Adjacent sequences: A295569 A295570 A295571 * A295573 A295574 A295575

Keyword

nonn

Author

N. J. A. Sloane, Nov 30 2017, following a suggestion from Loren Booda

Extensions

More terms from Jinyuan Wang, Feb 20 2020

Status

approved