A278118 Irregular triangle T(n,k) = A278113(n,k) for 1 <= k <= A278116(n), read by rows.

1, 2, 1, 3, 2, 1, 4, 3, 2, 5, 4, 3, 2, 6, 4, 3, 7, 5, 8, 6, 9, 7, 5, 4, 3, 10, 8, 6, 5, 11, 8, 6, 5, 12, 9, 13, 10, 14, 11, 8, 15, 12, 9, 16, 13, 10, 8, 6, 17, 13, 10, 18, 14, 19, 15, 20, 16, 12, 10, 21, 17, 13, 22, 17, 13, 23, 18, 24, 19, 25, 20, 15, 26, 21, 16, 13, 27, 22, 17, 14, 11, 10

Offset

1,2

Comments

This triangle lists the "descending sequences for rank 1" of Eggleton et al.

References

R. B. Eggleton, J. S. Kimberley and J. A. MacDougall, Square-free rank of integers, submitted.

Links

Jason Kimberley, Table of i, a(i) for i = 1..10104 (T(n,k) for n = 1..3333)

Formula

From A278113: T(n,k) sqrt(prime(k)) <= n sqrt(2) < (T(n,k)+1) sqrt(prime(k)).

Here, we also have:

T(n,1) sqrt(2) > T(n,2) sqrt(3) > ... > T(n,A278116(n)) sqrt(prime(A278116(n))).

Example

For example, 6 sqrt(2) > 4 sqrt(3) > 3 sqrt(5), because 72 > 48 > 45.
The first six rows are:
1;
2, 1;
3, 2, 1;
4, 3, 2;
5, 4, 3, 2;
6, 4, 3;

Mathematica

Function[w, MapIndexed[Take[w[[First@ #2, 1]], 1 + Length@ TakeWhile[ Differences@ #1, # < 0 &]] &, w[[All, -1]]]]@ Table[Function[k, Function[p, {#, p #^2} &@ Floor[n Sqrt[2/p]]]@ Prime@ k]@ Range@ PrimePi[2 n^2], {n, 27}] (* Michael De Vlieger, Feb 17 2017 *)

Prog

(Magma)
A278112:=func<n, k|Isqrt(2*n^2 div k)>;
A278115_row:=func<n|[A278112(n, p)^2*p:p in PrimesUpTo(2*n^2)]>;
A278116:=func<n|(exists(j){j:j in[1..#row-1]|row[j]le row[j+1]}select j else #row) where row is A278115_row(n)>;
A278118_row:=func<n|[A278112(n, NthPrime(k)):k in[1..A278116(n)]]>;
[A278118_row(n):n in[1..20]];

Crossrefs

Cf. A278104.

Sequence in context: A227539 A133334 A003603 * A255238 A212536 A188277

Adjacent sequences: A278115 A278116 A278117 * A278119 A278120 A278121

Keyword

nonn,tabf,easy

Author

Jason Kimberley, Feb 12 2017

Status

approved