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A278118 Irregular triangle T(n,k) = A278113(n,k) for 1 <= k <= A278116(n), read by rows. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 1 22:12 EST 2021. Contains 349435 sequences. (Running on oeis4.)