A278100 -id:A278100 - cp's OEIS frontend

A277648 Triangle T(n,k) = A277647(n, A005117(k)), read by rows.

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

Offset: 1

Jason Kimberley, Nov 10 2016

The columns of this triangle are the columns of A277647 with squarefree index.

Other that the first (with length 1), row n has length A278100(n).

			Triangle begins:
1;
2, 1, 1;
3, 2, 1, 1, 1, 1;
4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
...
where the first 11 terms of A005117(k) are
1, 2, 3, 5, 6, 7,10,11,13,14,15.

Jason Kimberley, Table of n, a(n) for n = 1..10716 (the first 37 rows of the triangle)

Cf. A277646, A277647.

A277647:=func;
A277648_row:=funcA277647(n,k):k in[1..n^2]|IsSquarefree(k)]>;
&cat[A277648_row(n):n in[1..8]];

DeleteCases[#, 0] & /@ Table[Boole[SquareFreeQ@ k] Floor[n/Sqrt@ k], {n, 8}, {k, n^2}] (* Michael De Vlieger, Nov 24 2016 *)

row(n)={apply(k->sqrtint(n^2\k), select(issquarefree,[1..n^2]))}
for(n=1, 6, print(row(n))) \\ Andrew Howroyd, Feb 28 2018

T(n,k) = A000196(A277646(n,A005117(k))).

T(n,k) sqrt(A005117(k)) <= n < (T(n,k)+1) sqrt(A005117(k)).

Missing a(3009) in b-file inserted by Andrew Howroyd, Feb 28 2018

A278101 Triangle T(n,k) = A277648(n,k)^2 * A005117(k), read by rows.

Original entry on oeis.org

1, 4, 2, 3, 9, 8, 3, 5, 6, 7, 16, 8, 12, 5, 6, 7, 10, 11, 13, 14, 15, 25, 18, 12, 20, 24, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 36, 32, 27, 20, 24, 28, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 49, 32, 48, 45, 24, 28, 40, 44, 13, 14, 15, 17, 19, 21, 22

Offset: 1

Jason Kimberley, Nov 15 2016

Other that the first (with length 1), row n has length A278100(n).

Equivalently, the surd sqrt(T(n,k)) = A277648(n,k) * sqrt(A005117(k)).

			The first five rows are:
1;
4,  2,  3;
9,  8,  3,  5,  6,  7;
16,  8, 12,  5,  6,  7, 10, 11, 13, 14, 15;
25, 18, 12, 20, 24,  7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23;

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

Jason Kimberley, Table of n, a(n) for n = 1..10716 (the first 37 rows of the triangle)

Cf. A278103.

A277647:=func;
A278101_row:=funcA277647(n,j):j in[1..n^2]|IsSquarefree(j)]>;
&cat[A278101_row(n):n in[1..8]];

DeleteCases[#, 0] & /@ Table[Boole[SquareFreeQ@ k] k Floor[n/Sqrt@ k]^2, {n, 7}, {k, n^2}] // Flatten (* Michael De Vlieger, Nov 24 2016 *)

A278114 Number of primes <= 2n^2.

Original entry on oeis.org

1, 4, 7, 11, 15, 20, 25, 31, 37, 46, 53, 61, 68, 77, 87, 97, 106, 118, 128, 139, 152, 163, 177, 190, 204, 217, 231, 247, 263, 278, 293, 309, 326, 344, 363, 377, 399, 418, 436, 452, 474, 492, 516, 536, 558, 580, 600, 623, 647, 669, 692, 713, 738, 765, 789, 816, 842, 867

Offset: 1

Jason Kimberley, Feb 09 2017

This is the row length sequence for both A278113 and A278115.

Jason Kimberley, Table of n, a(n) for n = 1..5000

Cf. A038107, A278100.

A278114:=func;
[A278114(n):n in[1..58]];

Table[PrimePi[2 n^2], {n, 58}] (* Michael De Vlieger, Feb 17 2017 *)

a(n) = primepi(2*n^2); \\ Michel Marcus, Feb 15 2017

a(n) = A000720(A001105(n)).

cp's OEIS Frontend

A277648 Triangle T(n,k) = A277647(n, A005117(k)), read by rows.

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A278101 Triangle T(n,k) = A277648(n,k)^2 * A005117(k), read by rows.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

A278114 Number of primes <= 2n^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula