Jason Kimberley - cp's OEIS frontend

A304484 a(n) = A033270(n)*A033270(2n), where A033270 counts the odd primes.

0, 0, 2, 3, 6, 8, 15, 15, 18, 21, 28, 32, 40, 40, 45, 50, 60, 60, 77, 77, 84, 91, 104, 112, 112, 112, 120, 120, 135, 144, 170, 170, 170, 180, 180, 190, 220, 220, 220, 231, 252, 264, 286, 286, 299, 299, 322, 322, 336, 336, 350, 364, 390, 405, 420, 420, 435, 435, 464, 464

Offset: 1

Jason Kimberley, May 15 2018

Jason Kimberley, Table of n, a(n) for n = 1..10000

Cf. A304483 = A000720(n)*A000720(2n).

A033270:=func; A:=[A033270(n):n in[1..120]]; [A[n]*A[2*n]:n in[1..#A div 2]];

Array[(PrimePi@ # - Boole[# > 1]) (PrimePi[2 #] - Boole[2 # > 1]) &, 60] (* Michael De Vlieger, May 27 2018 *)

a033270(n) = max(primepi(n)-1, 0);
a(n) = a033270(n)*a033270(2*n);

A304483 a(n) = pi(n)*pi(2n), where pi is A000720: the prime counting function.

Original entry on oeis.org

0, 2, 6, 8, 12, 15, 24, 24, 28, 32, 40, 45, 54, 54, 60, 66, 77, 77, 96, 96, 104, 112, 126, 135, 135, 135, 144, 144, 160, 170, 198, 198, 198, 209, 209, 220, 252, 252, 252, 264, 286, 299, 322, 322, 336, 336, 360, 360, 375, 375, 390, 405, 432, 448, 464, 464, 480, 480, 510

Offset: 1

Jason Kimberley, May 13 2018

Jason Kimberley, Table of n, a(n) for n = 1..10000

Cf. A304484(n) = A033270(n)*A033270(2n).

Cf. A291440, A291538, A291539, A291540,

A000720:=func;
A304483:=funcA000720(n)*A000720(2*n)>;
[A304483(n):n in[1..59]];

Array[PrimePi[#] PrimePi[2 #] &, 59] (* Michael De Vlieger, May 27 2018 *)

a(n) = primepi(n) * primepi(2*n); \\ Michel Marcus, May 27 2018

A290340 Numbers m such that each of the four consecutive integers m, m+1, m+2, m+3 has squarefree rank 1.

Original entry on oeis.org

17, 241, 242, 1249, 4049, 4799, 17297, 120049, 206081, 281249, 388961, 470447, 538721, 1462049, 1566449, 1808801, 1916881, 3302449, 3302450, 3693761, 3959297, 5385761, 5664976, 6118001, 6986321, 9305297, 10479041, 14268481, 16831601, 20110481, 22997761, 27661922, 28140001

Offset: 1

Jason Kimberley, Jul 27 2017

nonn

A162642(k) is the squarefree rank of k.
Numbers that are the first of four consecutive terms of A228056 form a subsequence: 242, 3302450, 22997761, 27661922, 28140001, 64866050, ... consisting of those numbers m in this sequence such that m, m+1, m+2, and m+3 are all composite. - Charles R Greathouse IV, Sep 30 2021
One of for positive integer m, m+1, m+2, m+3 is of the form 4*k + 2 = 2*(2*k + 1). As 2 has an odd exponent the exponents in the prime factorization and 2*k + 1 is odd, the number of odd exponents in the prime factorization of 2*k + 1 must be 0 i.e., 2*k + 1 is a perfect square and so one of m, m+1, m+2, m+3 is of the form 2*t^2 where t is an odd square. - David A. Corneth, Nov 09 2023

			m = 17 is in the sequence as the number of odd prime exponents of each of the numbers m = 17 through m + 3 = 20 is 1. - _David A. Corneth_, Nov 06 2023

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 79 terms from Jason Kimberley, a(n) for n = 80..347 from Charles R Greathouse IV)
David A. Corneth, PARI program

Cf. A162642, A229125, A277225, A228056.

A162642:=func;
c:=func;
c(c(c([n:n in[1..10^6]|A162642(n)eq 1])));

list(lim)=my(u=vectorsmall(4),v=List(),s,i); forfactored(n=2,lim\1+3, if(i++>4,i=1); s-=u[i]; s+=u[i]=(vecsum(n[2][,2]%2)==1); if(s==4, listput(v,n[1]-3))); Vec(v); \\ Charles R Greathouse IV, Sep 30 2021

PARI
```
\\ See PARI link
```

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

Original entry on oeis.org

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

Jason Kimberley, Feb 12 2017

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

			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;

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

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

Cf. A278104.

A278112:=func;
A278115_row:=funcA278112(n,p)^2*p:p in PrimesUpTo(2*n^2)]>;
A278116:=funcA278115_row(n)>;
A278118_row:=funcA278112(n,NthPrime(k)):k in[1..A278116(n)]]>;
[A278118_row(n):n in[1..20]];

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 *)

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))).

A278117 Irregular triangle T(n,k) = A278115(n,k) for 1 <= k <= A278116(n), read by rows.

Original entry on oeis.org

2, 8, 3, 18, 12, 5, 32, 27, 20, 50, 48, 45, 28, 72, 48, 45, 98, 75, 128, 108, 162, 147, 125, 112, 99, 200, 192, 180, 175, 242, 192, 180, 175, 288, 243, 338, 300, 392, 363, 320, 450, 432, 405, 512, 507, 500, 448, 396, 578, 507, 500, 648, 588, 722, 675, 800, 768, 720

Offset: 1

Jason Kimberley, Feb 12 2017

Each row is the longest strictly decreasing prefix of the corresponding row of A278115.

			The first six rows are:
2;
8, 3;
18, 12, 5;
32, 27, 20;
50, 48, 45, 28;
72, 48, 45;

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

Cf. A278103, A278118.

A278112:=func;
A278115_row:=funcA278112(n,p)^2*p:p in PrimesUpTo(2*n^2)]>;
A278117_row:=funcA278115_row(n) >;
&cat[A278117_row(n):n in[1..20]];

Map[Take[#, 1 + Length@ TakeWhile[Differences@ #, # < 0 &]] &, #] &@ Table[# Floor[n Sqrt[2/#]]^2 &@ Prime@ k, {n, 20}, {k, PrimePi[2 n^2]}] // Flatten (* Michael De Vlieger, Feb 17 2017 *)

A278115 Triangle T(n,k) = A278113(n,k)^2 A000040(k) for 1 <= k <= A278114(n), read by rows.

Original entry on oeis.org

2, 8, 3, 5, 7, 18, 12, 5, 7, 11, 13, 17, 32, 27, 20, 28, 11, 13, 17, 19, 23, 29, 31, 50, 48, 45, 28, 44, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 72, 48, 45, 63, 44, 52, 68, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 98, 75, 80, 63, 44, 52, 68, 76, 92, 29, 31, 37, 41, 43, 47, 53

Offset: 1

Jason Kimberley, Feb 10 2017

			The first six rows are:
2;
8, 3, 5, 7;
18, 12, 5, 7, 11, 13, 17;
32, 27, 20, 28, 11, 13, 17, 19, 23, 29, 31;
50, 48, 45, 28, 44, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47;
72, 48, 45, 63, 44, 52, 68, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71;

Jason Kimberley, Table of i, a(i) for i = 1..10126 (T(n,k) for n = 1..46)

Cf. A278101.

A278112:=func;
A278115_row:=funcA278112(n,p)^2*p:p in PrimesUpTo(2*n^2)]>;
&cat[A278115_row(n):n in[1..7]];

Table[# Floor[n Sqrt[2/#]]^2 &@ Prime@ k, {n, 7}, {k, PrimePi[2 n^2]}] // Flatten (* Michael De Vlieger, Feb 17 2017 *)

T(n,k) = prime(k) * floor(n*sqrt(2/prime(k)))^2.

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)).

A278113 Triangle T(n,k) = A278112(n,A000040(k)) for 1 <= k <= A278114(n), read by rows.

Original entry on oeis.org

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

Offset: 1

Jason Kimberley, Feb 09 2017

This triangle consists of those columns of A278112 that have prime index.

			The first eight rows are:
  1;
  2, 1, 1, 1;
  3, 2, 1, 1, 1, 1, 1;
  4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1;
  5, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  6, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  7, 5, 4, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

Jason Kimberley, Table of n, a(n) for n = 1..10126 (rows 1..46)

Cf. A000040, A277648, A278112, A278114, A278115, A278118.

A278112:=func;
A278113_row:=funcA278112(n,p):p in PrimesUpTo(2*n^2)]>;
&cat[A278113_row(n):n in[1..8]];

Table[Floor[n Sqrt[2/Prime@ k]], {n, 8}, {k, PrimePi[2 n^2]}] // Flatten (* Michael De Vlieger, Feb 17 2017 *)

T(n,k) = floor(n*sqrt(2/prime(k))).

T(n,k) sqrt(A000040(k)) <= n sqrt(2) < (T(n,k)+1) sqrt(A000040(k)).

A278112 Triangle T(n,k) = floor(n sqrt(2/k)) for 1 <= k <= 2n^2, read by rows.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 4, 3, 2, 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, 1, 7, 5, 4, 3, 3, 2, 2, 2, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Jason Kimberley, Feb 09 2017

			The first five rows are:
1, 1;
2, 2, 1, 1, 1, 1, 1, 1;
4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
5, 4, 3, 2, 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, 1;
7, 5, 4, 3, 3, 2, 2, 2, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

Jason Kimberley, Table of n, a(n) for n = 1..11050 (rows 1..25, flattened)

Cf. A277647.

A278112:=func;
[[A278112(n,k):k in[1..2*n^2]]:n in[1..5]];

Table[Floor[n Sqrt[2/k]], {n, 5}, {k, 2 n^2}] // Flatten (* Michael De Vlieger, Feb 17 2017 *)

T(n,k) = A000196(A278111(n)).

T(n,k)sqrt(k) <= n sqrt(2) < (T(n,k)+1)sqrt(k).

A278111 Triangle T(n,k) = floor(2n^2/k) for 1 <= k <= 2n^2, read by rows.

Original entry on oeis.org

2, 1, 8, 4, 2, 2, 1, 1, 1, 1, 18, 9, 6, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 32, 16, 10, 8, 6, 5, 4, 4, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 50, 25, 16, 12, 10, 8, 7, 6, 5, 5, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Jason Kimberley, Feb 08 2017

			The first five rows are:
2, 1;
8, 4, 2, 2, 1, 1, 1, 1;
18, 9, 6, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1;
32, 16, 10, 8, 6, 5, 4, 4, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
50, 25, 16, 12, 10, 8, 7, 6, 5, 5, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 2, 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, 1, 1;

Jason Kimberley, Table of i, a(i) for i = 1..11050 (T(n,k) for n = 1..25)

Cf. A277646.

A278111:=func;
[A278111(n,k):k in[1..2*n^2],n in[1..5]];

T(n,k) = A010766(2n^2,k).

cp's OEIS Frontend

User: Jason Kimberley

A304484 a(n) = A033270(n)*A033270(2n), where A033270 counts the odd primes.

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A304483 a(n) = pi(n)*pi(2n), where pi is A000720: the prime counting function.

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A290340 Numbers m such that each of the four consecutive integers m, m+1, m+2, m+3 has squarefree rank 1.

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

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A278117 Irregular triangle T(n,k) = A278115(n,k) for 1 <= k <= A278116(n), read by rows.

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A278115 Triangle T(n,k) = A278113(n,k)^2 A000040(k) for 1 <= k <= A278114(n), read by rows.

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Examples

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Magma

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Formula

A278114 Number of primes <= 2n^2.

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Magma

Mathematica

PARI