Anthony Hernandez - cp's OEIS frontend

A323552 Denominators of the partial Euler product representation of Pi/4.

4, 16, 128, 512, 6144, 98304, 393216, 9437184, 37748736, 1207959552, 43486543872, 1739461754880, 6957847019520, 333976656936960, 1335906627747840, 80154397664870400, 4809263859892224000, 19237055439568896000, 1385067991648960512000, 99724895398725156864000

Offset: 1

Anthony Hernandez, Jan 17 2019

The Euler product representation follows from the classical Leibniz series representation of Pi/4 interpreted as a Dirichlet L-series using the unique non-principal Dirichlet characters modulo 4, whose (infinite) Euler product representation can be written as (3/4) * (5/4) * (7/8) * (11/12) * (13/12) * (17/16) * (19/20) * ... with each term in the product being the ratio of a prime number to its nearest multiple of 4. The sequence consists of the denominators of the partial products.

			a(3) = 128 = denominator((3/4) * (5/4) * (7/8)).

N. Elkies, Introduction to Analytic Number Theory: Primes in Arithmetic Progression, Dirichlet Characters and L-Functions
Leonhard Euler, On the sums of series of reciprocals, arXiv:math/0506415 [math.HO], 2005-2008.
Wikipedia, Superparticular ratio
Wikipedia, Wallis Product
Wikipedia, Gregory Series
Wikipedia, Madhava Series
Wikipedia, Machin-like Formula
Wikipedia, Inverse Trigonometric Functions

Cf. A003881 (Decimal Expansion of Pi/4).
Cf. A101455 (Dirichlet L-series of The Non-Principal Dirichlet Characters Mod 4).
Cf. A323551 (Numerators of the Partial Euler Product Representation of Pi/4).

a(n) = denominator(prod(k=2, n+1, my(p=prime(k)); if(p%4==1, p/(p-1), p/(p+1)))); \\ Daniel Suteu, Jan 22 2019

More terms from Daniel Suteu, Jan 22 2019

A323551 Numerators of the partial Euler product representation of Pi/4.

Original entry on oeis.org

3, 15, 105, 385, 5005, 85085, 323323, 7436429, 30808063, 955049953, 35336848261, 1448810778701, 5663533044013, 266186053068611, 1085220062510491, 64027983688118969, 3905707004975257109, 15393080549020130959, 1092908718980429298089, 79782336485571338760497

Offset: 1

Anthony Hernandez, Jan 16 2019

The Euler product representation follows from the classical Leibniz series representation of Pi/4 interpreted as a Dirichlet L-series using the unique non-principal Dirichlet characters modulo 4, whose (infinite) Euler product representation can be written as (3/4) * (5/4) * (7/8) * (11/12) * (13/12) * ..., with each term in the product being the ratio of a prime number to its nearest multiple of 4. The sequence consists of the numerators of the partial products.

			a(3) = 105 = numerator((3/4) * (5/4) * (7/8)).

N. Elkies, Introduction to Analytic Number Theory: Primes in Arithmetic Progression, Dirichlet Characters and L-Functions
Leonhard Euler, On the sums of series of reciprocals, arXiv:math/0506415 [math.HO], 2005-2008.
Wikipedia, Superparticular ratio
Wikipedia, Wallis Product
Wikipedia, Gregory Series
Wikipedia, Madhava Series
Wikipedia, Machin-like Formula
Wikipedia, Inverse Trigonometric Functions

Cf. A003881 (Decimal expansion of Pi/4).
Cf. A101455 (Dirichlet L-series of The Non-Principal Dirichlet Characters Mod 4).
Cf. A323552 (Denominators of the Partial Euler Product Representation of Pi/4).
Cf. A236436 (Denominators of the Product (1 + 1/p), where p is prime).
Cf. A002144 (Primes of the form 4n+1; Pythagorean primes).
Cf. A002145 (Primes of the form 4n+3).

a(n) = numerator(prod(k=2, n+1, my(p=prime(k)); if(p%4==1, p/(p-1), p/(p+1)))); \\ Daniel Suteu, Jan 22 2019

More terms from Daniel Suteu, Jan 22 2019

A293462 Let A_n be a square n X n matrix with entries A_n(i,j)=1 if i+j is a perfect power, and A_n(i,j)=0 otherwise. Then a(n) counts the 1's in A_n.

Original entry on oeis.org

0, 1, 3, 4, 8, 12, 16, 19, 21, 23, 25, 27, 31, 37, 43, 48, 54, 61, 69, 77, 85, 93, 101, 109, 117, 125, 131, 137, 143, 149, 155, 160, 166, 172, 178, 182, 186, 190, 194, 198, 204, 210, 216, 222, 228, 234, 240, 246, 250, 255, 261, 267, 273, 279, 285, 291, 297, 303, 309, 315, 323, 331, 341, 350, 360

Offset: 1

Anthony Hernandez, Oct 09 2017

nonn

A_n is a binary symmetric Hankel matrix.
Lim_{n->infinity} a(n)/n^2 = 0.
Proof: It can be seen from the formula that a(n) is bound from above by n*[number of perfect powers <= 2*n]. Powers of any particular number contribute no more than log_2(n) each, and there are no more than sqrt(2n) numbers that contribute anything at all, so a(n) <= n*log_2(n)*sqrt(2n), and a(n)/n^2 <= sqrt(2)*log_2(n)/sqrt(n), which goes to 0 at infinity. - Andrey Zabolotskiy, Oct 16 2017
Conjecture: The golden ratio/golden conjugate are eigenvalues of A_n if and only if n=6, 8 or 9. This has been verified up to n=500.
Conjecture: the sequence increases monotonically. - Robert G. Wilson v, Oct 09 2017
Where the parity of a(n) switches: 2, 4, 8, 16, 18, 32, 50, 64, 72, 98, 108, 128, 162, 200, 242, 256, 288, 338, 392, 450, 500, 512, 578, 648, 722, 800, 864, 882, 968, etc. Each number that is twice a square is present. - Robert G. Wilson v, Oct 09 2017

			         |0 0 1 0 0|
         |0 1 0 0 0|
   A_5 = |1 0 0 0 1| and so a(5) = 8.
         |0 0 0 1 1|
         |0 0 1 1 0|

Robert G. Wilson v, Table of n, a(n) for n = 1..10000
Mathematics StackExchange, A One Third Conjecture

Cf. A075802, A292918 (analogous sequence built upon A010051 instead of A075802).

perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; f[n_] := Plus @@ Flatten@ Table[ Boole[ perfectPowerQ[i + j]], {i, n}, {j, n}]; Array[f, 60] (* Robert G. Wilson v, Oct 09 2017 *)
f[n_] := f[n] = f[n - 1] + 2 Plus @@ Flatten@ Table[ Boole[ perfectPowerQ[i + n]], {i, n - 1}] + If[ perfectPowerQ[2 n], 1, 0]; f[1] = 0; Array[f, 60] (* Robert G. Wilson v, Nov 23 2017 *)

a(n) = sum(i=1, n, sum(j=1, n, ispower(i+j) != 0)); \\ Michel Marcus, Oct 09 2017

a(n) ~ (2/3)*n^(3/2). - Robert G. Wilson v, Oct 10 2017
a(n) = Sum_{k=1..n} k*A075802(k+1) + Sum_{k=1..n-1} k*A075802(2*n-k+1). - Andrey Zabolotskiy, Oct 16 2017
a(n) = a(n-1) + 2*(i+n) is a perfect power for i=1..n, + 1 if 2n is a perfect power. - Robert G. Wilson v, Nov 23 2017

a(31) onward from Robert G. Wilson v, Oct 09 2017

A292918 Let A_n be a square n X n matrix with entries A_n(i,j)=1 if i+j is prime, and A_n(i,j)=0 otherwise. Then a(n) counts the 1's in A_n.

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 19, 23, 29, 37, 43, 51, 57, 63, 71, 81, 89, 97, 105, 113, 123, 135, 145, 157, 169, 181, 195, 209, 221, 235, 249, 263, 277, 293, 309, 327, 345, 363, 381, 401, 419, 439, 457, 475, 495, 515, 533, 551, 571, 591, 613, 637, 659, 683, 709, 735

Offset: 1

Anthony Hernandez, Sep 26 2017

nonn

Bertrand's postulate guarantees for every integer n the existence of at least one prime q with n < q < 2n. Equivalently, A(n) has at least one skew diagonal below the main skew diagonal whose entries will be equal to 1.

			         |1 1 0 1 0|
         |1 0 1 0 1|
   A_5 = |0 1 0 1 0| and so a(5) = 11.
         |1 0 1 0 0|
         |0 1 0 0 0|

David A. Corneth, Table of n, a(n) for n = 1..10000
William Dowling and Nadia Lafreniere, Homomesy on permutations with toggling actions, arXiv:2312.02383 [math.CO], 2023. See page 10.

Cf. A000040, A000720, A069879, A071917.

sol:=[]; for n in [1..56] do k:=0; for i,j in [1..n] do if IsPrime(i+j) then k:=k+1; end if; end for; Append(~sol,k);end for; sol; // Marius A. Burtea, Aug 29 2019

with(numtheory):
a:= proc(n) option remember; `if`(n=1, 1,
      a(n-1)+2*(pi(2*n-1)-pi(n)))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 29 2017

A[n_] := Table[Boole[PrimeQ[i + j]], {i, 1, n}, {j, 1, n}]; a[n_] := Count[Flatten[A[n]], 1];
(* or, after Alois P. Heinz (200 times faster): *)
a[1] = 1; a[n_] := a[n] = a[n-1] + 2(PrimePi[2n-1] - PrimePi[n]);
Array[a, 80] (* Jean-François Alcover, Sep 29 2017 *)

first(n) = {my(res = vector(n), pn = 0, p2n1 = 1); res[1] = 1; for(i = 2, n,
if(isprime(i), pn++); if(isprime(2*i-1), p2n1++); res[i] = res[i-1] + 2*(p2n1 - pn)); res} \\ David A. Corneth, Aug 31 2019

from sympy import primepi
from sympy.core.cache import cacheit
@cacheit
def a(n): return 1 if n==1 else a(n - 1) + 2*(primepi(2*n - 1) - primepi(n))
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Dec 13 2017, after Alois P. Heinz

From Alois P. Heinz, Sep 29 2017: (Start)
a(n) = a(n-1) + 2 * (pi(2*n-1) - pi(n)) for n > 1, a(1) = 1.
a(n) = A069879(n) + 1 = 2*A071917(n) + 1. (End)
a(n) = Sum_{i=1..n} (pi(n+i) - pi(i)), where pi = A000720. - Ridouane Oudra, Aug 29 2019
a(n) = Sum_{p <= 2n+1, p prime} min(p-1, 2n+1-p). - Ridouane Oudra, Oct 30 2023

A276305 Primes p such that d(p*(2p+1)) = 12 where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

31, 37, 73, 103, 137, 139, 181, 193, 211, 269, 373, 433, 463, 541, 563, 571, 587, 733, 751, 859, 887, 929, 1021, 1129, 1151, 1381, 1399, 1489, 1637, 1723, 1993, 2053, 2083, 2087, 2237, 2521, 2621, 2731, 2837, 2843, 2909, 3109, 3137, 3209, 3271, 3313, 3323, 3343, 3541, 4091

Offset: 1

Anthony Hernandez, Aug 29 2016

nonn

Conjecture: this sequence is infinite.

Each number p * (2p + 1) is of the form p * q * r^2 but not of the form p * q^5. - David A. Corneth, Aug 30 2016

			Consider 31. Then 31*((2*31)+1) = 2*(31^2) + 31 = 1953 = 3*3*7*31 and d(1953) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A005384, A030630, A054753, A276045, A276307.

[n: n in [0..5000] | NumberOfDivisors(2*n+1) eq 6 and IsPrime(n)]; // Vincenzo Librandi, Aug 30 2016

Select[Prime@ Range@ 576, DivisorSigma[0, # (2 # + 1)] == 12 &] (* Michael De Vlieger, Aug 30 2016 *)

is(n) = ispseudoprime(n) && numdiv(n*(2*n+1))==12 \\ Felix Fröhlich, Aug 29 2016

is(n)=numdiv(2*n+1)==6 && isprime(n) \\ Charles R Greathouse IV, Aug 29 2016

More terms from Antti Karttunen, Aug 29 2016

A276307 Primes p such that d(p*(2p+1)) = 16 where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

67, 97, 127, 199, 227, 229, 241, 277, 307, 313, 331, 379, 397, 457, 467, 499, 547, 617, 619, 647, 709, 727, 739, 757, 773, 797, 823, 829, 857, 883, 977, 1033, 1069, 1093, 1117, 1123, 1171, 1187, 1193, 1201, 1277, 1297, 1303, 1319, 1423, 1447, 1459, 1471, 1483, 1609

Offset: 1

Anthony Hernandez, Aug 29 2016

nonn

Conjecture: this sequence is infinite.

Or, primes p such that d(2p+1)=8. - Zak Seidov, Sep 07 2016

			Consider 67. Then 67*(2*67+1) = 9045 and d(9045) = 16.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A005384, A030634, A276045, A276305.

[n: n in [0..2000] | NumberOfDivisors(2*n+1) eq 8 and IsPrime(n)]; // Vincenzo Librandi, Aug 30 2016

Select[Prime@ Range@ 256, DivisorSigma[0, # (2 # + 1)] == 16 &] (* Michael De Vlieger, Aug 30 2016 *)

lista(nn) = forprime (p=2, nn, if (numdiv(p*(2*p+1)) == 16, print1(p, ", "))); \\ Michel Marcus, Aug 29 2016

is(n)=numdiv(2*n+1)==8 && isprime(n) \\ Charles R Greathouse IV, Aug 29 2016

Corrected and extended by Michel Marcus, Aug 29 2016

A276045 Primes p such that d(p*(2p+1)) = 8 where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

7, 13, 17, 19, 43, 47, 59, 61, 71, 79, 101, 107, 109, 149, 151, 163, 167, 197, 223, 257, 263, 271, 311, 317, 347, 349, 353, 383, 389, 401, 421, 439, 449, 461, 479, 503, 521, 523, 557, 569, 599, 601, 613, 631, 673, 677, 691, 701, 811, 821, 827, 839, 853, 863, 881, 919

Offset: 1

Anthony Hernandez, Aug 17 2016

Primes p such that 2p+1 is in A030513. - Robert Israel, Aug 17 2016
From Anthony Hernandez, Aug 29 2016: (Start)
Conjecture: this sequence is infinite.
It appears that the prime numbers in this sequence which have 7 for as final digit form the sequence A104164.
Conjecture: this sequence contains infinitely many twin primes. The first few twin primes in this sequence are 17,19,59,61,107,109,521,523,599,601,... (End)
From Bernard Schott, Apr 28 2020: (Start)
This sequence equals the union of {13} and A234095; proof by double inclusion:
-> 1st inclusion: {13} Union A234095 is included in A276045.
1) if p = 13, then 13*27 = 351 = 3^3 * 13, hence d(351) = 8 and 13 belongs to A276045.
2) if p is in A234095, then p*(2*p+1) = p*r*s (p,r,s primes) and d(p*r*s) = 8, hence p is in 276045.
-> 2nd inclusion: A276045 is included in {13} Union A234095.
If p is in A276095, then m=p*(2*p+1) has 8 divisors and there are only three possibilities: m = u*v*w, or m = u^3*v or m = u^7 with u, v, w are distinct primes.
1st case: if p*(2*p+1) = u*v*w then u=p, and 2p+1=v*w is semiprime; hence, p is in A234095 Union {13}.
2nd case: if p*(2p+1) = u^3*v then  p=v and 2*p+1=u^3 ==> 2*p = u^3-1 = (u-1)*(u^2+u+1) with 2 and p are primes; then (u-1=2, u^2+u+1=p) so u=3, and p=3^2+3+1=13; hence p = 13 belongs to {13} Union A234095.
3rd case: p*(2p+1) = u^7 is impossible.
Conclusion: this sequence = {13} Union A234095. (End)

			d(7*(2*7+1))=d(105)=8 so 7 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A030513, A030626.

Equals {13} Union A234095.

select(n -> isprime(n) and numtheory:-tau(n*(2*n+1))=8,
[seq(i, i=3..1000, 2)]); # Robert Israel, Aug 17 2016

Select[Prime@ Range@ 160, DivisorSigma[0, # (2 # + 1)] == 8 &] (* Michael De Vlieger, Aug 28 2016 *)

lista(nn) = forprime(p=2, nn, if (numdiv(p*(2*p+1))==8, print1(p, ", "))); \\ Michel Marcus, Aug 17 2016

cp's OEIS Frontend

User: Anthony Hernandez

A323552 Denominators of the partial Euler product representation of Pi/4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A323551 Numerators of the partial Euler product representation of Pi/4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A293462 Let A_n be a square n X n matrix with entries A_n(i,j)=1 if i+j is a perfect power, and A_n(i,j)=0 otherwise. Then a(n) counts the 1's in A_n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A292918 Let A_n be a square n X n matrix with entries A_n(i,j)=1 if i+j is prime, and A_n(i,j)=0 otherwise. Then a(n) counts the 1's in A_n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

A276305 Primes p such that d(p*(2p+1)) = 12 where d(n) is the number of divisors of n (A000005).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Extensions

A276307 Primes p such that d(p*(2p+1)) = 16 where d(n) is the number of divisors of n (A000005).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Extensions