A047233 -id:A047233 - cp's OEIS frontend

A378096 Decimal expansion of Product_{k>=2} (1 - 1/A047233(k)^2).

8, 7, 5, 1, 8, 0, 0, 6, 8, 5, 7, 3, 0, 1, 8, 1, 3, 4, 6, 6, 0, 5, 4, 0, 8, 3, 0, 0, 8, 2, 7, 3, 9, 4, 7, 0, 4, 3, 2, 8, 7, 6, 1, 7, 4, 3, 9, 8, 4, 9, 3, 3, 4, 1, 5, 4, 4, 2, 4, 0, 7, 5, 2, 2, 9, 0, 1, 9, 9, 2, 1, 5, 3, 9, 4, 3, 0, 2, 6, 9, 4, 4, 7, 0, 9, 3, 5, 0, 1, 4, 1, 5, 2, 0, 4, 5, 9, 7, 9, 5, 3, 5, 2, 5, 3

Offset: 0

Frank Richter, Nov 16 2024

Infinite product (1-1/((q-1)^2)) where q = 5, 7, 11, 13, 17, 19, ... (A007310), are integers of the form sqrt(24*k+1), where k are terms of A001318.

			0.87518006857301813466054083008273947043287617439849...

Cf. A001318, A007310, A047233.

RealDigits[2^(2/3)*Sqrt[3]/Pi, 10, 120][[1]] (* Amiram Eldar, Nov 17 2024 *)

Equals 2^(2/3)*sqrt(3)/Pi. - Amiram Eldar, Nov 17 2024

A047241 Numbers that are congruent to {1, 3} mod 6.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 19, 21, 25, 27, 31, 33, 37, 39, 43, 45, 49, 51, 55, 57, 61, 63, 67, 69, 73, 75, 79, 81, 85, 87, 91, 93, 97, 99, 103, 105, 109, 111, 115, 117, 121, 123, 127, 129, 133, 135, 139, 141, 145, 147, 151, 153, 157, 159, 163, 165, 169, 171, 175, 177, 181, 183

Offset: 1

N. J. A. Sloane

Also the numbers k such that 10^p+k could possibly be prime. - Roderick MacPhee, Nov 20 2011 This statement can be written as follows. If 10^m + k = prime, for any m >= 1, then k is in this sequence. See the pink box comments by Roderick MacPhee from Dec 09 2014. - Wolfdieter Lang, Dec 09 2014
The odd-indexed terms are one more than the arithmetic mean of their neighbors; the even-indexed terms are one less than the arithmetic mean of their neighbors. - Amarnath Murthy, Jul 29 2003
Partial sums are A212959. - Philippe Deléham, Mar 16 2014
12*a(n) is conjectured to be the length of the boundary after n iterations of the hexagon and square expansion shown in the link. The squares and hexagons have side length 1 in some units. The pattern is supposed to become the planar Archimedean net 4.6.12 when n -> infinity. - Kival Ngaokrajang, Nov 30 2014
Positive numbers k for which 1/2 + k/3 + k^2/6 is an integer. - Bruno Berselli, Apr 12 2018

L. Lovasz, J. Pelikan, K. Vesztergombi, Discrete Mathematics, Springer (2003); 14.4, p. 225.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
L. Lovász, J. Pelikán and K. Vesztergombi, Discrete Mathematics, Elementary and Beyond, Springer (2003); 14.4, p. 225.
Kival Ngaokrajang, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A047233, A056970, A007310, A047228, A047261, A047273, A212959.
Subsequence of A186422.
Union of A016921 and A016945. - Wesley Ivan Hurt, Sep 28 2013

a047241 n = a047241_list !! (n-1)
a047241_list = 1 : 3 : map (+ 6) a047241_list
-- Reinhard Zumkeller, Feb 19 2013

seq(3*k-2-((k+1) mod 2), k=1..100); # Wesley Ivan Hurt, Sep 28 2013

Table[{2, 4}, {30}] // Flatten // Prepend[#, 1]& // Accumulate (* Jean-François Alcover, Jun 10 2013 *)
Select[Range[200], MemberQ[{1, 3}, Mod[#, 6]]&] (* or *) LinearRecurrence[{1, 1, -1}, {1, 3, 7}, 70] (* Harvey P. Dale, Oct 01 2013 *)

a(n)=bitor(3*n-3,1) \\ Charles R Greathouse IV, Sep 28 2013

for n in range(1,10**5):print(3*n-2-((n+1)%2)) # Soumil Mandal, Apr 14 2016

From Paul Barry, Sep 04 2003: (Start)
O.g.f.: (1 + 2*x + 3*x^2)/((1 + x)*(1 - x)^2) = (1 + 2*x + 3*x^2)/((1 - x)*(1 - x^2)).
E.g.f.: (6*x + 1)*exp(x)/2 + exp(-x)/2;
a(n) = 3*n - 5/2 - (-1)^n/2. (End)
a(n) = 2*floor((n-1)/2) + 2*n - 1. - Gary Detlefs, Mar 18 2010
a(n) = 6*n - a(n-1) - 8 with n > 1, a(1)=1. - Vincenzo Librandi, Aug 05 2010
a(n) = 3*n - 2 - ((n+1) mod 2). - Wesley Ivan Hurt, Jun 29 2013
a(1)=1, a(2)=3, a(3)=7; for n>3, a(n) = a(n-1) + a(n-2) - a(n-3). - Harvey P. Dale, Oct 01 2013
From Benedict W. J. Irwin, Apr 13 2016: (Start)
A005408(a(n)+1) = A016813(A001651(n)),
A007310(a(n)) = A005408(A087444(n)-1),
A007310(A005408(a(n)+1)) = A017533(A001651(n)). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)) + log(3)/4. - Amiram Eldar, Dec 11 2021

Formula corrected by Bruno Berselli, Jun 24 2010

A047255 Numbers that are congruent to {1, 2, 3, 5} mod 6.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23, 25, 26, 27, 29, 31, 32, 33, 35, 37, 38, 39, 41, 43, 44, 45, 47, 49, 50, 51, 53, 55, 56, 57, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 77, 79, 80, 81, 83, 85, 86, 87, 89, 91, 92, 93, 95, 97, 98, 99, 101, 103, 104

Offset: 1

N. J. A. Sloane

Each element is coprime to preceding two elements. - Amarnath Murthy, Jun 12 2001

The sequence is the interleaving of A047241 with A016789. - Guenther Schrack, Feb 16 2019

			After 21 and 23 the next term is 25 as 24 has a common divisor with 21.

Guenther Schrack, Table of n, a(n) for n = 1..10012
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A007310, A016789, A047228, A047237, A047241, A047251, A047261, A047273, A062062, A062063.

Complement: A047233

a047255 n = a047255_list !! (n-1)
a047255_list = 1 : 2 : 3 : 5 : map (+ 6) a047255_list
-- Reinhard Zumkeller, Jan 17 2014

[n : n in [0..100] | n mod 6 in [1, 2, 3, 5]]; // Wesley Ivan Hurt, May 21 2016

A047255:=n->(6*n-4+I^(1-n)+I^(n-1))/4: seq(A047255(n), n=1..100); # Wesley Ivan Hurt, May 20 2016

Select[Range[100], MemberQ[{1, 2, 3, 5}, Mod[#, 6]] &]
LinearRecurrence[{2,-2,2,-1},{1,2,3,5},100] (* Harvey P. Dale, May 14 2020 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,2,-2,2]^(n-1)*[1;2;3;5])[1,1] \\ Charles R Greathouse IV, Feb 11 2017

a=(x*(1+x^2+x^3)/((1+x^2)*(1-x)^2)).series(x, 80).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 16 2019

{k | k == 1, 2, 3, 5 (mod 6)}.
G.f.: x*(1 + x^2 + x^3) / ((1+x^2)*(1-x)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 20 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4), for n>4.
a(n) = (6*n - 4 + i^(1-n) + i^(n-1))/4, where i = sqrt(-1).
a(2*n) = A016789(n-1) for n>0, a(2*n-1) = A047241(n). (End)
E.g.f.: (2 + sin(x) + (3*x - 2)*exp(x))/2. - Ilya Gutkovskiy, May 21 2016
a(1-n) = - A047251(n). - Wesley Ivan Hurt, May 21 2016
From Guenther Schrack, Feb 16 2019: (Start)
a(n) = (6*n - 4 + (1 - (-1)^n)*(-1)^(n*(n-1)/2))/4.
a(n) = a(n-4) + 6, a(1)=1, a(2)=2, a(3)=3, a(4)=5, for n > 4.
a(n) = A047237(n) + 1. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*sqrt(3)*Pi/36 + log(2)/3 - log(3)/4. - Amiram Eldar, Dec 17 2021
a(n) = 2*n - 1 - floor(n/2) + floor(n/4) - floor((n+1)/4). - Ridouane Oudra, Feb 21 2023

More terms from Larry Reeves (larryr(AT)acm.org), Jun 15 2001

A352475 Numbers m such that gcd(d(m),6) = 1.

Original entry on oeis.org

1, 16, 64, 81, 625, 729, 1024, 1296, 2401, 4096, 5184, 10000, 11664, 14641, 15625, 28561, 38416, 40000, 46656, 50625, 59049, 65536, 82944, 83521, 117649, 130321, 153664, 194481, 234256, 250000, 262144, 279841, 331776, 455625, 456976, 531441, 640000, 707281, 746496

Offset: 1

Michael De Vlieger, Mar 26 2022

All terms are square since numbers coprime to 6 are odd.
The square roots of terms are in A001694.
Intersection of A000290 and A336590, i.e., numbers whose prime factorization has only exponents that are congruent to {0, 4} mod 6 (A047233). - Amiram Eldar, Mar 31 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000
Titus Hilberdink, How often is d(n) a power of a given integer?, Journal of Number Theory, Vol. 236 (2022), pp. 261-279.

Cf. A000005, A000290, A001694, A007310, A047233, A076400, A100044, A336590, A350014.

Select[Range[864]^2, GCD[DivisorSigma[0, #], 6] == 1 &] (* or, more efficiently, *)
With[{nn = 864}, Select[Union[Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}]]^2, Mod[DivisorSigma[0, #], 3] > 0 &]]

isok(m) = gcd(numdiv(m), 6) == 1; \\ Michel Marcus, Mar 29 2022

m = 100000; seq = direuler(p=2, m, (1 - X^8)/(1 - X^4)/(1 - X^6)); for(n=1, m, if(seq[n] != 0, print1(n, ", "))) \\ Vaclav Kotesovec, May 19 2022

a(n) = A350014(n)^2.
Sum_{n>=1} 1/a(n) = Pi^2/9 (A100044). - Amiram Eldar, Mar 31 2022
The number of terms <= x is (zeta(3/2)/zeta(2))*x^(1/4) + (zeta(2/3)/zeta(4/3))*x^(1/6) + O(x^(1/8 + eps)), for all eps > 0 (Hilberdink, 2022). - Amiram Eldar, May 18 2022

A047260 Numbers that are congruent to {0, 1, 4, 5} mod 6.

Original entry on oeis.org

0, 1, 4, 5, 6, 7, 10, 11, 12, 13, 16, 17, 18, 19, 22, 23, 24, 25, 28, 29, 30, 31, 34, 35, 36, 37, 40, 41, 42, 43, 46, 47, 48, 49, 52, 53, 54, 55, 58, 59, 60, 61, 64, 65, 66, 67, 70, 71, 72, 73, 76, 77, 78, 79, 82, 83, 84, 85, 88, 89, 90, 91, 94, 95, 96, 97

Offset: 1

N. J. A. Sloane

Numbers x which are not a solution to 3^x - 2^x == 5 mod 7. - Cino Hilliard, May 14 2003

The sequence is the interleaving of A047233 with A007310. - Guenther Schrack, Feb 13 2019

Guenther Schrack, Table of n, a(n) for n = 1..10006
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A007310, A047233, A047269.

Complement: A047243.

Filtered([0..100],n->n mod 6 = 0 or n mod 6 = 1 or n mod 6 = 4 or n mod 6 = 5); # Muniru A Asiru, Feb 19 2019

[n : n in [0..100] | n mod 6 in [0, 1, 4, 5]]; // Wesley Ivan Hurt, May 21 2016

A047260:=n->(6*n-5-I^(2*n)+(1-I)*I^(-n)+(1+I)*I^n)/4: seq(A047260(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Table[(6n-5-I^(2n)+(1-I)*I^(-n)+(1+I)*I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 21 2016 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,4,5,6},70] (* Harvey P. Dale, Sep 20 2023 *)

my(x='x+O('x^70)); concat([0], Vec(x^2*(1+3*x+x^2+x^3)/((1-x)*(1-x^4)))) \\ G. C. Greubel, Feb 16 2019

a=(x^2*(1+3*x+x^2+x^3)/((1-x)*(1-x^4))).series(x, 72).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 16 2019

G.f.: x^2*(1+3*x+x^2+x^3) / ((1+x)*(1+x^2)*(1-x)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (6*n - 5 - i^(2*n) + (1-i)*i^(-n) + (1+i)*i^n)/4 where i=sqrt(-1).
a(2*n) = A007310(n), a(2*n-1) = A047233(n). (End)
From Guenther Schrack, Feb 13 2019: (Start)
a(n) = (6*n - 5 - (-1)^n + 2*(-1)^(n*(n + 1)/2))/4.
a(n) = a(n-4) + 6, a(1)=0, a(2)=1, a(3)=4, a(4)=5, for n > 4.
a(-n) = -A047269(n+2). (End)
Sum_{n>=2} (-1)^n/a(n) = sqrt(3)*Pi/36 + log(3)/4 + 2*log(2)/3. - Amiram Eldar, Dec 16 2021

More terms from Wesley Ivan Hurt, May 21 2016

A047262 Numbers that are congruent to {0, 2, 4, 5} mod 6.

Original entry on oeis.org

0, 2, 4, 5, 6, 8, 10, 11, 12, 14, 16, 17, 18, 20, 22, 23, 24, 26, 28, 29, 30, 32, 34, 35, 36, 38, 40, 41, 42, 44, 46, 47, 48, 50, 52, 53, 54, 56, 58, 59, 60, 62, 64, 65, 66, 68, 70, 71, 72, 74, 76, 77, 78, 80, 82, 83, 84, 86, 88, 89, 90, 92, 94, 95, 96, 98

Offset: 1

N. J. A. Sloane

The sequence is the interleaving of A047233 with A016789(n-1). - Guenther Schrack, Feb 14 2019

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).
Index entries for two-way infinite sequences.

Cf. A016789, A047233, A047237, A047273.

Complement: A047241.

[n : n in [0..100] | n mod 6 in [0, 2, 4, 5]]; // Wesley Ivan Hurt, May 21 2016

A047262:=n->(6*n-4-I^(1-n)+I^(1+n))/4: seq(A047262(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Select[Range[0,100], MemberQ[{0,2,4,5}, Mod[#,6]]&] (* or *) LinearRecurrence[{2,-2,2,-1}, {0,2,4,5}, 70] (* Harvey P. Dale, Dec 09 2015 *)

my(x='x+O('x^70)); concat([0], Vec(x^2*(2+x^2)/((1+x^2)*(1-x)^2))) \\ G. C. Greubel, Feb 16 2019

a=(x^2*(2+x^2)/((1+x^2)*(1-x)^2)).series(x, 72).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 16 2019

From R. J. Mathar, Oct 08 2011: (Start)
G.f.: x^2*(2+x^2) / ( (1+x^2)*(1-x)^2 ).
a(n) = 3*n/2 - 1 - sin(Pi*n/2)/2. (End)
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n > 4.
a(n) = (6*n - 4 - i^(1-n) + i^(1+n))/4, where i = sqrt(-1).
a(2*n) = A016789(n-1) for n>0, a(2*n-1) = A047233(n).
a(2-n) = - A047237(n), a(n-1) = A047273(n) - 1 for n > 1. (End)
From Guenther Schrack, Feb 14 2019: (Start)
a(n) = (6*n - 4 - (1 - (-1)^n)*(-1)^(n*(n-1)/2))/4.
a(n) = a(n-4) + 6, a(1)=0, a(2)=2, a(3)=4, a(4)=5, for n > 4. (End)
Sum_{n>=2} (-1)^n/a(n) = log(3)/4 + log(2)/3 - sqrt(3)*Pi/36. - Amiram Eldar, Dec 17 2021

More terms from Wesley Ivan Hurt, May 21 2016

A047250 Numbers that are congruent to {0, 3, 4, 5} (mod 6).

Original entry on oeis.org

0, 3, 4, 5, 6, 9, 10, 11, 12, 15, 16, 17, 18, 21, 22, 23, 24, 27, 28, 29, 30, 33, 34, 35, 36, 39, 40, 41, 42, 45, 46, 47, 48, 51, 52, 53, 54, 57, 58, 59, 60, 63, 64, 65, 66, 69, 70, 71, 72, 75, 76, 77, 78, 81, 82, 83, 84, 87, 88, 89, 90, 93, 94, 95, 96, 99

Offset: 1

N. J. A. Sloane

The sequence is the interleaving of A047233 with A047270. - Guenther Schrack, Feb 15 2019

Guenther Schrack, Table of n, a(n) for n = 1..10010
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047233, A047246, A047270.

Complement: A047239.

[n : n in [0..150] | n mod 6 in [0, 3, 4, 5]]; // Wesley Ivan Hurt, Jun 02 2016

A047250:=n->(6*n-3+I^(2*n)-(1+I)*I^(-n)-(1-I)*I^n)/4: seq(A047250(n), n=1..100); # Wesley Ivan Hurt, Jun 02 2016

Select[Range[0,100], MemberQ[{0,3,4,5}, Mod[#,6]]&] (* or *) LinearRecurrence[{1,0,0,1,-1}, {0,3,4,5,6}, 60] (* Harvey P. Dale, Apr 01 2013 *)

my(x='x+O('x^70)); concat([0], Vec(x^2*(3+x+x^2+x^3)/((1+x)*(1+x^2)*(1-x)^2))) \\ G. C. Greubel, Feb 16 2019

a=(x^2*(3+x+x^2+x^3)/((1+x)*(1+x^2)*(1-x)^2)).series(x, 72).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 16 2019

G.f.: x^2*(3+x+x^2+x^3)/((1+x)*(1+x^2)*(1-x)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, Jun 02 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (6*n - 3 + i^(2*n) - (1+i)*i^(-n) - (1-i)*i^n)/4 where i=sqrt(-1).
a(2*k) = A047270(k), a(2*k-1) = A047233(k). (End)
E.g.f.: (2 - sin(x) - cos(x) + (3*x - 2)*sinh(x) + (3*x - 1)*cosh(x))/2. - Ilya Gutkovskiy, Jun 02 2016
From Guenther Schrack, Feb 15 2019: (Start)
a(n) = (6*n - 3 + (-1)^n - 2*(-1)^(n*(n-1)/2))/4.
a(n) = a(n-4) + 6, a(1)=0, a(2)=3, a(3)=4, a(4)=5, for n > 4.
a(-n) = -A047246(n+2). (End)
Sum_{n>=2} (-1)^n/a(n) = 2*log(2)/3 - Pi/(6*sqrt(3)). - Amiram Eldar, Dec 17 2021

A229489 Conjecturally, possible differences between prime(k)^2 and the next prime for some k.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 16, 18, 22, 24, 28, 30, 34, 36, 40, 42, 46, 48, 52, 54, 58, 60, 64, 66, 70, 72, 76, 78, 82, 84, 88, 90, 94, 96, 100, 102, 106, 108, 112, 114, 118, 120, 124, 126, 130, 132, 136, 138, 142, 144, 148, 150, 154, 156, 160, 162, 166, 168, 172, 174

Offset: 1

T. D. Noe, Oct 21 2013

nonn

Are there any missing terms? The first 10^7 primes were examined. All these differences occur for some k < 10^5. Note that the first differences of these terms is 1, 2, or 4.

The similarity to A047233 is understood by a comment in A091666. - R. J. Mathar, Oct 28 2013

Cf. A000040 (primes), A001248 (primes squared).
Cf. A004277 (conjecturally, possible gaps between adjacent primes).
Cf. A091666 (prime greater than prime(n)^2).
Cf. A229488 (possible differences between prime(k)^2 and the previous prime).

t = Table[p2 = Prime[k]^2; NextPrime[p2] - p2, {k, 100000}]; Take[Union[t], 60]
NextPrime[#]-#&/@(Prime[Range[100000]]^2)//Union (* Harvey P. Dale, Apr 19 2020 *)

cp's OEIS Frontend

A378096 Decimal expansion of Product_{k>=2} (1 - 1/A047233(k)^2).

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A047241 Numbers that are congruent to {1, 3} mod 6.

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A047255 Numbers that are congruent to {1, 2, 3, 5} mod 6.

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A352475 Numbers m such that gcd(d(m),6) = 1.

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A047260 Numbers that are congruent to {0, 1, 4, 5} mod 6.

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A047262 Numbers that are congruent to {0, 2, 4, 5} mod 6.

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