A290732 -id:A290732 - cp's OEIS frontend

A290730 Analog of A084848, replacing "quadratic residue" (X^2) with "value of X(3X-1)/2". a(n) = A290732(A290729(n)).

1, 3, 4, 6, 7, 9, 10, 12, 11, 12, 18, 21, 24, 28, 36, 40, 42, 44, 66, 77, 72, 84, 108, 120, 126, 162, 168, 216, 240, 252, 280, 264, 308, 396, 440, 462, 594, 504, 648, 720, 756, 840, 1008, 1080, 1134, 1260, 1512, 1512, 1680, 2016

Offset: 1

N. J. A. Sloane, Aug 10 2017

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..182
Andreas Enge, William Hart, Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], 2016-2018. See Table 6.

Cf. A000326, A085635, A084848, A290727, A290728, A290729, A290732.

a290732[n_] := Product[{p, e} = pe; If[p <= 3, p^e, (p^e - p^(e-1))/2 + (p^(e-1) - p^(Mod[e+1, 2]))/(2*(p+1))+1], {pe, FactorInteger[n]}];
r = 2; Reap[For[j = 1, j <= 24001, j = j+1, w = a290732[j]; t = w/j; If[t < r, r = t; Sow[w]]]][[2, 1]] (* Jean-François Alcover, Oct 03 2018, after Hugo Pfoertner *)

a290732(n)={my(f=factor(n));prod(k=1,#f~,my([p,e]=f[k, ]); if(p<=3,p^e,(p^e-p^(e-1))/2+(p^(e-1)-p^((e+1)%2))/(2*(p+1))+1))}
my(r=2);for(j=1,24001,my(w=a290732(j),t=w/j);if(tHugo Pfoertner, Aug 26 2018

More terms from Hugo Pfoertner, Aug 23 2018

a(1), a(19) and a(38) corrected by Hugo Pfoertner, Aug 26 2018

A290731 Number of distinct values of X*(X+1) mod n.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 4, 4, 3, 6, 4, 7, 4, 6, 8, 9, 4, 10, 6, 8, 6, 12, 8, 11, 7, 11, 8, 15, 6, 16, 16, 12, 9, 12, 8, 19, 10, 14, 12, 21, 8, 22, 12, 12, 12, 24, 16, 22, 11, 18, 14, 27, 11, 18, 16, 20, 15, 30, 12, 31, 16, 16, 32, 21, 12, 34, 18, 24, 12, 36, 16, 37, 19, 22, 20, 24, 14, 40, 24

Offset: 1

N. J. A. Sloane, Aug 10 2017

Also the number of distinct values of X^2+X+1 mod n. - N. J. A. Sloane, Oct 05 2024

			The values taken by X^2+X mod n for small n are:
1, [0]
2, [0]
3, [0, 2]
4, [0, 2]
5, [0, 1, 2]
6, [0, 2]
7, [0, 2, 5, 6]
8, [0, 2, 4, 6]
9, [0, 2, 3, 6]
10, [0, 2, 6]
11, [0, 1, 2, 6, 8, 9]
12, [0, 2, 6, 8]
...

Andrew Howroyd, Table of n, a(n) for n = 1..10000
Andreas Enge, William Hart, Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], (24-August-2016). See Table 5.

Cf. A000224 (analog for X^2), A290732.

a:=[]; M:=80;
for n from 1 to M do
q1:={};
for i from 0 to n-1 do q1:={op(q1), (i^2+i) mod n}; od;
s1:=sort(convert(q1,list));
a:=[op(a),nops(s1)];
od:
a;

a[n_] := Product[{p, e} = pe; If[p==2, 2^(e-1), 1+Quotient[p^(e+1), (2p+2)]], {pe, FactorInteger[n]}];
Array[a, 100] (* Jean-François Alcover, Aug 05 2018, after Andrew Howroyd *)

a(n)={my(v=vector(n)); for(i=0, n-1, v[i*(i+1)%n + 1]=1); vecsum(v)} \\ Andrew Howroyd, Aug 01 2018

a(n)={my(f=factor(n)); prod(i=1, #f~, my([p,e]=f[i,]); if(p==2, 2^(e-1), 1 + p^(e+1)\(2*p+2)))} \\ Andrew Howroyd, Aug 01 2018

from math import prod
from sympy import factorint
def A290731(n): return prod((p**(e+1)//(p+(q:=p>2))>>1)+q for p, e in factorint(n).items()) # Chai Wah Wu, Oct 07 2024

Multiplicative with a(2^e) = 2^(e-1), a(p^2) = 1 + floor(p^(e+1)/(2*p+2)) for odd prime p. - Andrew Howroyd, Aug 01 2018

A014113 a(n) = a(n-1) + 2*a(n-2) with a(0)=0, a(1)=2.

Original entry on oeis.org

0, 2, 2, 6, 10, 22, 42, 86, 170, 342, 682, 1366, 2730, 5462, 10922, 21846, 43690, 87382, 174762, 349526, 699050, 1398102, 2796202, 5592406, 11184810, 22369622, 44739242, 89478486, 178956970, 357913942, 715827882, 1431655766, 2863311530, 5726623062, 11453246122

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 457
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A001045, A078008, A163271, A290732.

a014113 n = a014113_list !! n
a014113_list = 0 : 2 : zipWith (+)
                       (map (* 2) a014113_list) (tail a014113_list)
-- Reinhard Zumkeller, Jul 20 2013

LinearRecurrence[{1,2},{0,2},50] (* Vincenzo Librandi, Feb 19 2012 *)

a(0) = 0 and if n>=1, a(n) = 2^n - a(n-1).
a(n) = A078008(n+1). - Reinhard Zumkeller, Jun 10 2005
a(n) = 2*A001045(n). - Benoit Jubin, Dec 01 2011
a(n) = (2^(n+1) - 2*(-1)^n)/3. - Ralf Stephan, Jul 18 2013
G.f.: 2*x/(1+x)/(1-2*x). - Colin Barker, Feb 19 2012
G.f.: 1/Q(0) -1, where Q(k) = 1 + 2*x^2 - (2*k+3)*x + x*(2*k+1 - 2*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 05 2013
Consider a Pascal-like triangle T(n,k) in which T(n,0) = T(n,n) = 0 if n even, 1 if n odd, and each interior entry T(n,k) is the sum of the two entries "above" it: T(n,k) = T(n-1,k-1) + T(n-1,k) for 0 < k < n. Then, a(n) is the sum of the entries in the n-th row of T(n,k). - Greg Dresden, May 24 2024

A290729 Analog of A085635, replacing "quadratic residue" (X^2) with "value of X(3X-1)/2".

Original entry on oeis.org

1, 5, 7, 11, 13, 17, 19, 23, 25, 35, 55, 65, 77, 91, 119, 133, 143, 175, 275, 325, 385, 455, 595, 665, 715, 935, 1001, 1309, 1463, 1547, 1729, 1925, 2275, 2975, 3325, 3575, 4675, 5005, 6545, 7315, 7735, 8645

Offset: 1

N. J. A. Sloane, Aug 10 2017

nonn

Positions k where R(k) = A290732(k)/k, achieves a new minimum.

Hugo Pfoertner, Table of n, a(n) for n = 1..182
Andreas Enge, William Hart, Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], 2016-2018. See Table 6.

Cf. A000326, A085635, A084848, A290727, A290728, A290730, A290732.

a[n_] := Product[{p, e} = pe; If[p <= 3, p^e, (p^e - p^(e-1))/2 + (p^(e-1) - p^(Mod[e+1, 2]))/(2*(p+1)) + 1], {pe, FactorInteger[n]}];
r = 2; Reap[For[j=1, j <= 10^4, j = j+1, t = a[j]/j; If[tJean-François Alcover, Oct 02 2018, after Hugo Pfoertner *)

a290732(n)={my(f=factor(n));prod(k=1,#f~,my([p,e]=f[k,]);if(p<=3,p^e,(p^e-p^(e-1))/2+(p^(e-1)-p^((e+1)%2))/(2*(p+1))+1))}
my(r=2);for(j=1,10001,my(t=a290732(j)/j);if(tHugo Pfoertner, Aug 26 2018

a(1) corrected by Hugo Pfoertner, Aug 26 2018

A317623 Number of distinct values of X(3X-1) mod n.

Original entry on oeis.org

1, 1, 3, 2, 3, 3, 4, 4, 9, 3, 6, 6, 7, 4, 9, 8, 9, 9, 10, 6, 12, 6, 12, 12, 11, 7, 27, 8, 15, 9, 16, 16, 18, 9, 12, 18, 19, 10, 21, 12, 21, 12, 22, 12, 27, 12, 24, 24, 22, 11, 27, 14, 27, 27, 18, 16, 30, 15, 30, 18, 31, 16, 36, 32, 21, 18, 34, 18, 36, 12, 36

Offset: 1

Andrew Howroyd, Aug 01 2018

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

Cf. A000224, A290731, A290732.

f[2, e_] := 2^(e-1); f[3, e_] := 3^e; f[p_, e_] := 1 + Floor[p^(e+1)/(2*p+2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 13 2020 *)

a(n)={my(v=vector(n)); for(i=0, n-1, v[i*(3*i-1)%n + 1]=1); vecsum(v)}

a(n)={my(f=factor(n)); prod(i=1, #f~, my([p,e]=f[i,]); if(p<=3, if(p==2, 2^(e-1), 3^e), 1 + p^(e+1)\(2*p+2)))}

Multiplicative with a(2^e) = 2^(e-1), a(3^e) = 3^e, a(p^e) = 1 + floor( p^(e+1)/(2*p+2) ) for prime p >= 5.

cp's OEIS Frontend

A290730 Analog of A084848, replacing "quadratic residue" (X^2) with "value of X(3X-1)/2". a(n) = A290732(A290729(n)).

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A290731 Number of distinct values of X*(X+1) mod n.

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A014113 a(n) = a(n-1) + 2*a(n-2) with a(0)=0, a(1)=2.

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A290729 Analog of A085635, replacing "quadratic residue" (X^2) with "value of X(3X-1)/2".

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Mathematica

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A317623 Number of distinct values of X*(3*X-1) mod n.

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A317623 Number of distinct values of X(3X-1) mod n.