cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A290728 Analog of A084848, replacing "quadratic residue" (X^2) with "value of X^2+X".

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 6, 8, 12, 12, 12, 16, 24, 24, 36, 42, 44, 48, 72, 84, 96, 112, 144, 144, 168, 176, 264, 308, 288, 336, 432, 480, 504, 648, 672, 864, 960, 1008, 1008, 1056, 1232, 1584, 1760, 1848, 2376, 2016, 2592
Offset: 1

Views

Author

N. J. A. Sloane, Aug 10 2017

Keywords

Links

Crossrefs

Cf. A085635, A084848, A290727, A290731.

Programs

  • Mathematica
    a290731[n_] := Product[{p, e} = pe; If[p==2, 2^(e-1), 1 + Quotient[p^(e+1), (2p + 2)]], {pe, FactorInteger[n]}];
Reap[For[r = 2; k = 1, k <= 200000, k++, v = a290731[k]; t = v/k; If[t < r, r = t; Sow[v]]]][[2, 1]] (* Jean-François Alcover, Sep 13 2018, from PARI *)
  • PARI
    a290731(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)))} \\ from Andrew Howroyd
r=2;for(k=1,200000,v=a290731(k);t=v/k;if(tHugo Pfoertner, Aug 23 2018

Formula

a(n) = A290731(A290727(n)) - Hugo Pfoertner, Aug 23 2018

Extensions

More terms from Hugo Pfoertner, Aug 22 2018
Initial term added by Hugo Pfoertner, Aug 23 2018

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Aug 10 2017

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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

Extensions

More terms from Hugo Pfoertner, Aug 23 2018
a(1), a(19) and a(38) corrected by Hugo Pfoertner, Aug 26 2018

A085635 Compute S, the number of different quadratic residues modulo B for every base B. If the density S/B is smaller for B than for every B' less than B, then B is added to the sequence.

Original entry on oeis.org

1, 3, 4, 8, 12, 16, 32, 48, 80, 96, 112, 144, 240, 288, 336, 480, 560, 576, 720, 1008, 1440, 1680, 2016, 2640, 2880, 3600, 4032, 5040, 7920, 9360, 10080, 15840, 18480, 20160, 25200, 31680, 37440, 39600, 44352, 50400, 55440, 65520, 85680, 95760
Offset: 1

Views

Author

Jose R. Brox (tautocrona(AT)terra.es), Jul 10 2003

Keywords

Comments

After 2880, 3360 has exactly the same density (5%).

Examples

			a(3)=4 because for B=4 the different quadratic residues are {0,1}, so S=2, the density is D_4=50%, which is smaller than D_2=100% and D_3=66.67%.

Links

Crossrefs

Cf. A000224, A084848.

Programs

  • Mathematica
    Block[{s = Range[0, 2^14 + 1]^2, t}, t = Array[#/Length@ Union@ Mod[Take[s, # + 1], #] &, Length@ s - 1]; Map[FirstPosition[t, #][[1]] &, Union@ FoldList[Max, t]]] (* Michael De Vlieger, Sep 10 2018 *)
  • PARI
    r=-1;for(n=1,1e6,t=1-sum(k=1,n,issquare(Mod(k,n)))/n;if(t>r,r=t;print1(n", "))) \\ Charles R Greathouse IV, Jul 15 2011
  • PARI
    sq1(m)=sum(n=0,m-1,issquare(Mod(n,m)))
sq(n,f=factor(n))=prod(i=1,#f~,my(p=f[i,1],e=f[i,2]); if(e>1,sq1(p^e),p\2+1))
r=2;for(n=1,1e6, t=sq(n)/n; if(tCharles R Greathouse IV, Mar 30 2018

Extensions

More terms from Jud McCranie, Jul 12 2003
a(1) and PARI programs corrected by Hugo Pfoertner, Aug 23 2018

A290727 Analog of A085635, replacing "quadratic residue" (X^2) with "value of X^2+X".

Original entry on oeis.org

1, 2, 6, 10, 14, 18, 30, 42, 66, 70, 90, 126, 198, 210, 330, 390, 450, 630, 990, 1170, 1386, 1638, 2142, 2310, 2730, 3150, 4950, 5850, 6930, 8190, 10710, 11970, 12870, 16830, 18018, 23562, 26334, 27846, 30030, 34650
Offset: 1

Views

Author

N. J. A. Sloane, Aug 10 2017

Keywords

Comments

Positions where R(k) = A290731(k)/k achieves a new minimum, i.e., R(k) < R(j), j = 0..k-1, R(0) = 2.

Links

Crossrefs

Cf. A085635, A084848, A290728, A290731.

Programs

  • Mathematica
    a290731[n_] := Product[{p, e} = pe; If[p == 2, 2^(e-1), 1+Quotient[p^(e+1), (2p+2)]], {pe, FactorInteger[n]}];
Reap[For[r = 2; k = 1, k <= 35000, k++, t = a290731[k]/k; If[tJean-François Alcover, Sep 03 2018, from PARI *)
  • PARI
    a290731(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)))} \\ from Andrew Howroyd
r=2;for(k=1,40000,t=a290731(k)/k;if(tHugo Pfoertner, Aug 23 2018

Extensions

More terms from Hugo Pfoertner, Aug 22 2018
Initial term added by Hugo Pfoertner, Aug 23 2018

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

Views

Author

N. J. A. Sloane, Aug 10 2017

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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

Extensions

a(1) corrected by Hugo Pfoertner, Aug 26 2018
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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