A085635 -id:A085635 - cp's OEIS frontend

A084848 a(n) is the number of quadratic residues of A085635(n).

1, 2, 2, 3, 4, 4, 7, 8, 12, 14, 16, 16, 24, 28, 32, 42, 48, 48, 48, 64, 84, 96, 112, 144, 144, 176, 192, 192, 288, 336, 336, 504, 576, 576, 704, 864, 1008, 1056, 1152, 1232, 1152, 1344, 1728, 1920, 2016, 2016, 2352

Offset: 1

Jose R. Brox (tautocrona(AT)terra.es), Jul 12 2003

nonn

Note that the terms are not all distinct.

			a(2)=2 because there are 2 different quadratic residues modulo 3, so 3 has 66.67% of quadratic residues density, while 2 has a 100%, so 3 has the least quadratic residues density up to 3.

Keith F. Lynch, Table of n, a(n) for n = 1..200 (first 111 terms from Hugo Pfoertner).
Andreas Enge, William Hart, Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], 2016-2018.

Cf. A000224, A085635.

Block[{s = Range[0, 2^15 + 1]^2, t}, t = Array[{#1/#2, #2} & @@ {#, Length@ Union@ Mod[Take[s, # + 1], #]} &, Length@ s - 1]; Map[t[[All, -1]][[FirstPosition[t[[All, 1]], #][[1]] ]] &, Union@ FoldList[Max, t[[All, 1]] ] ] ] (* Michael De Vlieger, Sep 10 2018 *)

a000224(n)=my(f=factor(n));prod(i=1,#f[,1],if(f[i,1]==2,2^f[1,2]\6+2,f[i,1]^(f[i,2]+1)\(2*f[i,1]+2)+1)) \\ from Charles R Greathouse IV
r=2;for(k=1,1e6,v=a000224(k);t=v/k;if(tHugo Pfoertner, Aug 24 2018

a(n) = A000224(A085635(n)). - Hugo Pfoertner, Aug 24 2018

More terms from Jud McCranie, Jul 18 2003

a(1) 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

N. J. A. Sloane, Aug 10 2017

nonn

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

Hugo Pfoertner, Table of n, a(n) for n = 1..116
Andreas Enge, William Hart, Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], 2016-2018. See Table 5.

Cf. A085635, A084848, A290728, A290731.

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 *)

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

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

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

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

N. J. A. Sloane, Aug 10 2017

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..116
Andreas Enge, William Hart, Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], 2016-2018. See Table 5.

Cf. A085635, A084848, A290727, A290731.

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 *)

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

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

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

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

A359195 Positive integers k with a smaller fraction of powers (mod k) than any smaller positive integers.

Original entry on oeis.org

1, 4, 16, 32, 36, 72, 144, 288, 432, 864, 1728, 3456, 3600, 5400, 7200, 10800, 21600, 43200, 86400, 151200, 172800, 216000, 302400

Offset: 1

Charles R Greathouse IV, Dec 19 2022

It seems that a(n) <= 2*a(n-1) for n > 3.

Conjecture: terms are products of primorials (A025487). A proof would greatly speed the search for more terms. On this conjecture, the next terms are 352800, 529200, 1058400, 2116800, 4233600, 6350400, 10584000, 19051200, 21168000, 31752000, 63504000, ....

			2 is not a square or a cube mod 4, while 0, 1, and 3 are all cubes mod 4. 1/4 is a record, so 4 is in the sequence.
None of 2, 4, 6, 10, 12, 14 are cubes mod 16 and of those only 4 is a square and none are 5th powers, for a 5/16 fraction, which is a record, so 16 is in the sequence.

A085635 is the analogous sequence for squares.

Cf. A025487, A337868.

cp's OEIS Frontend

A084848 a(n) is the number of quadratic residues of A085635(n).

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

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

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

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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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A359195 Positive integers k with a smaller fraction of powers (mod k) than any smaller positive integers.

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