A085635 -id:A085635

Search: a085635 -id:a085635

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A084848

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

+20
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Note that the terms are not all distinct.

LINKS

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.

FORMULA

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

EXAMPLE

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.

MATHEMATICA

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 *)

PROG

(PARI) 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(t<r, r=t; print1(v, ", "))) \\ Hugo Pfoertner, Aug 24 2018

CROSSREFS

Cf. A000224, A085635.

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

More terms from Jud McCranie, Jul 18 2003

a(1) corrected by Hugo Pfoertner, Aug 23 2018

STATUS

approved

A290727

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

+20
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

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.

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[t<r, r = t; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Sep 03 2018, from PARI *)

PROG

(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(t<r, r=t; print1(k, ", "))) \\ Hugo Pfoertner, Aug 23 2018

CROSSREFS

Cf. A085635, A084848, A290728, A290731.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Aug 10 2017

EXTENSIONS

More terms from Hugo Pfoertner, Aug 22 2018

Initial term added by Hugo Pfoertner, Aug 23 2018

STATUS

approved

A290729

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

+20
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

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.

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[t<r, r = t; Sow[j] ]]][[2, 1]] (* Jean-François Alcover, Oct 02 2018, after Hugo Pfoertner *)

PROG

(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(t<r, r=t; print1(j, ", "))) \\ Hugo Pfoertner, Aug 26 2018

CROSSREFS

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

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Aug 10 2017

EXTENSIONS

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

STATUS

approved

A290728

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

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.

FORMULA

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

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 *)

PROG

(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(t<r, r=t; print1(v, ", "))) \\ Hugo Pfoertner, Aug 23 2018

CROSSREFS

Cf. A085635, A084848, A290727, A290731.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Aug 10 2017

EXTENSIONS

More terms from Hugo Pfoertner, Aug 22 2018

Initial term added by Hugo Pfoertner, Aug 23 2018

STATUS

approved

A290730

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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.

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 *)

PROG

(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(t<r, r=t; print1(w, ", "))) \\ Hugo Pfoertner, Aug 26 2018

CROSSREFS

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

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Aug 10 2017

EXTENSIONS

More terms from Hugo Pfoertner, Aug 23 2018

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

STATUS

approved

A359195

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

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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, ....

LINKS

Table of n, a(n) for n=1..23.

EXAMPLE

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.

CROSSREFS

A085635 is the analogous sequence for squares.

Cf. A025487, A337868.

KEYWORD

nonn,more

AUTHOR

Charles R Greathouse IV, Dec 19 2022

STATUS

approved

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