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A331008 Numbers n such that (11*prime(n)) mod Pi > (11*prime(n+1)) mod Pi. 1
71, 179, 274, 367, 452, 539, 623, 705, 786, 869, 943, 1024, 1106, 1183, 1262, 1335, 1405, 1483, 1562, 1636, 1705, 1780, 1860, 1929, 2000, 2074, 2146, 2214, 2286, 2355, 2431, 2502, 2576, 2645, 2717, 2781, 2849, 2918, 2990, 3059, 3130, 3201, 3262, 3330, 3399, 3462, 3538 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

It seems that sequences of the form b(n) = (k*prime(n)) mod x exhibit a quasi-periodic sawtooth-like trend with slightly decreasing period when x is a positive irrational and k is the numerator (or a multiple of it) of a convergent to x. The commented part of the Mathematica program allows an easy experimentation on this feature. Average distance between consecutive elements a(n) decreases very slowly, and this pattern can be observed in this sequence up to values of n as high as 2^42 where average distance is about four times lower than at the beginning of the sequence.

LINKS

Table of n, a(n) for n=1..47.

EXAMPLE

First element a(1) is 71 because(11*prime(71)) mod Pi = ~3.133072, a larger value than (11*prime(72)) mod Pi = ~0.018034. For any other primes p and q, such that p < q < prime(71) we can see that (11*prime(p)) mod Pi < (11*prime(q)) mod Pi.

Second element a(2) is 179 because(11*prime(179)) mod Pi = ~3.133735, a larger value than (11*prime(180)) mod Pi = ~0.018697. For any other primes p and q, such that prime(71) < p < q < prime(179) we can see that (11*prime(p)) mod Pi < (11*prime(q)) mod Pi.

MATHEMATICA

Flatten@Position[Differences[N[Mod[11*Prime[Range[2^13]], Pi], 24]],

   x_ /; x < 0]

(*Uncomment following lines to experiment with different irrational constants "x", constant factors "k", offsets and length of subsequences *)

(*

Manipulate[Column@{Column@{

Style["("<> ToString[k] <>  "*prime(n)) mod (" <> ToString[ x]<>")"],

ListPlot[N[Mod[k*Prime[Range[min, min+delta]], x], 20], Joined->TF, AxesLabel->{n " - offset", }, ImageSize->Medium, ImageSize->Medium, ImagePadding->{{10, 60}, {30, 20}}]},

Column@{

Style["First Differences of ("<> ToString[k] <>  "*prime(n)) mod (" <> ToString[ x]<>")"],

ListPlot[Differences[N[Mod[k*Prime[Range[min, min+delta]], x], 20]], PlotRange->All, AxesLabel->{n " - offset", }, ImageSize->Medium, ImagePadding->{{10, 60}, {30, 10}}]}}

, {{k, 11, "constant factor"}, 1, 2^20, 1, Appearance->"Labeled"}, {{x, Pi, "irrational modulus"}, {Pi, 2Pi, Sqrt[2], Sqrt[3], Sqrt[5], Sqrt[7], GoldenRatio, Sqrt[GoldenRatio], E }, Appearance->"Labeled"}, {{min, 1, "offset"}, 1, 2^32, 2^20, Appearance->"Labeled"}, {{delta, 2^10, "subsequence length"}, 1, 2^16, 2^7, Appearance->"Labeled"}, {{TF, False, "Points joined"}, {False, True}, Appearance->"Labeled"}]

*)

CROSSREFS

Cf. A002485, A046947, A276435.

Sequence in context: A044403 A044784 A142488 * A068364 A142612 A295835

Adjacent sequences:  A331005 A331006 A331007 * A331009 A331010 A331011

KEYWORD

nonn

AUTHOR

Andres Cicuttin, Jan 06 2020

STATUS

approved

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Last modified December 4 02:40 EST 2021. Contains 349469 sequences. (Running on oeis4.)