A072608 -id:A072608 - cp's OEIS frontend

A072609 Changing of parity of remainder A072608(n) from alternation [..010101..] to steadily 1-range [...1111..]. AC-range corresponds to 0, while DC-range labeled by 1.

0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Labos Elemer, Jun 24 2002

			Take n = 11,12,13,14: A004648[n]=9,1,2,1. Parity A072608(n) = 1,1,0,1. So ..11.. transforms into 01 between n = 11 and n = 12: a(11) = 1, a(12)=0. With increasing n, A072609(n) changes from ..0000.. into ...1111. reflected by this sequence. by a range consisting only of 1-s. This secondary alternation also goes on.

Cf. A004648, A072608.

mm[x_] := Mod[Mod[Prime[x], x], 2] Table[mm[w]*mm[w+1], {w, 1, 256}]
Times@@@Partition[Table[Mod[Mod[Prime[n],n],2],{n,110}],2,1] (* Harvey P. Dale, Dec 21 2014 *)

a(n)=Mod[A004648(n), 2]*Mod[A004648(n+1), 2]= A072608(n)*A072608(n+1)

A072610 Values of transition of A072608(n) from alternating behavior (0,1,0,1,..) into steadily-1 (1,1,1,..) behavior or changing back. Expressing in terms of A072609(n): at n values it switches from steadily 0 into steadily 1 successive values or back.

Original entry on oeis.org

3, 11, 29, 67, 69, 71, 179, 181, 189, 441, 1059, 2699, 6453, 6459, 6471, 15927, 40071, 40083, 40121, 100363, 251705, 251707, 251709, 251721, 251723, 251735, 251737, 251741, 251761, 637233, 637235, 637319, 637321, 637323, 637325, 637329, 637333

Offset: 1

Labos Elemer, Jun 24 2002

nonn

Values of n such that A072609(n) != A072609(n+1).

Switching zones appear in clusters of n. Remainder A004648 either drops or starts to increase at these values of n.

			At n=637330...637370 the change of remainder A004648 is as follows: {..637323, 4, 13, 4, 637333, 637327, 637335, 637323, 637319, 637325, 637321, 4, 11, 4, 7, 34, 29, 26, 17, 44, 43, 46, 41, 38, 49, 52, 49, 44, 37, 28, 37..}

Cf. A004648, A072608, A072609.

mm[x_] := Mod[Mod[Prime[x], x], 2];
pm[x_] := mm[x]*mm[x+1];
Do[s1=pm[n]; s2=pm[n+1]; If[ !Equal[s1, s2], Print[n]], {n, 10^9}]

A072630 Values of n where A072629 switches from 01010.. into 0000.. or back.

Original entry on oeis.org

1, 7, 19, 53, 147, 403, 1095, 2979, 8103, 22025, 59873, 162753, 442413, 1202603, 3269017, 8886109, 24154951, 65659969, 178482299, 485165195, 1318815733, 3584912845, 9744803445, 26489122129, 72004899337, 195729609427

Offset: 1

Labos Elemer, Jun 28 2002

nonn

See also A004648, A072608, A072609, A072610, A072630.

m[x_] := Mod[x*Floor[Log[x]//N],2]; Do[s=m[n]+m[n+1]; s1=m[n+1]+m[n+2]; If[ !Equal[s1,s],Print[n]],{n,1,1000000}]

See program below.

a(n) = A000149(n) or A000149(n)-1 whichever is odd. [From Max Alekseyev, Feb 06 2010]

More terms from Max Alekseyev, Feb 06 2010

A072631 Floor( n*log(n) ) mod n.

Original entry on oeis.org

0, 1, 0, 1, 3, 4, 6, 0, 1, 3, 4, 5, 7, 8, 10, 12, 14, 16, 17, 19, 0, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 14, 16, 17, 19, 21, 22, 24, 25, 27, 29, 30, 32, 34, 36, 38, 39, 41, 43, 45, 47, 49, 51, 53, 0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29

Offset: 1

Labos Elemer, Jun 28 2002

nonn

Compare with A004648 because prime(n) ~ n * log(n).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A004648, A072608, A072609, A072610, A072630.

[Floor(n*Log(n)) mod n: n in [1..100]]; // Vincenzo Librandi, Jun 30 2014

Table[Floor[Mod[n*Log[n],n]],{n,80}] (* Harvey P. Dale, Jun 29 2014 *)

a(n)=floor(n*log(n))%n \\ Charles R Greathouse IV, Aug 01 2011

A072629 Parity of n*floor(log n).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Labos Elemer, Jun 28 2002

nonn

a(n)=1 for n: 3, 5, 7, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 149, 151, .... - Robert G. Wilson v, Feb 01 2015

			Parity either alternates or it is steadily 0. Intervals of such kind also change and return: 01010...0000....0101.., etc.

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for characteristic functions

Cf. A004648, A072608, A072609, A072610, A072630.

f[n_] := Mod[n*Floor@ Log@ n, 2]; Array[f, 105] (* Robert G. Wilson v, Feb 01 2015 *)

a(n) = (log(n)\1*n)%2; \\ Charles R Greathouse IV, Sep 04 2015, corrected by Antti Karttunen, Feb 06 2019

a(n) = n*floor(log(n)) mod 2.

A072623 Numbers n such that A065863(n) = 1, i.e., prime(n) mod (n - Pi(n)) = 1.

Original entry on oeis.org

4, 5, 6, 11, 19, 25, 34, 36, 75, 82, 87, 90, 94, 237, 604, 609, 614, 1583, 1592, 10466, 10467, 10498, 10504, 10505, 70501, 70511, 180227, 180294, 180358, 180443, 180447, 466078, 8103422, 21058343, 21058649, 143052872, 143052877, 143053068

Offset: 1

Labos Elemer, Jun 26 2002

nonn

A004648, A065134 and A065863 behave similarly; they grow relatively slowly and drop suddenly at unexpected values of n. Parity of A004648 behaves most regularly.

Each cluster of entries exceeds the previous cluster by a power of e.

			For the cluster started at n = 10466 the remainders of A065863(n) are as follows: {9089, 9092, 9117, 9127, 9148, 9159, 1, 1, 9180, 9183, 9182, 9179, 9172, 9169, 9168, 9177, 9176, 9178, 9183, 9192, 43}. It behaves like A004648 or A065134.

Cf. A065860-A065863, A065133-A065135, A072608-A072610, A004648, A057809.

Do[ If[ Mod[ Prime[n], n-PrimePi[n]] == 1, Print[n]], {n, 1, 150000000}]
(* Second program: *)
Position[Table[Mod[Prime[n], n - PrimePi[n]], {n, 10^6}], 1] // Flatten (* Michael De Vlieger, Jul 30 2017 *)

Edited by Robert G. Wilson v, Jun 27 2002

A072632 Solutions to A072631[n]=0.

Original entry on oeis.org

1, 3, 8, 21, 55, 149, 404, 1097, 2981, 8104, 22027, 59875, 162755, 442414, 1202605, 3269018

Offset: 1

Labos Elemer, Jun 28 2002

Essentially the same as A001671.

			Compare with A072610 [related to A004648].

Cf. A004648, A072608, A072609, A072610, A072630, A072631, A072610.

Do[s=Floor[Mod[Floor[n*Log[n]]//N, n]]; If[s==0, Print[n]], {n, 1, 10000000}]

cp's OEIS Frontend

A072609 Changing of parity of remainder A072608(n) from alternation [..010101..] to steadily 1-range [...1111..]. AC-range corresponds to 0, while DC-range labeled by 1.

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A072610 Values of transition of A072608(n) from alternating behavior (0,1,0,1,..) into steadily-1 (1,1,1,..) behavior or changing back. Expressing in terms of A072609(n): at n values it switches from steadily 0 into steadily 1 successive values or back.

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A072630 Values of n where A072629 switches from 01010.. into 0000.. or back.

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A072631 Floor( n*log(n) ) mod n.

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PARI

A072629 Parity of n*floor(log n).

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A072623 Numbers n such that A065863(n) = 1, i.e., prime(n) mod (n - Pi(n)) = 1.

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A072632 Solutions to A072631[n]=0.

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