A065134 -id:A065134 - cp's OEIS frontend

A065133 Remainder when n-th prime is divided by the number of primes not exceeding n.

0, 1, 1, 2, 1, 1, 3, 3, 1, 1, 2, 5, 1, 5, 5, 3, 5, 3, 7, 1, 7, 2, 8, 7, 2, 4, 8, 9, 3, 6, 10, 5, 7, 6, 8, 1, 7, 11, 5, 10, 12, 9, 11, 1, 3, 1, 13, 2, 4, 8, 14, 1, 11, 1, 7, 13, 15, 5, 9, 13, 5, 1, 5, 7, 11, 8, 14, 5, 7, 13, 19, 10, 16, 1, 5, 11, 19, 5, 13, 1, 3, 17, 19, 2, 6, 12, 20, 5, 7, 11, 23

Offset: 2

Labos Elemer, Oct 15 2001

nonn

			n = 2: pi(2) = 1, prime(2) = 3, 3 mod 1 = 0, the first term = a(2);
n = 100: pi(100) = 25, prime(100) = 541, 541 mod 25 = 16 = a(100). [corrected by _Jon E. Schoenfield_, Jun 18 2018]

Harry J. Smith, Table of n, a(n) for n = 2..1000

Cf. A000720, A000040, A065134, A004648.

Table[Mod[Prime[n],PrimePi[n]],{n,2,100}] (* Harvey P. Dale, Nov 28 2013 *)

{ for (n=2, 1000, write("b065133.txt", n, " ", prime(n)%primepi(n)) ) } \\ Harry J. Smith, Oct 11 2009

a(n) = prime(n) mod pi(n) = A000040(n) mod A000720(n), n > 1.

A065864 Remainder when n is divided by the number of nonprimes not exceeding n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 21, 21, 21, 21, 21, 21

Offset: 1

Labos Elemer, Nov 26 2001

nonn

			For n=100, pi(100)=25, so a(100) = 100 mod (100-25) = 25.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040, A004648, A062298, A065133, A065134, A065858-A065864.

Table[Mod[n, n - PrimePi@ n], {n, 78}] (* or *)
Table[Mod[n, Count[Range@ n, k_ /; ! PrimeQ@ k]], {n, 78}] (* Michael De Vlieger, Jan 01 2017 *)

{ for (n = 1, 1000, a=n%(n - primepi(n)); write("b065864.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 02 2009

a(n) = n mod (n-pi(n)) = n mod (n-A000720(n)) = n mod A062298(n).

A073436 Smallest k such that k mod pi(k) = n.

Original entry on oeis.org

2, 3, 5, 7, 16, 21, 22, 25, 26, 29, 32, 65, 66, 70, 77, 78, 82, 86, 87, 88, 92, 93, 94, 95, 99, 106, 116, 117, 118, 119, 218, 219, 220, 221, 222, 247, 248, 249, 250, 255, 256, 261, 262, 267, 268, 289, 290, 291, 292, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306

Offset: 0

Labos Elemer, Jul 31 2002

nonn

a(n) > a(n-1) except for 68, 180, 1051, 6454, 6456, 6459, 40073, 40078, ..., . - Robert G. Wilson v, Feb 24 2023

			Remainder 7 appears first as 25 mod pi(25) = 25 mod 9 = 7, so a(7) = 25.

Giovanni Resta, Table of n, a(n) for n = 0..10000

Cf. A000720, A038625, A057809, A065134, A073435, A073437, A360789.

with(numtheory); f:=proc(n) local i,j,k; for i from 2 to 10000 do if i mod pi(i) = n then RETURN(i); fi; od: RETURN(-1); end; # N. J. A. Sloane, Sep 01 2008

a = Compile[{{n, Integer}}, Block[{k = 2}, While[ Mod[k, PrimePi@ k] != n, k++]; k]]; Array[a, 59, 0] (* _Robert G. Wilson v, Feb 24 2023 *)

a(n)={my(q=0, k=2);forprime(p=3, oo, q++; while(kAndrew Howroyd, Feb 23 2023

a(n) = Min{k: k mod A000720(k) = n} = Min{k: A065134(k) = n}.

a(0) from Robert G. Wilson v, Feb 23 2023

A065863 Remainder when n-th prime is divided by the number of nonprimes not exceeding n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 3, 3, 5, 1, 2, 6, 3, 2, 3, 9, 6, 1, 11, 8, 9, 13, 14, 1, 16, 13, 12, 14, 13, 7, 5, 5, 1, 5, 1, 7, 7, 5, 5, 11, 7, 17, 13, 11, 7, 19, 25, 23, 19, 17, 17, 19, 23, 23, 23, 23, 19, 25, 23, 25, 29, 37, 35, 31, 29, 43, 43, 47, 43, 47, 47, 3, 2, 1, 53, 53, 55, 2, 3, 6, 1, 11, 6

Offset: 1

Labos Elemer, Nov 26 2001

nonn

			For n=25, prime(25)=97, n - pi(n) = 25 - 9 = 16, a(25)=1 because 97 = 6*16 + 1.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040, A062298, A065858-A065864, A065134, A004648, A065133, A065134.

Table[Mod[Prime[n],n-PrimePi[n]],{n,90}] (* Harvey P. Dale, Aug 04 2015 *)

a(n) = { prime(n)%(n - primepi(n)) } \\ Harry J. Smith, Nov 02 2009

a(n) = prime(n) mod (n - pi(n)) = A000040(n) mod A062298(n).

A065859 Remainder when the n-th prime is divided by the n-th composite number.

Original entry on oeis.org

2, 3, 5, 7, 1, 1, 3, 4, 7, 11, 11, 16, 19, 19, 22, 1, 5, 5, 7, 7, 7, 11, 13, 17, 21, 23, 23, 23, 21, 23, 35, 35, 39, 39, 47, 47, 49, 53, 55, 2, 5, 1, 5, 4, 5, 4, 13, 19, 20, 19, 17, 17, 16, 23, 26, 29, 29, 28, 31, 29, 28, 35, 46, 47, 43, 44, 55, 58, 65, 64, 65, 65, 70, 73, 73, 71

Offset: 1

Labos Elemer, Nov 26 2001

nonn

			n=100, p(100)=541, c(100)=133, a(100)=9 because 541 = 4*133 + 9.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040, A002808, A004648, A065133, A065134, A065855-A065864, A065134.

a[n]=Mod[p(n), c(n)]=Mod[A000040(n), A002808(n)]
With[{nn=80},Module[{prs=Prime[Range[nn]],comps},comps=Take[Complement[ Range[2,Prime[nn]+1],prs],Length[prs]];Mod[#[[1]],#[[2]]]&/@ Thread[ {prs,comps}]]] (* Harvey P. Dale, Apr 18 2012 *)

Composite(n) = { local(k); k=n + primepi(n) + 1; while (k != n + primepi(k) + 1, k = n + primepi(k) + 1); return(k) } { for (n = 1, 1000, a=prime(n)%Composite(n); write("b065859.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 01 2009

A065136 Numbers n such that n = pi(n)*k + 1 for some k.

Original entry on oeis.org

3, 9, 11, 13, 28, 34, 37, 43, 121, 336, 341, 351, 356, 361, 1081, 1087, 1135, 3060, 3074, 3081, 3088, 3095, 8409, 8425, 8441, 8457, 8465, 8473, 23527, 23536, 24301, 64541, 64581, 64591, 64601, 64611, 64651, 64661, 64691, 64701, 64711, 64721

Offset: 1

Labos Elemer, Oct 15 2001

nonn

Solutions to Mod[n,PrimePi[n]] = 1, i.e. A065134(n) = 1.

			n=28: Pi(28)=9 and 28=3*Pi(28)+1, so 28 is here; n=27 is present in A057809. A large proportion of A057809(m)+1 numbers (but not all of them) arise in this sequence. Numbers from A057809 arise in clusters [see grouping around 8450, 64650, 480900 etc.]

Giovanni Resta, Table of n, a(n) for n = 1..1172

A000720, A065134, A057809.

A065862 Remainder when n-th composite number is divided by the number of nonprimes not exceeding n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 3, 1, 0, 2, 0, 1, 0, 7, 6, 7, 6, 8, 8, 7, 6, 7, 6, 6, 5, 4, 4, 6, 5, 6, 6, 5, 4, 3, 2, 4, 3, 2, 1, 2, 2, 4, 3, 2, 1, 2, 2, 1, 0, 0, 0, 1, 0, 38, 38, 39, 39, 40, 41, 42, 42, 42, 42, 43, 43, 44, 44, 44, 44, 45, 46, 47, 47, 48, 49, 49, 49, 51, 52, 52, 52, 54, 54, 54, 54, 54

Offset: 1

Labos Elemer, Nov 26 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A002808, A000720, A062298, A065858-A065864, A065134.

Module[{nn=150,cmps,len},cmps=Select[Range[nn],CompositeQ];len=Length[ cmps];Mod[#[[1]],#[[2]]-PrimePi[#[[2]]]]&/@Thread[{cmps,Range[len]}]] (* Harvey P. Dale, Feb 21 2020 *)

Composite(n) = { local(k); k=n + primepi(n) + 1; while (k != n + primepi(k) + 1, k = n + primepi(k) + 1); return(k) } { for (n = 1, 1000, a=Composite(n)%(n - primepi(n)); write("b065862.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 02 2009

a(n) = c(n) mod (n - pi(n)) = A002808(n) mod (n - A000720(n)) = A002808(n) mod A062298(n).

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

A072624 a(n) = prime(n^2) mod n^2.

Original entry on oeis.org

0, 3, 5, 5, 22, 7, 31, 55, 14, 41, 56, 107, 164, 17, 77, 83, 145, 199, 271, 341, 437, 73, 100, 179, 262, 319, 416, 519, 594, 697, 846, 993, 25, 93, 131, 259, 369, 497, 575, 699, 879, 989, 1085, 1259, 1409, 1533, 1799, 1961, 2183, 2307, 2519, 23, 188, 329, 514

Offset: 1

Labos Elemer, Jun 28 2002

nonn

			For n = 10: a(10) = prime(100) mod 100 = 541 mod 100 = 41.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A004648, A065863, A065134, A065875, A072623.

a[n_] := Mod[Prime[n^2], n^2]; Array[a, 100] (* Amiram Eldar, Mar 17 2025 *)

a(n) = prime(n^2) % (n^2); \\ Amiram Eldar, Mar 17 2025

a(n) = A004648(n^2).

cp's OEIS Frontend

A065133 Remainder when n-th prime is divided by the number of primes not exceeding n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A065864 Remainder when n is divided by the number of nonprimes not exceeding n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A073436 Smallest k such that k mod pi(k) = n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A065863 Remainder when n-th prime is divided by the number of nonprimes not exceeding n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A065859 Remainder when the n-th prime is divided by the n-th composite number.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A065136 Numbers n such that n = pi(n)*k + 1 for some k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A065862 Remainder when n-th composite number is divided by the number of nonprimes not exceeding n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords