cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A065134 Remainder when n is divided by the number of primes not exceeding n.

Original entry on oeis.org

0, 1, 0, 2, 0, 3, 0, 1, 2, 1, 2, 1, 2, 3, 4, 3, 4, 3, 4, 5, 6, 5, 6, 7, 8, 0, 1, 9, 0, 9, 10, 0, 1, 2, 3, 1, 2, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 6, 7, 5, 6, 7, 8, 9, 10, 8, 9, 7, 8, 9, 10, 11, 12, 10, 11, 12, 13, 11, 12, 10, 11, 12, 13, 14, 15, 13, 14, 15, 16, 14, 15, 16, 17, 18, 19, 17, 18, 19
Offset: 2

Views

Author

Labos Elemer, Oct 15 2001

Keywords

Comments

Also remainder when the number of nonprimes is divided by the number of primes (not exceeding n).

Examples

			n = 2: Pi[2] = 1,Mod[1,1] = 0, the first term = a(2) = 0; n = 100: Pi[100] = 25, Mod[100,25] = 0 = a(100); n = 20: Pi[20] = 8, Mod[20,8] = 4 = a(20).

Links

Crossrefs

Cf. A000720, A057809, A057810, A000040, A065134, A004648, A062298, A065858-A065864, A065133.

Programs

  • Mathematica
    Table[Last@ QuotientRemainder[n, PrimePi[n]], {n, 2, 91}] (* Michael De Vlieger, Jul 04 2016 *)
  • PARI
    { for (n=2, 1000, write("b065134.txt", n, " ", n%primepi(n)) ) } \\ Harry J. Smith, Oct 11 2009

Formula

a(n) = n (mod pi(n)).

Extensions

Term a(1) removed so OFFSET changed from 1,5 to 2,4 by Harry J. Smith, Oct 11 2009
Since OFFSET is 2,4; Term a(1) removed and a(91) added by Harry J. Smith, Oct 11 2009

A065860 Remainder when the n-th composite number is divided by n.

Original entry on oeis.org

0, 0, 2, 1, 0, 0, 0, 7, 7, 8, 9, 9, 9, 10, 10, 10, 10, 10, 11, 12, 12, 12, 12, 12, 13, 13, 13, 14, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18, 19, 19, 19, 19, 19, 20, 20, 20, 21, 22, 22, 22, 22, 22, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25, 25, 26, 26
Offset: 1

Views

Author

Labos Elemer, Nov 26 2001

Keywords

Examples

			n=100, c(100)=133, a(100)=33.

Links

Crossrefs

Cf. A002808, A065855-A065864.

Programs

  • Mathematica
    Module[{nn=100,cmps,len},cmps=Select[Range[nn],CompositeQ];len=Length[ cmps]; Mod[#[[1]],#[[2]]]&/@Thread[{cmps,Range[len]}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 29 2020 *)
  • PARI
    Composite(n) = { my(k=n + primepi(n) + 1); while (k != n + primepi(k) + 1, k = n + primepi(k) + 1); k }
a(n) = { Composite(n)%n } \\ Harry J. Smith, Nov 02 2009

Formula

a(n) = A002808(n) mod n.

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

Views

Author

Labos Elemer, Nov 26 2001

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    Table[Mod[Prime[n],n-PrimePi[n]],{n,90}] (* Harvey P. Dale, Aug 04 2015 *)
  • PARI
    a(n) = { prime(n)%(n - primepi(n)) } \\ Harry J. Smith, Nov 02 2009

Formula

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

Views

Author

Labos Elemer, Nov 26 2001

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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

A065861 Remainder when the n-th composite number is divided by pi(n), the number of primes not exceeding n.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 3, 0, 2, 0, 1, 4, 0, 1, 2, 6, 0, 6, 0, 1, 2, 8, 0, 2, 3, 4, 6, 4, 5, 2, 4, 5, 6, 7, 8, 6, 7, 8, 9, 6, 8, 6, 7, 8, 9, 6, 8, 9, 10, 12, 14, 11, 12, 13, 14, 0, 1, 14, 16, 13, 14, 15, 16, 0, 1, 16, 17, 18, 0, 16, 18, 15, 16, 18, 20, 0, 1, 20, 0, 1, 2, 22, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4
Offset: 2

Views

Author

Labos Elemer, Nov 26 2001

Keywords

Examples

			n=100, c(100)=133, pi(100)=25, a(100)=8 because 133 = 5*25 + 8.

Links

Crossrefs

Cf. A002808, A000720, A065855-A065864.

Programs

  • PARI
    Composite(n) = { local(k); k=n + primepi(n) + 1; while (k != n + primepi(k) + 1, k = n + primepi(k) + 1); return(k) } { for (n = 2, 1000, a=Composite(n)%primepi(n); write("b065861.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 02 2009

Formula

a(n) = A002808(n) mod A000720(n).

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

Views

Author

Labos Elemer, Nov 26 2001

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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

Formula

a(n) = c(n) mod (n - pi(n)) = A002808(n) mod (n - A000720(n)) = A002808(n) mod A062298(n).
Showing 1-6 of 6 results.

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