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.

Previous Showing 81-89 of 89 results.

A065891 The a(n)-th composite number is 2^n.

Original entry on oeis.org

1, 3, 9, 20, 45, 96, 201, 414, 851, 1738, 3531, 7163, 14483, 29255, 58993, 118820, 239143, 480897, 966550, 1941540, 3898356, 7824444, 15699344, 31490742, 63151054, 126614174, 253804612, 508678161, 1019341795, 2042386082, 4091687074, 8196318785, 16416930072
Offset: 2

Views

Author

Labos Elemer, Nov 28 2001

Keywords

Comments

Index of n-th power of 2 in A002808.
Remainder of division 2^n/c(n) equals zero, where c(n) = A002808(n), the n-th composite number.
Exponential increase with a factor > 2 and approaching two.

Examples

			For n = 4, 2^4 = 16 is the 9th composite number: 4,6,8,9,10,12,14,15,16, so a(4) = 9.

Crossrefs

Cf. A000079, A000720, A002808, A015910, A065855, A073800.

Programs

  • Maple
    seq(2^k - numtheory:-pi(2^k)-1, k=2..28); # Robert Israel, Dec 10 2024
  • Mathematica
    Do[s=Mod[2^n, c[n]]; If[s==0, Print[n]], {n, 2, 1000000}]
Table[2^n-(PrimePi[2^n])-1, {n, 2, 31}]
  • PARI
    lista(kmax) = {my(c = 0); forcomposite(k = 1, kmax, c++; if(k >> valuation(k, 2) == 1, print1(c, ", ")));} \\ Amiram Eldar, Jun 04 2024

Formula

a(n) = 2^n - A065855(2^n) - 1. - Robert Israel, Dec 10 2024

Extensions

Edited by Robert G. Wilson v, Jun 18 2002
a(32)-a(34) from Amiram Eldar, Jun 04 2024

A065896 Number of composites <= 2*n.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 16, 18, 19, 20, 22, 24, 25, 27, 28, 29, 31, 32, 34, 36, 37, 39, 41, 42, 43, 45, 47, 48, 50, 51, 52, 54, 56, 57, 59, 60, 62, 64, 65, 67, 69, 71, 72, 74, 75, 76, 78, 79, 80, 82, 83, 85, 87, 89, 91, 93, 95, 96, 98, 99, 101, 103, 104, 105
Offset: 1

Views

Author

Labos Elemer, Nov 28 2001

Keywords

Examples

			a(9) = 10, since 18 is the 10th composite number: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18.

Links

Crossrefs

Cf. A002808, A000720, A005843, A065855.

Programs

  • Haskell
    a065896 = a065855 . (* 2)  -- Reinhard Zumkeller, Oct 14 2014
  • Mathematica
    Table[2*n-(PrimePi[2*n])-1, {n, 1, 128}]
  • PARI
    { for (n=1, 1000, f=2*n; a=f - primepi(f) - 1; write("b065896.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 04 2009

Formula

a(n) = A065855(2*n). - Reinhard Zumkeller, Oct 14 2014

Extensions

More precise definition from Michel Marcus, Mar 07 2021~

A065899 a(n) is the index of the n-th compositorial number, A036691(n), in the sequence of composites (A002808).

Original entry on oeis.org

1, 14, 148, 1458, 15293, 188782, 2692726, 40909988, 660637057, 11976280879, 240871231369, 5080851687840, 112183659405198, 2700581280109040, 67686358108129808, 1763651979163805444, 47707175694652299653, 1337959106215345951164, 40196133912310028013721, 1287910861213828031657392
Offset: 1

Views

Author

Labos Elemer, Nov 28 2001

Keywords

Examples

			a(2) = 14 because 4*6 = 24, the 2nd compositorial number is the 14th composite number: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24.

Crossrefs

Cf. A002808, A000720, A036691, A065855.

Programs

  • Mathematica
    Table[A036691[n]-(PrimePi[A036691[n]])-1, {n, 1, 9}]
Composite[n_] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; Table[c = Product[ Composite[i], {i, 1, n} ]; c - PrimePi[c] - 1, {n, 1, 10} ]
  • Python
    from sympy import factorial, primepi, composite, primorial, compositepi
def A065899(n):
    return compositepi(factorial(composite(n))//primorial(primepi(composite(n)))) # Chai Wah Wu, Sep 08 2020

Formula

a(n) = A036691(n) - primepi(A036691(n))-1.
a(n) = A065855(A036691(n)). - Chai Wah Wu, Sep 08 2020

Extensions

One more term from Robert G. Wilson v, Nov 29 2001
a(11)-a(19) from Chai Wah Wu, Sep 08 2020
a(20) from Chai Wah Wu, Sep 09 2020
Name rewritten by Felix Fröhlich, Jun 01 2021

A073438 Remainder of division G[n]/Pi[n], where G[n] is the number of composites not exceeding n.

Original entry on oeis.org

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

Views

Author

Labos Elemer, Jul 31 2002

Keywords

Examples

			n=100: G[100]=100-Pi[100]-1=100-25-1=74, Pi[100]=25, remainder=a(100)=Mod[74,25]=24.

Links

Crossrefs

Cf. A065855, A000720, A073437.

Programs

  • Mathematica
    Table[Mod[w-PrimePi[w]-1, PrimePi[w]], {w, 1, 128}]
With[{nn=100},Mod[#[[1]]-#[[2]]-1,#[[2]]]&/@Thread[{Range[2,nn],PrimePi[Range[2,nn]]}]] (* Harvey P. Dale, Feb 26 2025 *)

Formula

a(n)=Mod[A065855(n), A000720(n)]=Mod[n-Pi[n]-1, Pi[n]] for n>1.

A096533 Number of composite numbers not greater than the n-th composite number that divide at least one other number not greater than the n-th composite number.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 2, 3, 4, 5, 5, 5, 6, 6, 6, 6, 7, 8, 9, 9, 9, 9, 10, 10, 10, 11, 12, 13, 13, 13, 14, 14, 15, 15, 16, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 21, 22, 22, 23, 24, 24, 24, 25, 25, 26, 27, 27, 27, 28, 28, 28, 28, 29, 30, 30, 31, 31, 31, 31, 32, 33, 33, 34, 35, 36
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 23 2004

Keywords

Comments

a(n) = A065855(floor(A002808(n)/2));
A096532(n) + a(n) = n.

A119524 Decimal expansion of Sum_{k >= 1} ((-1)^A010051(k))/2^k.

Original entry on oeis.org

1, 7, 0, 6, 3, 4, 9, 8, 0, 2, 9, 7, 7, 7, 6, 6, 7, 9, 5, 0, 3, 7, 8, 0, 7, 5, 5, 6, 9, 1, 3, 8, 4, 5, 8, 3, 2, 6, 8, 4, 5, 1, 5, 2, 3, 7, 2, 4, 1, 6, 6, 0, 4, 4, 2, 6, 3, 5, 0, 9, 1, 7, 1, 0, 2, 2, 7, 1, 8, 0, 7, 8, 7, 6, 1, 2, 8, 5, 3, 3, 1, 6, 0, 7, 4, 1, 9, 9, 0, 3, 1, 4, 3, 0, 4, 8, 4, 4, 4, 1
Offset: 0

Views

Author

Roger L. Bagula, May 27 2006

Keywords

Examples

			0.17063498029777667950378075569138458326845152372416604426350917102271807...

Crossrefs

Cf. A010051, A065855, A119523, A275306.

Programs

  • PARI
    suminf(k=1, (-1)^isprime(k)/2^k) \\ Michel Marcus, Jan 16 2024

Formula

Equals Sum_{k >= 1} (k - 1 - PrimePi(k))/2^k = Sum_{k >= 1} A065855(k)/2^k.
From Antonio Graciá Llorente, Jan 14 2024: (Start)
Equals Sum_{k >= 2} A005171(k)/2^(k-1).
Equals Sum_{k >= 1} ((-1)^A010051(k))/2^k.
Equals 2*A275306. (End)

Extensions

Edited and corrected by N. J. A. Sloane, Nov 17 2006
Better name from Joerg Arndt, Jan 16 2024

A130042 A130041(n) is the a(n)-th composite number.

Original entry on oeis.org

1, 4, 2, 15, 9, 33, 58, 3, 510, 90, 45, 129, 2043, 12925, 6, 227, 599, 288, 13803, 25453, 75368, 429, 24, 5, 118133, 255419, 106550, 694, 74, 798, 1638477, 650183, 850528, 1733925, 32, 1149, 2620617, 3173990, 151, 1417, 176, 1565, 4489633, 22622373
Offset: 2

Views

Author

Leroy Quet, May 02 2007

Keywords

Comments

This sequence is a permutation of the positive integers.

Crossrefs

Cf. A130041.

Formula

a(n)=A065855(A130041(n)). - R. J. Mathar, Oct 15 2007

Extensions

More terms from R. J. Mathar, Oct 15 2007

A072553 Sigma of n-th composite number equals a(n)-th composite number if it is also a composite or equals zero if sigma[c] is prime.

Original entry on oeis.org

0, 6, 8, 0, 10, 18, 14, 14, 0, 26, 28, 20, 24, 42, 0, 28, 27, 39, 51, 44, 32, 37, 32, 66, 42, 39, 65, 71, 60, 56, 51, 93, 40, 68, 51, 72, 89, 51, 89, 57, 65, 128, 71, 76, 0, 60, 109, 95, 71, 109, 150, 83, 93, 105, 71, 128, 143, 90, 95, 175, 79, 99, 89, 138, 182, 82, 128, 96
Offset: 1

Views

Author

Labos Elemer, Aug 06 2002

Keywords

Examples

			n=1: c[1]=4, sigma[4]=1+2+4=7 prime, a(1)=0; n=10: c[10]=18, sigma[18]=1+2+3+6+9+18=39 composite and 39 is the 26th composite number, so a(10)=26.

Crossrefs

Cf. A065855, A000203, A002808, A073255.

Programs

  • Mathematica
    c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x] G[x_] := x-PrimePi[x]-1 Do[s=c[n]; s1=DivisorSigma[1, s]; s2=G[s1]; If[PrimeQ[s1], Print[0]]; If[ !PrimeQ[s1], Print[s2]], {n, 1, 128}]

Formula

a(n)=G[sigma[c[n]]]=A065855[A000203[A002808(n)]]]= A065855[A073255[n]] if sigma[c]=A000203[A002808(n)]] is composite and a(n)=0 if A073255[n]=A000203[A002808(n)]] is prime.

A073368 Remainder when n is divided by number of composites not exceeding n.

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 0, 1, 0, 1, 0, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 18, 18, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 22, 22, 22, 22, 22, 22
Offset: 4

Views

Author

Labos Elemer, Jul 30 2002

Keywords

Examples

			n=14: a(14)=Mod[14,14-Pi(14)-1]=Mod[14,14-7]=0; n=15: a(15)=Mod[15,15-Pi(15)-1]=Mod[15,7]=1.

Crossrefs

Cf. A065855, A073366, A073367.

Programs

  • Mathematica
    Table[Mod[w, w-PrimePi[w]-1], {w, 1, 128}]

Formula

a(n)=Mod[n, A065855(n)]
Previous Showing 81-89 of 89 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.