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 1-4 of 4 results.

A000469 1 together with products of 2 or more distinct primes.

Original entry on oeis.org

1, 6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158
Offset: 1

Views

Author

Dan Bentley (dtb(AT)research.att.com)

Keywords

Comments

Nonprime squarefree numbers.
Except for 1, composite n such that the squarefree part of n is greater than phi(n). - Benoit Cloitre, Apr 06 2002

Links

Crossrefs

Cf. A005117, A007913, A000010, A010051, A239508, A239509, A120944 (composite squarefree numbers, same sequence apart from the first term).

Programs

  • Haskell
    a000469 n = a000469_list !! (n-1)
a000469_list = filter ((== 0) . a010051) a005117_list
-- Reinhard Zumkeller, Mar 21 2014
  • Maple
    select(numtheory:-issqrfree and not isprime, [$1..1000]); # Robert Israel, Aug 06 2015
  • Mathematica
    lst={}; Do[If[SquareFreeQ[n], If[ !PrimeQ[n], AppendTo[lst,n]]], {n,200}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 20 2009 *)
With[{upto=200},Complement[Select[Range[upto],SquareFreeQ],Prime[ Range[ PrimePi[ upto]]]]] (* Harvey P. Dale, Oct 01 2011 *)
Select[Range[200], !PrimeQ[#] && PrimeOmega[#] == PrimeNu[#] &] (* Carlos Eduardo Olivieri, Aug 06 2015 *)
  • PARI
    for(n=0,64, if(isprime(n), n+1, if(issquarefree(n),print(n))))
  • PARI
    for(n=1,160,if(core(n)*(1-isprime(n))>eulerphi(n),print1(n,",")))
  • Python
    from math import isqrt
from sympy import primepi, mobius
def A000469(n):
    def f(x): return n+primepi(x)+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 02 2024

Formula

n such that A007913(n)>A000010(n). - Benoit Cloitre, Apr 06 2002
N-floor(N/p1) - floor(N/(p2) - ... - floor(N/p(i) + floor(N/(c2) + floor(N/(c3)+ ... + floor(N/c(j)-1 where N is any number; p1,p2 are the primes with p(i) being the first prime > square root of N and c2, c3 are the numbers other than 1 in this sequence with c(j) <= N will yield the number of primes less than or equal to N other than p1, p2, ..., p(i). - Ben Paul Thurston, Aug 15 2007
A005171(a(n))*A008966(a(n)) = 1. - Reinhard Zumkeller, Nov 01 2009
Sum(n=1, Infinity, 1/a(n)^s) = Zeta(s)/Zeta(2s) - PrimeZeta(s). - Enrique Pérez Herrero, Mar 31 2012
n such that A001221(n) = A001222(n), n nonprime. - Carlos Eduardo Olivieri, Aug 06 2015
a(n) = kn + O(n/log n) where k = Pi^2/6. - Charles R Greathouse IV, Aug 02 2024

A239508 Number of partitions of n into nonprime squarefree numbers, cf. A000469.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 6, 7, 7, 8, 8, 10, 12, 14, 14, 16, 17, 20, 22, 25, 26, 31, 33, 37, 40, 45, 49, 57, 60, 66, 71, 80, 86, 98, 104, 115, 125, 138, 147, 164, 175, 193, 209, 230, 244, 269, 289, 318, 343, 374, 398, 437, 468, 510, 548
Offset: 0

Views

Author

Reinhard Zumkeller, Mar 21 2014

Keywords

Examples

			a(10) = #{10, 6+1+1+1+1, 10x1} = 3;
a(11) = #{10+1, 6+1+1+1+1+1, 11x1} = 3;
a(12) = #{10+1+1, 6+6, 6+6x1, 12x1} = 4;
a(13) = #{10+1+1+1, 6+6+1, 6+7x1, 13x1} = 4;
a(14) = #{14, 10+1+1+1+1, 6+6+1+1, 6+8x1, 14x1} = 5;
a(15) = #{15, 14+1, 10+5x1, 6+6+1+1+1, 6+9x1, 15x1} = 6;
a(16) = #{15+1, 14+1+1, 10+6, 10+6x1, 6+6+4x1, 6+10x1, 16x1} = 7.

Crossrefs

Cf. A239509, A073576.

Programs

  • Haskell
    a239508 = p a000469_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

A290136 Positive numbers that are not the sum of two nonprime squarefree numbers (A000469).

Original entry on oeis.org

1, 3, 4, 5, 6, 8, 9, 10, 13, 14, 17, 18, 19, 26, 33, 38, 46, 62, 82
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 20 2017

Keywords

Comments

The sequence is conjectured to be complete.

Crossrefs

Cf. A000469, A014092, A071068, A071331, A098235, A239508, A239509.

Programs

  • Mathematica
    nmax = 82; f[x_] := Sum[Boole[SquareFreeQ[k] && PrimeNu[k] != 1] x^k, {k, 1, nmax}]^2; b = Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, nmax}]]; c = Complement[Range[nmax], b][[1 ;; 19]]

A290137 Number of compositions (ordered partitions) of n into nonprime squarefree parts (A000469).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 9, 12, 16, 22, 30, 41, 55, 73, 96, 128, 173, 235, 317, 426, 570, 763, 1023, 1375, 1848, 2484, 3337, 4482, 6017, 8077, 10843, 14562, 19560, 26276, 35292, 47392, 63632, 85443, 114741, 154098, 206957, 277941, 373254, 501244, 673121, 903945, 1213935, 1630246, 2189330
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 20 2017

Keywords

Examples

			a(8) = 4 because we have [6, 1, 1], [1, 6, 1], [1, 1, 6] and [1, 1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A000469, A239508, A239509, A280194.

Programs

  • Mathematica
    nmax = 53; CoefficientList[Series[1/(1 - Sum[Boole[SquareFreeQ[k] && PrimeNu[k] != 1] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - Sum_{k>=1} x^A000469(k)).
Showing 1-4 of 4 results.

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