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 1-3 of 3 results.

A285572 Number of finite sets of pairwise indivisible positive integers with least common multiple n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 9, 1, 1, 2, 2, 2, 6, 1, 2, 2, 4, 1, 9, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 23, 1, 2, 3, 1, 2, 9, 1, 3, 2, 9, 1, 10, 1, 2, 3, 3, 2, 9, 1, 5, 1, 2, 1, 23, 2, 2, 2, 4, 1, 23, 2, 3, 2, 2, 2, 6, 1, 3, 3, 6
Offset: 1

Views

Author

Gus Wiseman, Apr 21 2017

Keywords

Examples

			The a(72)=10 sets are {72}, {8,9}, {8,18}, {8,36}, {9,24}, {18,24}, {24,36}, {6,8,9}, {8,9,12}, {8,12,18}.

Links

Crossrefs

Cf. A000666, A006126, A048143, A054921, A076078, A076413, A285573.

Programs

  • Mathematica
    nn=50;
stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[Rest[stableSets[Divisors[n],Divisible]],LCM@@#===n&]],{n,1,nn}]

A076078 a(n) is the number of nonempty sets of distinct positive integers that have a least common multiple of n.

Original entry on oeis.org

1, 2, 2, 4, 2, 10, 2, 8, 4, 10, 2, 44, 2, 10, 10, 16, 2, 44, 2, 44, 10, 10, 2, 184, 4, 10, 8, 44, 2, 218, 2, 32, 10, 10, 10, 400, 2, 10, 10, 184, 2, 218, 2, 44, 44, 10, 2, 752, 4, 44, 10, 44, 2, 184, 10, 184, 10, 10, 2, 3748, 2, 10, 44, 64, 10, 218, 2, 44, 10, 218, 2, 3392, 2, 10
Offset: 1

Views

Author

Amarnath Murthy, Oct 05 2002

Keywords

Comments

a(n)=1 iff n=1, a(p^k)=2^k, a(p*q)=10; where p & q are unique primes. a(n) cannot equal an odd number >1. - Robert G. Wilson v
If m has more divisors than n, then a(m) > a(n). - Matthew Vandermast, Aug 22 2004
If n is of the form p^r*q^s where p & q are distinct primes and r & s are nonnegative integers then a(n)=2^(rs)*(2^(r+s+1) -2^r-2^s+1); for example f(1400846643)=f(3^5*7^8)=2^(5*8)*(2^ (5+8+1)-2^5-2^8+1)=17698838672310272. Also if n=p_1^r_1*p_2^r_2*...*p_k^r_k where p_1,p_2,...,p_k are distinct primes and r_1,r_2,...,r_k are natural numbers then 2^(r_1*r_2*...*r_k)||a(n). - Farideh Firoozbakht, Aug 06 2005
None of terms is divisible by Mersenne numbers 3 or 7. For any n, a(n) is congruent to A008836(n) mod 3. Since A008836(n) is always 1 or -1, this implies that A000225(2)=3 never divides a(n). - Matthew Vandermast, Oct 12 2010
There are terms divisible by larger Mersenne numbers. For example, a(2*3*5*7*11*13*19*23^3) is divisible by 31. - Max Alekseyev, Nov 18 2010

Examples

			a(6) = 10. The sets with LCM 6 are {6}, {1,6}, {2,3}, {2,6}, {3,6}, {1,2,3}, {1,2,6}, {1,3,6}, {2,3,6}, {1,2,3,6}.

Links

Crossrefs

Cf. A076413, A097210-A097218, A097416, A002235.
Cf. A069626.

Programs

  • Maple
    with(numtheory): seq(add(mobius(n/d)*(2^tau(d)-1), d in divisors(n)), n=1..80); # Ridouane Oudra, Mar 12 2024
  • Mathematica
    f[n_] := Block[{d = Divisors[n]}, Plus @@ (MoebiusMu[n/d](2^DivisorSigma[0, d] - 1))]; Table[ f[n], {n, 75}] (* Robert G. Wilson v *)
  • PARI
    a(n) = local(f, l, s, t, q); f = factor(n); l = matsize(f)[1]; s = 0; forvec(v = vector(l, i, [0, 1]), q = sum(i = 1, l, v[i]); t = (-1)^(l - q)*2^prod(i = 1, l, f[i, 2] + v[i]); s += t); s; \\ Definition corrected by David Wasserman, Dec 26 2007

Formula

2^d(n) - 1 = Sum_{m|n} a(m), where d(n) = A000005(n) is the number of divisors of n, so a(n) = Sum_{m|n} mu(n/m)*(2^d(m) - 1).
a(n) = 2*A069626(n), for n > 1. - Ridouane Oudra, Mar 12 2024

Extensions

Edited by Dean Hickerson, Oct 08 2002
Definition corrected by David Wasserman, Dec 26 2007
Edited by Charles R Greathouse IV, Aug 02 2010
Edited by Max Alekseyev, Nov 18 2010

A285573 Number of finite nonempty sets of pairwise indivisible divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 4, 3, 5, 2, 9, 2, 5, 5, 5, 2, 9, 2, 9, 5, 5, 2, 14, 3, 5, 4, 9, 2, 19, 2, 6, 5, 5, 5, 19, 2, 5, 5, 14, 2, 19, 2, 9, 9, 5, 2, 20, 3, 9, 5, 9, 2, 14, 5, 14, 5, 5, 2, 49, 2, 5, 9, 7, 5, 19, 2, 9, 5, 19, 2, 34, 2, 5, 9, 9, 5, 19, 2, 20, 5, 5, 2, 49, 5, 5, 5, 14, 2, 49, 5, 9, 5, 5, 5, 27, 2, 9, 9, 19
Offset: 1

Views

Author

Gus Wiseman, Apr 21 2017

Keywords

Comments

From Robert Israel, Apr 21 2017: (Start)
If n = p^k for prime p, a(n) = k+1.
If n = p^j*q^k for distinct primes p,q, a(n) = binomial(j+k+2,j+1)-1. (End)

Examples

			The a(12)=9 sets are: {1}, {2}, {3}, {4}, {6}, {12}, {2,3}, {3,4}, {4,6}.

Crossrefs

Cf. A006126, A048143, A076078, A076413, A198085, A285572.

Programs

  • Maple
    g:= proc(S) local x, Sx; option remember;
   if nops(S) = 0 then return {{}} fi;
   x:= S[1];
   Sx:= subsop(1=NULL,S);
   procname(Sx) union map(t -> t union {x}, procname(remove(s -> s mod x = 0 or x mod s = 0, Sx)))
end proc:
f:= proc(n) local F,D;
  F:= ifactors(n)[2];
  D:= numtheory:-divisors(mul(ithprime(i)^F[i,2],i=1..nops(F)));
  nops(g(D)) - 1;
end proc:
map(f, [$1..100]); # Robert Israel, Apr 21 2017
  • Mathematica
    nn=50;
stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Rest[stableSets[Divisors[n],Divisible]]],{n,1,nn}]
Showing 1-3 of 3 results.

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