A002035 Numbers that contain primes to odd powers only.

2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 96, 97, 101

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

Complement of the union of {1} and A072587. - Reinhard Zumkeller, Nov 15 2012, corrected version from Jun 23 2002

A036537 is a subsequence and this sequence is a subsequence of A162644. - Reinhard Zumkeller, Jul 08 2009

The asymptotic density of this sequence is A065463. - Amiram Eldar, Sep 18 2022

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Eckford Cohen, Quadratic congruences with an odd number of summands, Amer. Math. Monthly, 73 (1966), 138-143.

Crossrefs

Cf. A000203, A036537, A065463, A072586, A072587, A124010, A162644, A188999, A295316.

Programs

Haskell

a002035 n = a002035_list !! (n-1)
a002035_list = filter (all odd . a124010_row) [1..]
-- Reinhard Zumkeller, Nov 14 2012

Maple

isA002035 := proc(n)
    local pe;
    for pe in ifactors(n)[2] do
        if type(pe[2],'even') then
            return false;
        end if;
    end do:
    true ;
end proc:
A002035 := proc(n)
    option remember;
    if n =1 then
        2;
    else
        for a from procname(n-1)+1 do
            if isA002035(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A002035(n),n=1..100) ; # R. J. Mathar, Nov 27 2017

Mathematica

ok[n_] := And @@ OddQ /@ FactorInteger[n][[All, 2]];
Select[Range[2, 101], ok]
(* Jean-François Alcover, Apr 22 2011 *)
Select[Range[2,110],AllTrue[FactorInteger[#][[All,2]],OddQ]&] (* Harvey P. Dale, Nov 02 2022 *)

PARI

is(n)=Set(factor(n)[,2]%2)==[1] \\ Charles R Greathouse IV, Feb 07 2017

Extensions

More terms from Reinhard Zumkeller, Jun 23 2002