login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A030229 Product of an even number of distinct primes (including 1). 24
1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201, 202, 203, 205, 206, 209, 210, 213, 214 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

From Enrique Pérez Herrero, Jul 06 2012: (Start)

This sequence and A030059 form a partition of the squarefree numbers set: A005117.

Also solutions to equation mu(n)=1.

Sum(n>=1, 1/a(n)^s) =  (Zeta(s)^2 + Zeta(2*s))/(2*Zeta(s)*Zeta(2*s)).

(End)

A008683(a(n)) = 1; a(A220969(n)) mod 2 = 0; a(A220968(n)) mod 2 = 1. - Reinhard Zumkeller, Dec 27 2012

Characteristic function for values of a(n) = (mu(n)+1)! - 1, where mu(n) is the Mobius function (A008683). - Wesley Ivan Hurt, Oct 11 2013

Conjecture: For the matrix M(i,j) = 1 if j|i and 0 otherwise, Inverse(M)(a,1) = -1, for any a in this sequence. - Benedict W. J. Irwin, Jul 26 2016

REFERENCES

B. C. Berndt and R. A. Rankin, Ramanujan: Letters and Commentary, see p. 23; AMS Providence RI 1995

Ramanujan, Collected Papers, pp. xxiv, 21.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

S. Ramanujan, Irregular numbers, J. Indian Math. Soc. 5 (1913) 105-106.

Eric Weisstein's World of Mathematics, Prime Factor

Eric Weisstein's World of Mathematics, Moebius Function

Eric Weisstein's World of Mathematics, Prime Sums

FORMULA

a(n) < 2n infinitely often; a(n) > 2n infinitely often. - Charles R Greathouse IV, Oct 04 2011

MAPLE

a := n -> `if`(numtheory[mobius](n)=1, n, NULL); seq(a(i), i=1..214); # - Peter Luschny, May 04 2009

with(numtheory); t := [ ]: f := [ ]: for n from 1 to 250 do if mobius(n) = 1 then t := [ op(t), n ] else f := [ op(f), n ]; fi; od: t; # Wesley Ivan Hurt, Oct 11 2013

MATHEMATICA

Select[Range[214], MoebiusMu[#] == 1 &] (* Jean-François Alcover, Oct 04 2011 *)

PROG

(PARI) isA030229(n)= #(n=factor(n)[, 2]) % 2 == 0 && (!n || vecmax(n)==1 )

(PARI) is(n)=moebius(n)==1 \\ Charles R Greathouse IV, Jan 31 2017

for(n=1, 500, isA030229(n)&print1(n", ")) \\ M. F. Hasler

(Haskell)

import Data.List (elemIndices)

a030229 n = a030229_list !! (n-1)

a030229_list = map (+ 1) $ elemIndices 1 a008683_list

-- Reinhard Zumkeller, Dec 27 2012

CROSSREFS

Cf. A006881, A151797, A030059, A005117, A008683.

Sequence in context: A265693 A211484 A006881 * A201650 A201514 A201464

Adjacent sequences:  A030226 A030227 A030228 * A030230 A030231 A030232

KEYWORD

nonn,easy,nice

AUTHOR

David W. Wilson

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified July 23 06:43 EDT 2017. Contains 289686 sequences.