A030229 -id:A030229 - cp's OEIS frontend

A005117 Squarefree numbers: numbers that are not divisible by a square greater than 1.

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113

Offset: 1

N. J. A. Sloane

1 together with the numbers that are products of distinct primes.
Also smallest sequence with the property that a(m)*a(k) is never a square for k != m. - Ulrich Schimke (ulrschimke(AT)aol.com), Dec 12 2001
Numbers k such that there is only one Abelian group with k elements, the cyclic group of order k (the numbers such that A000688(k) = 1). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 25 2001
Numbers k such that A007913(k) > phi(k). - Benoit Cloitre, Apr 10 2002
a(n) is the smallest m with exactly n squarefree numbers <= m. - Amarnath Murthy, May 21 2002
k is squarefree <=> k divides prime(k)# where prime(k)# = product of first k prime numbers. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 30 2004
Numbers k such that omega(k) = Omega(k) = A072047(k). - Lekraj Beedassy, Jul 11 2006
The LCM of any finite subset is in this sequence. - Lekraj Beedassy, Jul 11 2006
This sequence and the Beatty Pi^2/6 sequence (A059535) are "incestuous": the first 20000 terms are bounded within (-9, 14). - Ed Pegg Jr, Jul 22 2008
Let us introduce a function D(n) = sigma_0(n)/2^(alpha(1) + ... + alpha(r)), sigma_0(n) number of divisors of n (A000005), prime factorization of n = p(1)^alpha(1) * ... * p(r)^alpha(r), alpha(1) + ... + alpha(r) is sequence (A001222). Function D(n) splits the set of positive integers into subsets, according to the value of D(n). Squarefree numbers (A005117) has D(n)=1, other numbers are "deviated" from the squarefree ideal and have 0 < D(n) < 1. For D(n)=1/2 we have A048109, for D(n)=3/4 we have A060687. - Ctibor O. Zizka, Sep 21 2008
Numbers k such that gcd(k,k')=1 where k' is the arithmetic derivative (A003415) of k. - Giorgio Balzarotti, Apr 23 2011
Numbers k such that A007913(k) = core(k) = k. - Franz Vrabec, Aug 27 2011
Numbers k such that sqrt(k) cannot be simplified. - Sean Loughran, Sep 04 2011
Indices m where A057918(m)=0, i.e., positive integers m for which there are no integers k in {1,2,...,m-1} such that k*m is a square. - John W. Layman, Sep 08 2011
It appears that these are numbers j such that Product_{k=1..j} (prime(k) mod j) = 0 (see Maple code). - Gary Detlefs, Dec 07 2011. - This is the same claim as Mohammed Bouayoun's Mar 30 2004 comment above. To see why it holds: Primorial numbers, A002110, a subsequence of this sequence, are never divisible by any nonsquarefree number, A013929, and on the other hand, the index of the greatest prime dividing any n is less than n. Cf. A243291. - Antti Karttunen, Jun 03 2014
Conjecture: For each n=2,3,... there are infinitely many integers b > a(n) such that Sum_{k=1..n} a(k)*b^(k-1) is prime, and the smallest such an integer b does not exceed (n+3)*(n+4). - Zhi-Wei Sun, Mar 26 2013
The probability that a random natural number belongs to the sequence is 6/Pi^2, A059956 (see Cesàro reference). - Giorgio Balzarotti, Nov 21 2013
Booker, Hiary, & Keating give a subexponential algorithm for testing membership in this sequence without factoring. - Charles R Greathouse IV, Jan 29 2014
Because in the factorizations into prime numbers these a(n) (n >= 2) have exponents which are either 0 or 1 one could call the a(n) 'numbers with a fermionic prime number decomposition'. The levels are the prime numbers prime(j), j >= 1, and the occupation numbers (exponents) e(j) are 0 or 1 (like in Pauli's exclusion principle). A 'fermionic state' is then denoted by a sequence with entries 0 or 1, where, except for the zero sequence, trailing zeros are omitted. The zero sequence stands for a(1) = 1. For example a(5) = 6 = 2^1*3^1 is denoted by the 'fermionic state' [1, 1], a(7) = 10 by [1, 0, 1]. Compare with 'fermionic partitions' counted in A000009. - Wolfdieter Lang, May 14 2014
From Vladimir Shevelev, Nov 20 2014: (Start)
The following is an Eratosthenes-type sieve for squarefree numbers. For integers > 1:
1) Remove even numbers, except for 2; the minimal non-removed number is 3.
2) Replace multiples of 3 removed in step 1, and remove multiples of 3 except for 3 itself; the minimal non-removed number is 5.
3) Replace multiples of 5 removed as a result of steps 1 and 2, and remove multiples of 5 except for 5 itself; the minimal non-removed number is 6.
4) Replace multiples of 6 removed as a result of steps 1, 2 and 3 and remove multiples of 6 except for 6 itself; the minimal non-removed number is 7.
5) Repeat using the last minimal non-removed number to sieve from the recovered multiples of previous steps.
Proof. We use induction. Suppose that as a result of the algorithm, we have found all squarefree numbers less than n and no other numbers. If n is squarefree, then the number of its proper divisors d > 1 is even (it is 2^k - 2, where k is the number of its prime divisors), and, by the algorithm, it remains in the sequence. Otherwise, n is removed, since the number of its squarefree divisors > 1 is odd (it is 2^k-1).
(End)
The lexicographically least sequence of integers > 1 such that each entry has an even number of proper divisors occurring in the sequence (that's the sieve restated). - Glen Whitney, Aug 30 2015
0 is nonsquarefree because it is divisible by any square. - Jon Perry, Nov 22 2014, edited by M. F. Hasler, Aug 13 2015
The Heinz numbers of partitions with distinct parts. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product_{j=1..r} prime(j) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] the Heinz number is 2*2*3*7*29 = 2436. The number 30 (= 2*3*5) is in the sequence because it is the Heinz number of the partition [1,2,3]. - Emeric Deutsch, May 21 2015
It is possible for 2 consecutive terms to be even; for example a(258)=422 and a(259)=426. - Thomas Ordowski, Jul 21 2015. [These form a subsequence of A077395 since their product is divisible by 4. - M. F. Hasler, Aug 13 2015]
There are never more than 3 consecutive terms. Runs of 3 terms start at 1, 5, 13, 21, 29, 33, ... (A007675). - Ivan Neretin, Nov 07 2015
a(n) = product of row n in A265668. - Reinhard Zumkeller, Dec 13 2015
Numbers without excess, i.e., numbers k such that A001221(k) = A001222(k). - Juri-Stepan Gerasimov, Sep 05 2016
Numbers k such that b^(phi(k)+1) == b (mod k) for every integer b. - Thomas Ordowski, Oct 09 2016
Boreico shows that the set of square roots of the terms of this sequence is linearly independent over the rationals. - Jason Kimberley, Nov 25 2016 (reference found by Michael Coons).
Numbers k such that A008836(k) = A008683(k). - Enrique Pérez Herrero, Apr 04 2018
The prime zeta function P(s) "has singular points along the real axis for s=1/k where k runs through all positive integers without a square factor". See Wolfram link. - Maleval Francis, Jun 23 2018
Numbers k such that A007947(k) = k. - Kyle Wyonch, Jan 15 2021
The Schnirelmann density of the squarefree numbers is 53/88 (Rogers, 1964). - Amiram Eldar, Mar 12 2021
Comment from Isaac Saffold, Dec 21 2021: (Start)
Numbers k such that all groups of order k have a trivial Frattini subgroup [Dummit and Foote].
Let the group G have order n. If n is squarefree and n > 1, then G is solvable, and thus by Hall's Theorem contains a subgroup H_p of index p for all p | n. Each H_p is maximal in G by order considerations, and the intersection of all the H_p's is trivial. Thus G's Frattini subgroup Phi(G), being the intersection of G's maximal subgroups, must be trivial. If n is not squarefree, the cyclic group of order n has a nontrivial Frattini subgroup. (End)
Numbers for which the squarefree divisors (A206778) and the unitary divisors (A077610) are the same; moreover they are also the set of divisors (A027750). - Bernard Schott, Nov 04 2022
0 = A008683(a(n)) - A008836(a(n)) = A001615(a(n)) - A000203(a(n)). - Torlach Rush, Feb 08 2023
From Robert D. Rosales, May 20 2024: (Start)
Numbers n such that mu(n) != 0, where mu(n) is the Möbius function (A008683).
Solutions to the equation Sum_{d|n} mu(d)*sigma(d) = mu(n)*n, where sigma(n) is the sum of divisors function (A000203). (End)
a(n) is the smallest root of x = 1 + Sum_{k=1..n-1} floor(sqrt(x/a(k))) greater than a(n-1). - Yifan Xie, Jul 10 2024
Number k such that A001414(k) = A008472(k). - Torlach Rush, Apr 14 2025
To elaborate on the formula from Greathouse (2018), the maximum of a(n) - floor(n*Pi^2/6 + sqrt(n)/17) is 10 at indices n = 48715, 48716, 48721, and 48760. The maximum is 11, at the same indices, if floor is taken individually for the two addends and the square root. If the value is rounded instead, the maximum is 9 at 10 indices between 48714 and 48765. - M. F. Hasler, Aug 08 2025

Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 165, p. 53, Ellipses, Paris, 2008.
David S. Dummit and Richard M. Foote, Abstract algebra. Vol. 1999. Englewood Cliffs, NJ: David S.Prentice Hall, 1991.
Ivan M. Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 251.
Michael Pohst and Hans J. Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, page 432.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel Forgues, Table of n, a(n) for n = 1..60794 (first 10000 terms from T. D. Noe)
Zenon B. Batang, Squarefree integers and the abc conjecture, arXiv:2109.10226 [math.GM], 2021.
Andrew R. Booker, Ghaith A. Hiary and Jon P. Keating, Detecting squarefree numbers, Duke Mathematical Journal, Vol. 164, No. 2 (2015), pp. 235-275; arXiv preprint, arXiv:1304.6937 [math.NT], 2013-2015.
Iurie Boreico, Linear independence of radicals, The Harvard College Mathematics Review 2(1), 87-92, Spring 2008.
Ernesto Cesàro, La serie di Lambert in aritmetica assintotica, Rendiconto della Reale Accademia delle Scienze di Napoli, Serie 2, Vol. 7 (1893), pp. 197-204.
Henri Cohen, Francois Dress, and Mohamed El Marraki, Explicit estimates for summatory functions linked to the Möbius μ-function, Functiones et Approximatio Commentarii Mathematici 37 (2007), part 1, pp. 51-63.
H. Gent, Letter to N. J. A. Sloane, Nov 27 1975.
Andrew Granville, ABC means we can count squarefrees, International Mathematical Research Notices 19 (1998), 991-1009.
Pentti Haukkanen, Mika Mattila, Jorma K. Merikoski and Timo Tossavainen, Can the Arithmetic Derivative be Defined on a Non-Unique Factorization Domain?, Journal of Integer Sequences, Vol. 16 (2013), Article 13.1.2.
Aaron Krowne, squarefree number, PlanetMath.org.
Louis Marmet, First occurrences of squarefree gaps and an algorithm for their computation.
Louis Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv preprint arXiv:1210.3829 [math.NT], 2012.
Srinivasa Ramanujan, Irregular numbers, J. Indian Math. Soc., Vol. 5 (1913), pp. 105-106.
Kenneth Rogers, The Schnirelmann density of the squarefree integers, Proceedings of the American Mathematical Society, Vol. 15, No. 4 (1964), pp. 515-516.
J. A. Scott, Square-freedom revisited, The Mathematical Gazette, Vol. 90, No. 517 (2006), pp. 112-113.
Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv preprint arXiv:1511.03860 [math.NT], 2015.
O. Trifonov, On the Squarefree Problem II, Math. Balkanica, Vol. 3 (1989), Fasc. 3-4.
Eric Weisstein's World of Mathematics, Squarefree.
Eric Weisstein's World of Mathematics, Prime Zeta Function.
Wikipedia, Squarefree integer.
Index entries for "core" sequences.

Complement of A013929. Subsequence of A072774 and A209061.
Characteristic function: A008966 (mu(n)^2, where mu = A008683).
Subsequences: A000040, A002110, A235488.
Cf. A001414, A008472, A076259 (first differences), A173143 (partial sums), A000688, A003277, A013928, A020753, A020754, A020755, A030059, A030229, A033197, A034444, A039956, A048672, A053797, A057918, A059956, A071403, A072284, A120992, A133466, A136742, A136743, A160764, A243289, A243347, A243348, A243351, A215366, A046660, A265668, A265675.
Cf. A027750, A077610, A206778.
Subsequences: numbers j such that j*a(k) is squarefree where k > 1: A056911 (k = 2), A261034 (k = 3), A274546 (k = 5), A276378 (k = 6).

a005117 n = a005117_list !! (n-1)
a005117_list = filter ((== 1) . a008966) [1..]
-- Reinhard Zumkeller, Aug 15 2011, May 10 2011

[ n : n in [1..1000] | IsSquarefree(n) ];

with(numtheory); a := [ ]; for n from 1 to 200 do if issqrfree(n) then a := [ op(a), n ]; fi; od:
t:= n-> product(ithprime(k),k=1..n): for n from 1 to 113 do if(t(n) mod n = 0) then print(n) fi od; # Gary Detlefs, Dec 07 2011
A005117 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if numtheory[issqrfree](a) then return a; end if; end do: end if; end proc:  # R. J. Mathar, Jan 09 2013

Select[ Range[ 113], SquareFreeQ] (* Robert G. Wilson v, Jan 31 2005 *)
Select[Range[150], Max[Last /@ FactorInteger[ # ]] < 2 &] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26 2006 *)
NextSquareFree[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sf = n + sgn; While[c < Abs[k], While[ ! SquareFreeQ@ sf, If[sgn < 0, sf--, sf++]]; If[ sgn < 0, sf--, sf++]; c++]; sf + If[ sgn < 0, 1, -1]]; NestList[ NextSquareFree, 1, 70] (* Robert G. Wilson v, Apr 18 2014 *)
Select[Range[250], MoebiusMu[#] != 0 &] (* Robert D. Rosales, May 20 2024 *)

bnd = 1000; L = vector(bnd); j = 1; for (i=1,bnd, if(issquarefree(i),L[j]=i; j=j+1)); L

{a(n)= local(m,c); if(n<=1,n==1, c=1; m=1; while( cMichael Somos, Apr 29 2005 */

list(n)=my(v=vectorsmall(n,i,1),u,j); forprime(p=2,sqrtint(n), forstep(i=p^2, n, p^2, v[i]=0)); u=vector(sum(i=1,n,v[i])); for(i=1,n,if(v[i],u[j++]=i)); u \\ Charles R Greathouse IV, Jun 08 2012

for(n=1, 113, if(core(n)==n, print1(n, ", "))); \\ Arkadiusz Wesolowski, Aug 02 2016

S(n) = my(s); forsquarefree(k=1,sqrtint(n),s+=n\k[1]^2*moebius(k)); s;
a(n) = my(min=1, max=231, k=0, sc=0); if(n >= 144, min=floor(zeta(2)*n - 5*sqrt(n)); max=ceil(zeta(2)*n + 5*sqrt(n))); while(min <= max, k=(min+max)\2; sc=S(k); if(abs(sc-n) <= sqrtint(n), break); if(sc > n, max=k-1, if(sc < n, min=k+1, break))); while(!issquarefree(k), k-=1); while(sc != n, my(j=1); if(sc > n, j = -1); k += j; sc += j; while(!issquarefree(k), k += j)); k; \\ Daniel Suteu, Jul 07 2022

first(n)=my(v=vector(n),i); forsquarefree(k=1,if(n<268293,(33*n+30)\20,(n*Pi^2/6+0.058377*sqrt(n))\1), if(i++>n, return(v)); v[i]=k[1]); v \\ Charles R Greathouse IV, Jan 10 2023

A5117=[1..3]; A005117(n)={if(n>#A5117, my(N=#A5117); A5117=Vec(A5117, max(n+999, N*5\4)); iferr(forsquarefree(k=A5117[N]+1, #A5117*Pi^2\6+sqrtint(#A5117)\17+11, A5117[N++]=k[1]),E,)); A5117[n]} \\ M. F. Hasler, Aug 08 2025

from sympy.ntheory.factor_ import core
def ok(n): return core(n, 2) == n
print(list(filter(ok, range(1, 114)))) # Michael S. Branicky, Jul 31 2021

from itertools import count, islice
from sympy import factorint
def A005117_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(x == 1 for x in factorint(n).values()),count(max(startvalue,1)))
A005117_list = list(islice(A005117_gen(),20)) # Chai Wah Wu, May 09 2022

from math import isqrt
from sympy import mobius
def A005117(n):
    def f(x): return n+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, Jul 22 2024

Limit_{n->oo} a(n)/n = Pi^2/6 (see A013661). - Benoit Cloitre, May 23 2002
Equals A039956 UNION A056911. - R. J. Mathar, May 16 2008
A122840(a(n)) <= 1; A010888(a(n)) < 9. - Reinhard Zumkeller, Mar 30 2010
a(n) = A055229(A062838(n)) and a(n) > A055229(m) for m < A062838(n). - Reinhard Zumkeller, Apr 09 2010
A008477(a(n)) = 1. - Reinhard Zumkeller, Feb 17 2012
A055653(a(n)) = a(n); A055654(a(n)) = 0. - Reinhard Zumkeller, Mar 11 2012
A008966(a(n)) = 1. - Reinhard Zumkeller, May 26 2012
Sum_{n>=1} 1/a(n)^s = zeta(s)/zeta(2*s). - Enrique Pérez Herrero, Jul 07 2012
A056170(a(n)) = 0. - Reinhard Zumkeller, Dec 29 2012
A013928(a(n)+1) = n. - Antti Karttunen, Jun 03 2014
A046660(a(n)) = 0. - Reinhard Zumkeller, Nov 29 2015
Equals {1} UNION A000040 UNION A006881 UNION A007304 UNION A046386 UNION A046387 UNION A067885 UNION A123321 UNION A123322 UNION A115343 ... - R. J. Mathar, Nov 05 2016
|a(n) - n*Pi^2/6| < 0.058377*sqrt(n) for n >= 268293; this result can be derived from Cohen, Dress, & El Marraki, see links. - Charles R Greathouse IV, Jan 18 2018
From Amiram Eldar, Jul 07 2021: (Start)
Sum_{n>=1} (-1)^(a(n)+1)/a(n)^2 = 9/Pi^2.
Sum_{k=1..n} 1/a(k) ~ (6/Pi^2) * log(n).
Sum_{k=1..n} (-1)^(a(k)+1)/a(k) ~ (2/Pi^2) * log(n).
(all from Scott, 2006) (End)

A006881 Squarefree semiprimes: Numbers that are the product of two distinct primes.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Robert Munafo, Simon Plouffe

Numbers k such that phi(k) + sigma(k) = 2*(k+1). - Benoit Cloitre, Mar 02 2002
Numbers k such that tau(k) = omega(k)^omega(k). - Benoit Cloitre, Sep 10 2002 [This comment is false. If k = 900 then tau(k) = omega(k)^omega(k) = 27 but 900 = (2*3*5)^2 is not the product of two distinct primes. - Peter Luschny, Jul 12 2023]
Could also be called 2-almost primes. - Rick L. Shepherd, May 11 2003
From the Goldston et al. reference's abstract: "lim inf [as n approaches infinity] [(a(n+1) - a(n))] <= 26. If an appropriate generalization of the Elliott-Halberstam Conjecture is true, then the above bound can be improved to 6." - Jonathan Vos Post, Jun 20 2005
The maximal number of consecutive integers in this sequence is 3 - there cannot be 4 consecutive integers because one of them would be divisible by 4 and therefore would not be product of distinct primes. There are several examples of 3 consecutive integers in this sequence. The first one is 33 = 3 * 11, 34 = 2 * 17, 35 = 5 * 7; (see A039833). - Matias Saucedo (solomatias(AT)yahoo.com.ar), Mar 15 2008
Number of terms less than or equal to 10^k for k >= 0 is A036351(k). - Robert G. Wilson v, Jun 26 2012
Are these the numbers k whose difference between the sum of proper divisors of k and the arithmetic derivative of k is equal to 1? - Omar E. Pol, Dec 19 2012
Intersection of A001358 and A030513. - Wesley Ivan Hurt, Sep 09 2013
A237114(n) (smallest semiprime k^prime(n)+1) is a term, for n != 2. - Jonathan Sondow, Feb 06 2014
a(n) are the reduced denominators of p_2/p_1 + p_4/p_3, where p_1 != p_2, p_3 != p_4, p_1 != p_3, and the p's are primes. In other words, (p_2*p_3 + p_1*p_4) never shares a common factor with p_1*p_3. - Richard R. Forberg, Mar 04 2015
Conjecture: The sums of two elements of a(n) forms a set that includes all primes greater than or equal to 29 and all integers greater than or equal to 83 (and many below 83). - Richard R. Forberg, Mar 04 2015
The (disjoint) union of this sequence and A001248 is A001358. - Jason Kimberley, Nov 12 2015
A263990 lists the subsequence of a(n) where a(n+1)=1+a(n). - R. J. Mathar, Aug 13 2019

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Zervos, Marie: Sur une classe de nombres composés. Actes du Congrès interbalkanique de mathématiciens 267-268 (1935)

T. D. Noe, Table of n, a(n) for n = 1..10000
D. A. Goldston, S. W. Graham, J. Pimtz and Y. Yildirim, "Small Gaps Between Primes or Almost Primes", arXiv:math/0506067 [math.NT], March 2005.
G. T. Leavens and M. Vermeulen, 3x+1 search programs, Computers and Mathematics with Applications, 24 (1992), 79-99. (Annotated scanned copy)
R. J. Mathar, Series of reciprocal powers of k-almost primes arXiv:0803.0900, table 6 k=2 shows sum 1/a(n)^s.
Eric Weisstein's World of Mathematics, Semiprime
Index to sequences related to prime signature

Products of exactly k distinct primes, for k = 1 to 6: A000040, A006881. A007304, A046386, A046387, A067885.
Cf. A030229, A051709, A001221 (omega(n)), A001222 (bigomega(n)), A001358 (semiprimes), A005117 (squarefree), A007304 (squarefree 3-almost primes), A213952, A039833, A016105 (subsequences), A237114 (subsequence, n != 2).
Subsequence of A007422.
Cf. A259758 (subsequence), A036351, A363923.

a006881 n = a006881_list !! (n-1)
a006881_list = filter chi [1..] where
   chi n = p /= q && a010051 q == 1 where
      p = a020639 n
      q = n `div` p
-- Reinhard Zumkeller, Aug 07 2011

[n: n in [1..210] | EulerPhi(n) + DivisorSigma(1,n) eq 2*(n+1)]; // Vincenzo Librandi, Sep 17 2015

N:= 1001: # to get all terms < N
Primes:= select(isprime, [2,seq(2*k+1,k=1..floor(N/2))]):
{seq(seq(p*q,q=Primes[1..ListTools:-BinaryPlace(Primes,N/p)]),p=Primes)} minus {seq(p^2,p=Primes)};
# Robert Israel, Jul 23 2014
# Alternative, using A001221:
isA006881 := proc(n)
     if numtheory[bigomega](n) =2 and A001221(n) = 2 then
        true ;
    else
        false ;
    end if;
end proc:
A006881 := proc(n) if n = 1 then 6; else for a from procname(n-1)+1 do if isA006881(a) then return a; end if; end do: end if;
end proc: # R. J. Mathar, May 02 2010
# Alternative:
with(NumberTheory): isA006881 := n -> is(NumberOfPrimeFactors(n, 'distinct') = 2 and NumberOfPrimeFactors(n) = 2):
select(isA006881, [seq(1..205)]); # Peter Luschny, Jul 12 2023

mx = 205; Sort@ Flatten@ Table[ Prime[n]*Prime[m], {n, Log[2, mx/3]}, {m, n + 1, PrimePi[ mx/Prime[n]]}] (* Robert G. Wilson v, Dec 28 2005, modified Jul 23 2014 *)
sqFrSemiPrimeQ[n_] := Last@# & /@ FactorInteger@ n == {1, 1}; Select[Range[210], sqFrSemiPrimeQ] (* Robert G. Wilson v, Feb 07 2012 *)
With[{upto=250},Select[Sort[Times@@@Subsets[Prime[Range[upto/2]],{2}]],#<=upto&]] (* Harvey P. Dale, Apr 30 2018 *)

for(n=1,214,if(bigomega(n)==2&&omega(n)==2,print1(n,",")))

for(n=1,214,if(bigomega(n)==2&&issquarefree(n),print1(n,",")))

list(lim)=my(v=List()); forprime(p=2,sqrt(lim), forprime(q=p+1, lim\p, listput(v,p*q))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

from sympy import factorint
def ok(n): f=factorint(n); return len(f) == 2 and sum(f[p] for p in f) == 2
print(list(filter(ok, range(1, 206)))) # Michael S. Branicky, Jun 10 2021

from math import isqrt
from sympy import primepi, primerange
def A006881(n):
    def f(x): return int(n+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 15 2024

def A006881_list(n) :
    R = []
    for i in (6..n) :
        d = prime_divisors(i)
        if len(d) == 2 :
            if d[0]*d[1] == i :
                R.append(i)
    return R
A006881_list(205)  # Peter Luschny, Feb 07 2012

A000005(a(n)^(k-1)) = A000290(k) for all k>0. - Reinhard Zumkeller, Mar 04 2007
A109810(a(n)) = 4; A178254(a(n)) = 6. - Reinhard Zumkeller, May 24 2010
A056595(a(n)) = 3. - Reinhard Zumkeller, Aug 15 2011
a(n) = A096916(n) * A070647(n). - Reinhard Zumkeller, Sep 23 2011
A211110(a(n)) = 3. - Reinhard Zumkeller, Apr 02 2012
Sum_{n >= 1} 1/a(n)^s = (1/2)*(P(s)^2 - P(2*s)), where P is Prime Zeta. - Enrique Pérez Herrero, Jun 24 2012
A050326(a(n)) = 2. - Reinhard Zumkeller, May 03 2013
sopf(a(n)) = a(n) - phi(a(n)) + 1 = sigma(a(n)) - a(n) - 1. - Wesley Ivan Hurt, May 18 2013
d(a(n)) = 4. Omega(a(n)) = 2. omega(a(n)) = 2. mu(a(n)) = 1. - Wesley Ivan Hurt, Jun 28 2013
a(n) ~ n log n/log log n. - Charles R Greathouse IV, Aug 22 2013
A089233(a(n)) = 1. - Reinhard Zumkeller, Sep 04 2013
From Peter Luschny, Jul 12 2023: (Start)
For k > 1: k is a term <=> k^A001221(k) = k*A007947(k).
For k > 1: k is a term <=> k^A001222(k) = k*A007947(k).
For k > 1: k is a term <=> A363923(k) = k. (End)
a(n) ~ n log n/log log n. - Charles R Greathouse IV, Jan 13 2025

Name expanded (based on a comment of Rick L. Shepherd) by Charles R Greathouse IV, Sep 16 2015

A035363 Number of partitions of n into even parts.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 7, 0, 11, 0, 15, 0, 22, 0, 30, 0, 42, 0, 56, 0, 77, 0, 101, 0, 135, 0, 176, 0, 231, 0, 297, 0, 385, 0, 490, 0, 627, 0, 792, 0, 1002, 0, 1255, 0, 1575, 0, 1958, 0, 2436, 0, 3010, 0, 3718, 0, 4565, 0, 5604, 0, 6842, 0, 8349, 0, 10143, 0, 12310, 0

Offset: 0

Olivier Gérard

Convolved with A036469 = A000070. - Gary W. Adamson, Jun 09 2009
Note that these partitions are located in the head of the last section of the set of partitions of n (see A135010). - Omar E. Pol, Nov 20 2009
Number of symmetric unimodal compositions of n+2 where the maximal part appears twice, see example. Also number of symmetric unimodal compositions of n where the maximal part appears an even number of times. - Joerg Arndt, Jun 11 2013
Number of partitions of n having parts of even multiplicity. These are the conjugates of the partitions from the definition. Example: a(8)=5 because we have [4,4],[3,3,1,1],[2,2,2,2],[2,2,1,1,1,1], and [1,1,1,1,1,1,1,1]. - Emeric Deutsch, Jan 27 2016
From Gus Wiseman, May 22 2021: (Start)
The Heinz numbers of the conjugate partitions described in Emeric Deutsch's comment above are given by A000290.
For n > 1, also the number of integer partitions of n-1 whose only odd part is the smallest. The Heinz numbers of these partitions are given by A341446. For example, the a(2) = 1 through a(14) = 15 partitions (empty columns shown as dots, A..D = 10..13) are:
  1  .  3   .  5    .  7     .  9      .  B       .  D
        21     41      43       63        65         85
               221     61       81        83         A3
                       421      441       A1         C1
                       2221     621       443        643
                                4221      641        661
                                22221     821        841
                                          4421       A21
                                          6221       4441
                                          42221      6421
                                          222221     8221
                                                     44221
                                                     62221
                                                     422221
                                                     2222221
Also the number of integer partitions of n whose greatest part is the sum of all the other parts. The Heinz numbers of these partitions are given by A344415. For example, the a(2) = 1 through a(12) = 11 partitions (empty columns not shown) are:
  (11)  (22)   (33)    (44)     (55)      (66)
        (211)  (321)   (422)    (532)     (633)
               (3111)  (431)    (541)     (642)
                       (4211)   (5221)    (651)
                       (41111)  (5311)    (6222)
                                (52111)   (6321)
                                (511111)  (6411)
                                          (62211)
                                          (63111)
                                          (621111)
                                          (6111111)
Also the number of integer partitions of n of length n/2. The Heinz numbers of these partitions are given by A340387. For example, the a(2) = 1 through a(14) = 15 partitions (empty columns not shown) are:
  (2)  (22)  (222)  (2222)  (22222)  (222222)  (2222222)
       (31)  (321)  (3221)  (32221)  (322221)  (3222221)
             (411)  (3311)  (33211)  (332211)  (3322211)
                    (4211)  (42211)  (333111)  (3332111)
                    (5111)  (43111)  (422211)  (4222211)
                            (52111)  (432111)  (4322111)
                            (61111)  (441111)  (4331111)
                                     (522111)  (4421111)
                                     (531111)  (5222111)
                                     (621111)  (5321111)
                                     (711111)  (5411111)
                                               (6221111)
                                               (6311111)
                                               (7211111)
                                               (8111111)
(End)

			From _Joerg Arndt_, Jun 11 2013: (Start)
There are a(12)=11 symmetric unimodal compositions of 12+2=14 where the maximal part appears twice:
01:  [ 1 1 1 1 1 2 2 1 1 1 1 1 ]
02:  [ 1 1 1 1 3 3 1 1 1 1 ]
03:  [ 1 1 1 4 4 1 1 1 ]
04:  [ 1 1 2 3 3 2 1 1 ]
05:  [ 1 1 5 5 1 1 ]
06:  [ 1 2 4 4 2 1 ]
07:  [ 1 6 6 1 ]
08:  [ 2 2 3 3 2 2 ]
09:  [ 2 5 5 2 ]
10:  [ 3 4 4 3 ]
11:  [ 7 7 ]
There are a(14)=15 symmetric unimodal compositions of 14 where the maximal part appears an even number of times:
01:  [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 2 2 1 1 1 1 1 ]
03:  [ 1 1 1 1 3 3 1 1 1 1 ]
04:  [ 1 1 1 2 2 2 2 1 1 1 ]
05:  [ 1 1 1 4 4 1 1 1 ]
06:  [ 1 1 2 3 3 2 1 1 ]
07:  [ 1 1 5 5 1 1 ]
08:  [ 1 2 2 2 2 2 2 1 ]
09:  [ 1 2 4 4 2 1 ]
10:  [ 1 3 3 3 3 1 ]
11:  [ 1 6 6 1 ]
12:  [ 2 2 3 3 2 2 ]
13:  [ 2 5 5 2 ]
14:  [ 3 4 4 3 ]
15:  [ 7 7 ]
(End)
a(8)=5 because we  have [8], [6,2], [4,4], [4,2,2], and [2,2,2,2]. - _Emeric Deutsch_, Jan 27 2016
From _Gus Wiseman_, May 22 2021: (Start)
The a(0) = 1 through a(12) = 11 partitions into even parts are the following (empty columns shown as dots, A = 10, C = 12). The Heinz numbers of these partitions are given by A066207.
  ()  .  (2)  .  (4)   .  (6)    .  (8)     .  (A)      .  (C)
                 (22)     (42)      (44)       (64)        (66)
                          (222)     (62)       (82)        (84)
                                    (422)      (442)       (A2)
                                    (2222)     (622)       (444)
                                               (4222)      (642)
                                               (22222)     (822)
                                                           (4422)
                                                           (6222)
                                                           (42222)
                                                           (222222)
(End)

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education, Vol. 31, No. 1, pp. 24-28, Winter 1997. MathEduc Database (Zentralblatt MATH, 1997c.01891).
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17. Zentralblatt MATH, Zbl 1071.05501.

Robert Price, Table of n, a(n) for n = 0..2001

Bisection (even part) gives the partition numbers A000041.
Column k=0 of A103919, A264398.
Cf. A036469, A000070.
Cf. A135010, A138121.
Note: A-numbers of ranking sequences are in parentheses below.
The version for odd instead of even parts is A000009 (A066208).
The version for parts divisible by 3 instead of 2 is A035377.
The strict case is A035457.
The Heinz numbers of these partitions are given by A066207.
The ordered version (compositions) is A077957 prepended by (1,0).
This is column k = 2 of A168021.
The multiplicative version (factorizations) is A340785.
A000569 counts graphical partitions (A320922).
A004526 counts partitions of length 2 (A001358).
A025065 counts palindromic partitions (A265640).
A027187 counts partitions with even length/maximum (A028260/A244990).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts partitions of even length and sum (A340784).
A340601 counts partitions of even rank (A340602).
The following count partitions of even length:
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A000041, A000290, A087897, A100484, A110618, A209816, A210249, A233771, A339004, A340385, A340387, A340786, A341447.

ZL:= [S, {C = Cycle(B), S = Set(C), E = Set(B), B = Prod(Z,Z)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=0..69); # Zerinvary Lajos, Mar 26 2008
g := 1/mul(1-x^(2*k), k = 1 .. 100): gser := series(g, x = 0, 80): seq(coeff(gser, x, n), n = 0 .. 78); # Emeric Deutsch, Jan 27 2016
# Using the function EULER from Transforms (see link at the bottom of the page).
[1,op(EULER([0,1,seq(irem(n,2),n=0..66)]))]; # Peter Luschny, Aug 19 2020
# next Maple program:
a:= n-> `if`(n::odd, 0, combinat[numbpart](n/2)):
seq(a(n), n=0..84);  # Alois P. Heinz, Jun 22 2021

nmax = 50; s = Range[2, nmax, 2];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 05 2020 *)

from sympy import npartitions
def A035363(n): return 0 if n&1 else npartitions(n>>1) # Chai Wah Wu, Sep 23 2023

G.f.: Product_{k even} 1/(1 - x^k).
Convolution with the number of partitions into distinct parts (A000009, which is also number of partitions into odd parts) gives the number of partitions (A000041). - Franklin T. Adams-Watters, Jan 06 2006
If n is even then a(n)=A000041(n/2) otherwise a(n)=0. - Omar E. Pol, Nov 20 2009
G.f.: 1 + x^2*(1 - G(0))/(1-x^2) where G(k) =  1 - 1/(1-x^(2*k+2))/(1-x^2/(x^2-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 23 2013
a(n) = A096441(n) - A000009(n), n >= 1. - Omar E. Pol, Aug 16 2013
G.f.: exp(Sum_{k>=1} x^(2*k)/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Aug 13 2018

cp's OEIS Frontend

A220968 Positions in A030229 where odd terms occur.

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A220969 Positions in A030229 where even terms occur.

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A151797 Numbers which belong to just one of the two sequences A030229, A161918.

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A074905 a(n) = number of terms in A030229 between prime(n) and prime(n+1).

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A080324 Union of even squarefree numbers (A039956) and squarefree numbers for which the number of prime factors is even (A030229).

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A075734 Fibonacci numbers F(k) when k is a product of an even number of distinct primes A030229 (mu(k)=1).

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A286971 Number of ways to write n as a sum of two numbers, one of which is the product of an even number of distinct primes (including 1) (A030229) and another is the product of an odd number of distinct primes (A030059).

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A005117 Squarefree numbers: numbers that are not divisible by a square greater than 1.

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