A000837 -id:A000837 - cp's OEIS frontend

A130162 Triangle read by rows: A051731 * A000837 as a diagonalized matrix.

1, 1, 1, 1, 0, 2, 1, 1, 0, 3, 1, 0, 0, 0, 6, 1, 1, 2, 0, 0, 7, 1, 0, 0, 0, 0, 0, 14, 1, 1, 0, 3, 0, 0, 0, 17, 1, 0, 2, 0, 0, 0, 0, 0, 27, 1, 1, 0, 0, 6, 0, 0, 0, 0, 34, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 55, 1, 1, 2, 3, 0, 7, 0, 0, 0, 0, 0, 63

Offset: 1

Gary W. Adamson, May 13 2007

Right border = A000837 (offset 1).

Row sums = partition numbers A000041 starting (1, 2, 3, 5, 7, ...).

			First few rows of the triangle:
  1;
  1,  1;
  1,  0,  2;
  1,  1,  0,  3;
  1,  0,  0,  0,  6;
  1,  1,  2,  0,  0,  7;
  1,  0,  0,  0,  0,  0, 14;
  1,  1,  0,  3,  0,  0,  0, 17;
  ...

Cf. A000041, A000837, A051731.

rows = 12; A000837[n_] := Sum[ MoebiusMu[n/d]*PartitionsP[d], {d, Divisors[n]}]; A000837diag = DiagonalMatrix[Array[A000837, rows]]; A051731 = Table[ If[Mod[n, k] == 0, 1, 0], {n, 1, rows}, {k, 1, rows}]; A130162 = A051731.A000837diag; Table[ A130162[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 03 2013 *)

A051731 * A000837 (starting at offset 1) as a diagonalized matrix M, where M = T(n,k) = A000837(n) * 0^(n-k), 1<=k<=n; i.e., (1; 0,1; 0,0,2; 0,0,0,3; 0,0,0,0,6;...).

A051731 = inverse Moebius transform.

More terms from Jean-François Alcover, Oct 03 2013

Offset changed to 1 by Georg Fischer, Jun 27 2023

A060034 Number of partitions of n such that all parts are neither relatively prime (cf. A000837) nor are they periodic with each part occurring the same number of times (cf. A024994).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 0, 4, 0, 9, 3, 12, 0, 22, 0, 28, 9, 43, 0, 63, 3, 82, 19, 107, 0, 170, 0, 189, 43, 258, 12, 372, 0, 435, 82, 557, 0, 808, 0, 900, 162, 1150, 0, 1599, 9, 1836, 258, 2252, 0, 3111, 46, 3476, 435, 4308, 0, 5827, 0, 6501, 727, 7917, 85

Offset: 1

Alford Arnold, Mar 16 2001

			a(15) = 3 because partitions 6+3+3+3, 6+6+3 and 9+3+3 satisfy the description and A000041(15) - (A000837(15) + A024994(15)) = 176 - (167 + 6) = 3.

A000041, A000837, A024994 and A055892.

A000837[n_] := Sum[ MoebiusMu[n/d]*PartitionsP[d], {d, Divisors[n]}]; A024994[n_] := Sum[ PartitionsQ[k], {k, Divisors[n] // Most}]; a[n_] := PartitionsP[n] - (A000837[n] + A024994[n]); Table[a[n], {n, 1, 65}] (* Jean-François Alcover, Oct 03 2013 *)

a(n) = A000041(n) - ( A000837(n) + A024994(n))

More terms from Naohiro Nomoto, Mar 01 2002

A047966 a(n) = Sum_{ d divides n } q(d), where q(d) = A000009 = number of partitions of d into distinct parts.

Original entry on oeis.org

1, 2, 3, 4, 4, 8, 6, 10, 11, 15, 13, 25, 19, 29, 33, 42, 39, 62, 55, 81, 84, 103, 105, 153, 146, 185, 203, 253, 257, 344, 341, 432, 463, 552, 594, 747, 761, 920, 1003, 1200, 1261, 1537, 1611, 1921, 2089, 2410, 2591, 3095, 3270, 3815, 4138, 4769, 5121, 5972, 6394, 7367, 7974, 9066, 9793, 11305, 12077, 13736, 14940

Offset: 1

N. J. A. Sloane

nonn

Number of partitions of n such that every part occurs with the same multiplicity. - Vladeta Jovovic, Oct 22 2004
Christopher and Christober call such partitions uniform. - Gus Wiseman, Apr 16 2018
Equals inverse Mobius transform (A051731) * A000009, where the latter begins (1, 1, 2, 2, 3, 4, 5, 6, 8, ...). -  Gary W. Adamson, Jun 08 2009

			The a(6) = 8 uniform partitions are (6), (51), (42), (33), (321), (222), (2211), (111111). - _Gus Wiseman_, Apr 16 2018

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp.1-12.

Cf. A000009, A000837, A024994, A047968, A063834, A279788, A289501, A300383, A301462, A302698.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
     `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> add(b(d), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 11 2016

b[n_] := b[n] = If[n==0, 1, Sum[DivisorSum[j, If[OddQ[#], #, 0]&]*b[n-j], {j, 1, n}]/n]; a[n_] := DivisorSum[n, b]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 06 2016 after Alois P. Heinz *)
Table[DivisorSum[n,PartitionsQ],{n,20}] (* Gus Wiseman, Apr 16 2018 *)

N = 66; q='q+O('q^N);
D(q)=eta(q^2)/eta(q); \\ A000009
Vec( sum(e=1,N,D(q^e)-1) ) \\ Joerg Arndt, Mar 27 2014

G.f.: Sum_{k>0} (-1+Product_{i>0} (1+z^(k*i))). - Vladeta Jovovic, Jun 22 2003
G.f.: Sum_{k>=1} q(k)*x^k/(1 - x^k), where q() = A000009. - Ilya Gutkovskiy, Jun 20 2018
a(n) ~ exp(Pi*sqrt(n/3)) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Aug 27 2018

A000740 Number of 2n-bead balanced binary necklaces of fundamental period 2n, equivalent to reversed complement; also Dirichlet convolution of b_n=2^(n-1) with mu(n); also number of components of Mandelbrot set corresponding to Julia sets with an attractive n-cycle.

Original entry on oeis.org

1, 1, 3, 6, 15, 27, 63, 120, 252, 495, 1023, 2010, 4095, 8127, 16365, 32640, 65535, 130788, 262143, 523770, 1048509, 2096127, 4194303, 8386440, 16777200, 33550335, 67108608, 134209530, 268435455, 536854005, 1073741823, 2147450880

Offset: 1

N. J. A. Sloane

Also number of compositions of n into relatively prime parts (that is, the gcd of all the parts is 1). Also number of subsets of {1,2,..,n} containing n and consisting of relatively prime numbers. - Vladeta Jovovic, Aug 13 2003
Also number of perfect parity patterns that have exactly n columns (see A118141). - Don Knuth, May 11 2006
a(n) is odd if and only if n is squarefree (Tim Keller). - Emeric Deutsch, Apr 27 2007
a(n) is a multiple of 3 for all n>=3 (see Problem 11161 link). - Emeric Deutsch, Aug 13 2008
Row sums of triangle A143424. - Gary W. Adamson, Aug 14 2008
a(n) is the number of monic irreducible polynomials with nonzero constant coefficient in GF(2)[x] of degree n. - Michel Marcus, Oct 30 2016
a(n) is the number of aperiodic compositions of n, the number of compositions of n with relatively prime parts, and the number of compositions of n with relatively prime run-lengths. - Gus Wiseman, Dec 21 2017

			For n=4, there are 6 compositions of n into coprime parts: <3,1>, <2,1,1>, <1,3>, <1,2,1>, <1,1,2>, and <1,1,1,1>.
From _Gus Wiseman_, Dec 19 2017: (Start)
The a(6) = 27 aperiodic compositions are:
  (11112), (11121), (11211), (12111), (21111),
  (1113), (1122), (1131), (1221), (1311), (2112), (2211), (3111),
  (114), (123), (132), (141), (213), (231), (312), (321), (411),
  (15), (24), (42), (51),
  (6).
The a(6) = 27 compositions into relatively prime parts are:
  (111111),
  (11112), (11121), (11211), (12111), (21111),
  (1113), (1122), (1131), (1212), (1221), (1311), (2112), (2121), (2211), (3111),
  (114), (123), (132), (141), (213), (231), (312), (321), (411),
  (15), (51).
The a(6) = 27 compositions with relatively prime run-lengths are:
  (11112), (11121), (11211), (12111), (21111),
  (1113), (1131), (1212), (1221), (1311), (2112), (2121), (3111),
  (114), (123), (132), (141), (213), (231), (312), (321), (411),
  (15), (24), (42), (51),
  (6).
(End)

H. O. Peitgen and P. H. Richter, The Beauty of Fractals, Springer-Verlag; contribution by A. Douady, p. 165.
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).

Seiichi Manyama, Table of n, a(n) for n = 1..3322 (terms 1..300 from T. D. Noe)
Hunki Baek, Sejeong Bang, Dongseok Kim, and Jaeun Lee, A bijection between aperiodic palindromes and connected circulant graphs, arXiv:1412.2426 [math.CO], 2014. See Table 2.
Donald Knuth, Robin Chapman and Reiner Martin, Problem 11243, Perfect Parity Patterns, Am. Math. Monthly 115 (7) (2008) p 668, function c(n).
Emeric Deutsch and Lafayette College Problem Group, Problem 11161: Compositions without Common Factors, American Mathematical Monthly, vol. 114, No. 4, 2007, p. 363.
H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2(4) (1964), 241-260.
J. E. Iglesias, A formula for the number of closest packings of equal spheres having a given repeat period, Z. Krist. 155 (1981) 121-127, Table 2.
Wolfdieter Lang, Cantor's List of Real Algebraic Numbers of Heights 1 to 7, arXiv:2307.10645 [math.NT], 2023.
Nicolae Mihalache and Francois Vigneron, Factorization of the quadratic Misiurewicz-Thurston polynomials, arXiv:2506.17662 [math.DS], 2025. See p. 8.
Robert Munafo, Enumeration of Period-N Mu-Atoms
Jeffrey Shallit and N. J. A. Sloane, Correspondence 1974-1975
François Vigneron and Nicolae Mihalache, How to split a tera-polynomial, arXiv:2402.06083 [math.NA], 2024.
Index entries for sequences related to Lyndon words

Cf. A000837, A003239, A008683, A008965, A022553, A034738, A035928, A038199, A051168, A054525, A056267, A059966, A143424, A167606, A178472, A216954, A228369, A294859, A296302.
Equals A027375/2.
See A056278 for a variant.
First differences of A085945.
Column k=2 of A143325.
Row sums of A101391.

with(numtheory): a[1]:=1: a[2]:=1: for n from 3 to 32 do div:=divisors(n): a[n]:=2^(n-1)-sum(a[n/div[j]],j=2..tau(n)) od: seq(a[n],n=1..32); # Emeric Deutsch, Apr 27 2007
with(numtheory); A000740:=n-> add(mobius(n/d)*2^(d-1), d in divisors(n)); # N. J. A. Sloane, Oct 18 2012

a[n_] := Sum[ MoebiusMu[n/d]*2^(d - 1), {d, Divisors[n]}]; Table[a[n], {n, 1, 32}] (* Jean-François Alcover, Feb 03 2012, after PARI *)

a(n) = sumdiv(n,d,moebius(n/d)*2^(d-1))

from sympy import mobius, divisors
def a(n): return sum([mobius(n // d) * 2**(d - 1) for d in divisors(n)])
[a(n) for n in range(1, 101)]  # Indranil Ghosh, Jun 28 2017

a(n) = Sum_{d|n} mu(n/d)*2^(d-1), Mobius transform of A011782. Furthermore, Sum_{d|n} a(d) = 2^(n-1).
a(n) = A027375(n)/2 = A038199(n)/2.
a(n) = Sum_{k=0..n} A051168(n,k)*k. - Max Alekseyev, Apr 09 2013
Recurrence relation: a(n) = 2^(n-1) - Sum_{d|n,d>1} a(n/d). (Lafayette College Problem Group; see the Maple program and Iglesias eq (6)). - Emeric Deutsch, Apr 27 2007
G.f.: Sum_{k>=1} mu(k)*x^k/(1 - 2*x^k). - Ilya Gutkovskiy, Oct 24 2018
G.f. satisfies Sum_{n>=1} A( (x/(1 + 2*x))^n ) = x. - Paul D. Hanna, Apr 02 2025

Connection with Mandelbrot set discovered by Warren D. Smith and proved by Robert Munafo, Feb 06 2000

Ambiguous term a(0) removed by Max Alekseyev, Jan 02 2012

A072774 Powers of squarefree numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97

Offset: 1

Reinhard Zumkeller, Jul 10 2002

nonn

Essentially the same as A062770. - R. J. Mathar, Sep 25 2008
Numbers m such that in canonical prime factorization all prime exponents are identical: A124010(m,k) = A124010(m,1) for k = 2..A000005(m). - Reinhard Zumkeller, Apr 06 2014
Heinz numbers of uniform partitions. An integer partition is uniform if all parts appear with the same multiplicity. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Apr 16 2018

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Complement of A059404.
Cf. A072775, A072776, A072777 (subsequence), A005117, A072778, A124010, A329332 (tabular arrangement), A384667 (characteristic function).
A subsequence of A242414.
Cf. A000005, A000009, A000837, A007916, A013661, A047966, A052409, A052410, A072774, A078374, A289023, A289509, A300486, A302491, A302796, A302979.

import Data.Map (empty, findMin, deleteMin, insert)
import qualified Data.Map.Lazy as Map (null)
a072774 n = a072774_list !! (n-1)
(a072774_list, a072775_list, a072776_list) = unzip3 $
   (1, 1, 1) : f (tail a005117_list) empty where
   f vs'@(v:vs) m
    | Map.null m || xx > v = (v, v, 1) :
                             f vs (insert (v^2) (v, 2) m)
    | otherwise = (xx, bx, ex) :
                  f vs' (insert (bx*xx) (bx, ex+1) $ deleteMin m)
    where (xx, (bx, ex)) = findMin m
-- Reinhard Zumkeller, Apr 06 2014

isA := n -> n=1 or is(1 = nops({seq(p[2], p in ifactors(n)[2])})):
select(isA, [seq(1..97)]);  # Peter Luschny, Jun 10 2025

Select[Range[100], Length[Union[FactorInteger[#][[All, 2]]]] == 1 &] (* Geoffrey Critzer, Mar 30 2015 *)

is(n)=ispower(n,,&n); issquarefree(n) \\ Charles R Greathouse IV, Oct 16 2015

from math import isqrt
from sympy import mobius, integer_nthroot
def A072774(n):
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))-1
    def f(x): return n-2+x-sum(g(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length()))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 19 2024

a(n) = A072775(n)^A072776(n).
Sum_{n>=1} 1/a(n)^s = 1 + Sum_{k>=1} (zeta(k*s)/zeta(2*k*s)-1) for s > 1. - Amiram Eldar, Mar 20 2025
a(n)/n ~ Pi^2/6 (A013661). - Friedjof Tellkamp, Jun 09 2025

A051424 Number of partitions of n into pairwise relatively prime parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 10, 12, 15, 18, 23, 27, 33, 38, 43, 51, 60, 70, 81, 92, 102, 116, 134, 153, 171, 191, 211, 236, 266, 301, 335, 367, 399, 442, 485, 542, 598, 649, 704, 771, 849, 936, 1023, 1103, 1185, 1282, 1407, 1535, 1662, 1790, 1917, 2063, 2245, 2436

Offset: 0

Hugo van der Sanden

nonn

			a(4) = 4 since all partitions of 4 consist of relatively prime numbers except 2+2.
The a(6) = 7 partitions with pairwise coprime parts: (111111), (21111), (3111), (321), (411), (51), (6). - _Gus Wiseman_, Apr 14 2018

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..750 (terms 0..400 from Alois P. Heinz)
Eric Schmutz, Partitions whose parts are pairwise relatively prime, Discrete Math. 81(1) (1990), 87-89.
Temba Shonhiwa, Compositions with pairwise relatively prime summands within a restricted setting, Fibonacci Quart. 44(4) (2006), 316-323.

Number of partitions of n into relatively prime parts = A000837.
Row sums of A282749.
Cf. A000837, A007359, A007360, A051424, A289509, A298748, A302569, A302696, A302797.

a051424 = length . filter f . partitions where
   f [] = True
   f (p:ps) = (all (== 1) $ map (gcd p) ps) && f ps
   partitions n = ps 1 n where
     ps x 0 = [[]]
     ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
-- Reinhard Zumkeller, Dec 16 2013

with(numtheory):
b:= proc(n, i, s) option remember; local f;
      if n=0 or i=1 then 1
    elif i<2 then 0
    else f:= factorset(i);
         b(n, i-1, select(x->is(xis(x b(n, n, {}):
seq(a(n), n=0..80);  # Alois P. Heinz, Mar 14 2012

b[n_, i_, s_] := b[n, i, s] = Module[{f}, If[n == 0 || i == 1, 1, If[i < 2, 0, f = FactorInteger[i][[All, 1]]; b[n, i-1, Select[s, # < i &]] + If[i <= n && f ~Intersection~ s == {}, b[n-i, i-1, Select[s ~Union~ f, # < i &]], 0]]]]; a[n_] := b[n, n, {}]; Table[a[n], {n, 0, 54}] (* Jean-François Alcover, Oct 03 2013, translated from Maple, after Alois P. Heinz *)

log a(n) ~ (2*Pi/sqrt(6)) sqrt(n/log n). - Eric M. Schmidt, Jul 04 2013

Apparently no formula or recurrence is known. - N. J. A. Sloane, Mar 05 2017

More precise definition from Vladeta Jovovic, Dec 11 2004

A302696 Numbers whose prime indices (with repetition) are pairwise coprime. Nonprime Heinz numbers of integer partitions with pairwise coprime parts.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 14, 15, 16, 20, 22, 24, 26, 28, 30, 32, 33, 34, 35, 38, 40, 44, 46, 48, 51, 52, 55, 56, 58, 60, 62, 64, 66, 68, 69, 70, 74, 76, 77, 80, 82, 85, 86, 88, 92, 93, 94, 95, 96, 102, 104, 106, 110, 112, 116, 118, 119, 120, 122, 123, 124, 128, 132

Offset: 1

Gus Wiseman, Apr 11 2018

nonn

A prime index of n is a number m such that prime(m) divides n. Two or more numbers are coprime if no pair has a common divisor other than 1. A single number is not considered coprime unless it is equal to 1.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Number 36 = prime(1)*prime(1)*prime(2)*prime(2) is not included in the sequence, because the pair of prime indices {2,2} is not coprime. - Gus Wiseman, Dec 06 2021

			Sequence of integer partitions with pairwise coprime parts begins: (), (1), (11), (21), (111), (31), (211), (41), (32), (1111), (311), (51), (2111), (61), (411), (321).
Missing from this list are: (2), (3), (4), (22), (5), (6), (7), (221), (8), (42), (9), (33), (222).

Cf. A000837, A000961, A001222, A005117, A007359, A051424, A275024, A289508, A289509, A298748, A302568, A302569, A302697, A302698, A327512, A327513.

filter:= proc(n) local F;
   F:= ifactors(n)[2];
   if nops(F)=1 then if F[1][1] = 2 then return true else return false fi fi;
   if ormap(t -> t[2]>1 and t[1] <> 2, F) then return false fi;
   F:= map(t -> numtheory:-pi(t[1]), F);
   ilcm(op(F))=convert(F,`*`)
end proc:
select(filter, [$1..200]); # Robert Israel, Sep 10 2020

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[200],Or[#===1,CoprimeQ@@primeMS[#]]&]

isA302696(n) = if(isprimepower(n),!(n%2), if(!issquarefree(n>>valuation(n,2)), 0, my(pis=apply(primepi,factor(n)[,1])); (lcm(pis)==factorback(pis)))); \\ Antti Karttunen, Dec 06 2021

Clarification (with repetition) added to the definition by Antti Karttunen, Dec 06 2021

A078374 Number of partitions of n into distinct and relatively prime parts.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 6, 7, 11, 10, 17, 17, 23, 26, 37, 36, 53, 53, 70, 77, 103, 103, 139, 147, 184, 199, 255, 260, 339, 358, 435, 474, 578, 611, 759, 810, 963, 1045, 1259, 1331, 1609, 1726, 2015, 2200, 2589, 2762, 3259, 3509, 4058, 4416, 5119, 5488, 6364, 6882

Offset: 1

Vladeta Jovovic, Dec 24 2002

nonn

The Heinz numbers of these partitions are given by A302796, which is the intersection of A005117 (strict) and A289509 (relatively prime). - Gus Wiseman, Oct 18 2020

			From _Gus Wiseman_, Oct 18 2020: (Start)
The a(1) = 1 through a(13) = 17 partitions (empty column indicated by dot, A = 10, B = 11, C = 12):
  1   .  21   31   32   51    43    53    54    73     65     75     76
                   41   321   52    71    72    91     74     B1     85
                              61    431   81    532    83     543    94
                              421   521   432   541    92     651    A3
                                          531   631    A1     732    B2
                                          621   721    542    741    C1
                                                4321   632    831    643
                                                       641    921    652
                                                       731    5421   742
                                                       821    6321   751
                                                       5321          832
                                                                     841
                                                                     931
                                                                     A21
                                                                     5431
                                                                     6421
                                                                     7321
(End)

Seiichi Manyama, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms

Cf. A047966.
A000837 is the not necessarily strict version.
A302796 gives the Heinz numbers of these partitions.
A305713 is the pairwise coprime instead of relatively prime version.
A332004 is the ordered version.
A337452 is the case without 1's.
A000009 counts strict partitions.
A000740 counts relatively prime compositions.
Cf. A007359, A101268, A289508, A289509, A291166, A298748, A337451, A337485, A337451, A337561, A337563.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&GCD@@#==1&]],{n,15}] (* Gus Wiseman, Oct 18 2020 *)

Moebius transform of A000009.

G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = Product_{n>=1} (1 + x^n). - Ilya Gutkovskiy, Apr 26 2017

A007359 Number of partitions of n into pairwise coprime parts that are >= 2.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 3, 2, 3, 3, 5, 4, 6, 5, 5, 8, 9, 10, 11, 11, 10, 14, 18, 19, 18, 20, 20, 25, 30, 35, 34, 32, 32, 43, 43, 57, 56, 51, 55, 67, 78, 87, 87, 80, 82, 97, 125, 128, 127, 128, 127, 146, 182, 191, 185, 184, 193, 213, 263, 290, 279, 258, 271, 312, 354, 404, 402

Offset: 0

N. J. A. Sloane and Mira Bernstein, following a suggestion from Marc LeBrun, Apr 28 1994

This sequence is of interest for group theory. The partitions counted by a(n) correspond to conjugacy classes of optimal order of the symmetric group of n elements: they have no fixed point, their order is the direct product of their cycle lengths and they are not contained in a subgroup of Sym_p for p < n. A123131 gives the maximum order (LCM) reachable by these partitions.

			The a(17) = 9 strict partitions into pairwise coprime parts that are greater than 1 are (17), (15,2), (14,3), (13,4), (12,5), (11,6), (10,7), (9,8), (7,5,3,2). - _Gus Wiseman_, Apr 14 2018

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

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..750 (terms 0..400 from Alois P. Heinz)
M. LeBrun & D. Hoey, Emails

Cf. A000837, A007359, A007360, A051424, A101268, A123131, A184956, A187718, A289508, A289509, A298748, A302569, A302696, A302698, A302797.

with(numtheory):
b:= proc(n, i, s) option remember; local f;
      if n=0 then 1
    elif i<2 then 0
    else f:= factorset(i);
         b(n, i-1, select(x-> is(x is(x b(n, n, {}):
seq(a(n), n=0..80);  # Alois P. Heinz, Mar 14 2012

b[n_, i_, s_] := b[n, i, s] = Module[{f}, If[n == 0 || i == 1, 1, If[i<2, 0, f = FactorInteger[i][[All, 1]]; b[n, i-1, Select[s, #Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&(Length[#]===1||CoprimeQ@@#)&]],{n,20}] (* Gus Wiseman, Apr 14 2018 *)

a(n) = A051424(n) - A051424(n-1). - Vladeta Jovovic, Dec 11 2004

More precise definition from Vladeta Jovovic, Dec 11 2004

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 13 2005

A018783 Number of partitions of n into parts having a common factor.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 4, 1, 5, 3, 8, 1, 14, 1, 16, 9, 22, 1, 38, 1, 45, 17, 57, 1, 94, 7, 102, 30, 138, 1, 218, 1, 231, 58, 298, 21, 451, 1, 491, 103, 644, 1, 919, 1, 1005, 203, 1256, 1, 1784, 15, 1993, 299, 2439, 1, 3365, 62, 3735, 492, 4566, 1, 6252, 1, 6843, 819, 8349, 107, 11096

Offset: 0

David W. Wilson

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000
L. Naughton, G. Pfeiffer, Integer Sequences Realized by the Subgroup Pattern of the Symmetric Group, J. Int. Seq. 16 (2013) #13.5.8

Cf. A000041, A000837, A083710.

with(numtheory): with(combinat):
a:= n-> `if`(n=0, 0,
         numbpart(n) -add(mobius(n/d)*numbpart(d), d=divisors(n))):
seq(a(n), n=0..100); # Alois P. Heinz, Nov 29 2011

A000837[n_] := Sum[ MoebiusMu[n/d]*PartitionsP[d], {d, Divisors[n]}]; a[0] = 0; a[n_] := PartitionsP[n] - A000837[n]; Table[a[n], {n, 0, 66}] (* Jean-François Alcover, Oct 03 2013, after Vladeta Jovovic *)

a(n) = - sumdiv(n, d, (dMichel Marcus, Oct 07 2017

a(n) = -Sum_{d|n, dA000041(d) = A000041(n) - A000837(n). - Vladeta Jovovic, Jun 17 2003

cp's OEIS Frontend

A130162 Triangle read by rows: A051731 * A000837 as a diagonalized matrix.

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A060034 Number of partitions of n such that all parts are neither relatively prime (cf. A000837) nor are they periodic with each part occurring the same number of times (cf. A024994).

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A047966 a(n) = Sum_{ d divides n } q(d), where q(d) = A000009 = number of partitions of d into distinct parts.

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A000740 Number of 2n-bead balanced binary necklaces of fundamental period 2n, equivalent to reversed complement; also Dirichlet convolution of b_n=2^(n-1) with mu(n); also number of components of Mandelbrot set corresponding to Julia sets with an attractive n-cycle.

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A072774 Powers of squarefree numbers.

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A051424 Number of partitions of n into pairwise relatively prime parts.

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A302696 Numbers whose prime indices (with repetition) are pairwise coprime. Nonprime Heinz numbers of integer partitions with pairwise coprime parts.

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