A289508 -id:A289508 - cp's OEIS frontend

A034729 a(n) = Sum_{ k, k|n } 2^(k-1).

1, 3, 5, 11, 17, 39, 65, 139, 261, 531, 1025, 2095, 4097, 8259, 16405, 32907, 65537, 131367, 262145, 524827, 1048645, 2098179, 4194305, 8390831, 16777233, 33558531, 67109125, 134225995, 268435457, 536887863, 1073741825, 2147516555, 4294968325, 8590000131

Offset: 1

Erich Friedman

nonn

Dirichlet convolution of b_n=1 with c_n = 2^(n-1).
Equals row sums of triangle A143425, & inverse Möbius transform (A051731) of [1, 2, 4, 8, ...]. - Gary W. Adamson, Aug 14 2008
Number of constant multiset partitions of normal multisets of size n, where a multiset is normal if it spans an initial interval of positive integers. - Gus Wiseman, Sep 16 2018

			From _Gus Wiseman_, Sep 16 2018: (Start)
The a(4) = 11 constant multiset partitions:
  (1)(1)(1)(1)
    (11)(11)
    (12)(12)
     (1111)
     (1222)
     (1122)
     (1112)
     (1233)
     (1223)
     (1123)
     (1234)
(End)

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

Cf. A000005, A002033, A003238, A008480, A018783, A047968, A051731.
Cf. A052409, A055895, A078392, A143425, A215366, A245282, A248906.
Cf. A289508.
Sums of the form Sum_{d|n} q^(d-1): this sequence (q=2), A034730 (q=3), A113999 (q=10), A339684 (q=4), A339685 (q=5), A339686 (q=6), A339687 (q=7), A339688 (q=8), A339689 (q=9).

A034729:= func< n | (&+[2^(d-1): d in Divisors(n)]) >;
[A034729(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024

seq(add(2^(k-1),k=numtheory:-divisors(n)), n = 1 .. 100); # Robert Israel, Aug 22 2014

Rest[CoefficientList[Series[Sum[x^k/(1-2*x^k),{k,1,30}],{x,0,30}],x]] (* Vaclav Kotesovec, Sep 08 2014 *)

A034729(n) = sumdiv(n,k,2^(k-1)) \\ Michael B. Porter, Mar 11 2010

{a(n)=polcoeff(sum(m=1,n,2^(m-1)*x^m/(1-x^m +x*O(x^n))),n)}
for(n=1,40,print1(a(n),", ")) \\ Paul D. Hanna, Aug 21 2014

{a(n)=local(A=x+x^2);A=sum(m=1,n,x^m*sumdiv(m,d,1/(1 - x^(m/d) +x*O(x^n))^d) );polcoeff(A,n)}
for(n=1,40,print1(a(n),", ")) \\ Paul D. Hanna, Aug 21 2014

from sympy import divisors
def A034729(n): return sum(1<<(d-1) for d in divisors(n,generator=True)) # Chai Wah Wu, Jul 15 2022

def A034729(n): return sum(2^(k-1) for k in (1..n) if (k).divides(n))
[A034729(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

G.f.: Sum_{n>0} x^n/(1-2*x^n). - Vladeta Jovovic, Nov 14 2002
a(n) = 1/2 * A055895(n). - Joerg Arndt, Aug 14 2012
G.f.: Sum_{n>=1} 2^(n-1) * x^n / (1 - x^n). - Paul D. Hanna, Aug 21 2014
G.f.: Sum_{n>=1} x^n * Sum_{d|n} 1/(1 - x^d)^(n/d). - Paul D. Hanna, Aug 21 2014
a(n) ~ 2^(n-1). - Vaclav Kotesovec, Sep 09 2014
a(n) = Sum_{k in row n of A215366} A008480(k) * A000005(A289508(k)). - Gus Wiseman, Sep 16 2018
a(n) = Sum_{c is a composition of n} A000005(gcd(c)). - Gus Wiseman, Sep 16 2018

A291166 Connected Haar graph numbers.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Eric W. Weisstein, Aug 19 2017

nonn

Complement of A291165.

These appear to be numbers whose positions of 1's in their reversed binary expansion are relatively prime. If so, this sequence lists all positions of 1's in A326674. Numbers whose positions of 1's in their reversed binary expansion are pairwise coprime (as opposed to relatively prime) are A326675. - Gus Wiseman, Jul 19 2019

Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Haar Graph

Cf. A000120, A029931, A048793, A059519, A070939, A289508, A291165, A295235, A319826, A326031, A326669, A326672, A326673.

A302698 Number of integer partitions of n into relatively prime parts that are all greater than 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 3, 2, 5, 4, 13, 7, 23, 18, 32, 33, 65, 50, 104, 92, 148, 153, 252, 226, 376, 376, 544, 570, 846, 821, 1237, 1276, 1736, 1869, 2552, 2643, 3659, 3887, 5067, 5509, 7244, 7672, 10086, 10909, 13756, 15168, 19195, 20735, 26237, 28708, 35418, 39207

Offset: 1

Gus Wiseman, Apr 11 2018

nonn

Two or more numbers are relatively prime if they have no common divisor other than 1. A single number is not considered relatively prime unless it is equal to 1 (which is impossible in this case).

The Heinz numbers of these partitions are given by A302697.

			The a(5) = 1 through a(12) = 7 partitions (empty column indicated by dot):
  (32)  .  (43)   (53)   (54)    (73)    (65)     (75)
           (52)   (332)  (72)    (433)   (74)     (543)
           (322)         (432)   (532)   (83)     (552)
                         (522)   (3322)  (92)     (732)
                         (3222)          (443)    (4332)
                                         (533)    (5322)
                                         (542)    (33222)
                                         (632)
                                         (722)
                                         (3332)
                                         (4322)
                                         (5222)
                                         (32222)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

A000837 is the version allowing 1's.
A002865 does not require relative primality.
A302697 gives the Heinz numbers of these partitions.
A337450 is the ordered version.
A337451 is the ordered strict version.
A337452 is the strict version.
A337485 is the pairwise coprime instead of relatively prime version.
A000740 counts relatively prime compositions.
A078374 counts relatively prime strict partitions.
A212804 counts compositions with no 1's.
A291166 appears to rank relatively prime compositions.
A332004 counts strict relatively prime compositions.
A337561 counts pairwise coprime strict compositions.
A338332 is the case of length 3, with strict case A338333.
Cf. A007359, A018783, A051424, A101268, A289508, A289509, A302568, A337563, A337984, A338468.

b:= proc(n, i, g) option remember; `if`(n=0, `if`(g=1, 1, 0),
      `if`(i<2, 0, b(n, i-1, g)+b(n-i, min(n-i, i), igcd(g, i))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=1..60);  # Alois P. Heinz, Apr 12 2018

Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&GCD@@#===1&]],{n,30}]
(* Second program: *)
b[n_, i_, g_] := b[n, i, g] = If[n == 0, If[g == 1, 1, 0], If[i < 2, 0, b[n, i - 1, g] + b[n - i, Min[n - i, i], GCD[g, i]]]];
a[n_] := b[n, n, 0];
Array[a, 60] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

a(n) = A002865(n) - A018783(n).

Extended by Gus Wiseman, Oct 29 2020

A304711 Heinz numbers of integer partitions whose distinct parts are pairwise coprime.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 33, 34, 35, 36, 38, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 58, 60, 62, 64, 66, 68, 69, 70, 72, 74, 75, 76, 77, 80, 82, 85, 86, 88, 90, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 106, 108, 110

Offset: 1

Gus Wiseman, May 17 2018

nonn

Two parts are coprime if they have no common divisor greater than 1. For partitions of length 1 note that (1) is coprime but (x) is not coprime for x > 1.

First differs from A289509 at a(24) = 44, A289509(24) = 42.

			Sequence of all partitions whose distinct parts are pairwise coprime begins (1), (11), (21), (111), (31), (211), (41), (32), (1111), (221), (311), (51), (2111), (61), (411), (321), (11111), (52), (71), (43), (2211), (81), (3111).

Cf. A000837, A007359, A018783, A051424, A056239, A078374, A101268, A289508, A289509, A298748, A300486, A302569, A302696, A302698, A302796, A302797, A304709.

Select[Range[200],CoprimeQ@@PrimePi/@FactorInteger[#][[All,1]]&]

A355737 Number of ways to choose a sequence of divisors, one of each prime index of n (with multiplicity), such that the result has no common divisor > 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 2, 1, 3, 4, 1, 1, 4, 1, 2, 4, 2, 1, 2, 3, 4, 7, 3, 1, 4, 1, 1, 4, 2, 6, 4, 1, 4, 6, 2, 1, 6, 1, 2, 8, 3, 1, 2, 5, 4, 4, 4, 1, 8, 4, 3, 5, 4, 1, 4, 1, 2, 10, 1, 6, 4, 1, 2, 6, 6, 1, 4, 1, 6, 8, 4, 6, 8, 1, 2, 15, 2, 1, 6, 4, 4

Offset: 1

Gus Wiseman, Jul 17 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(2) = 1 through a(18) = 4 choices:
  1  1  11  1  11  1  111  11  11  1  111  1  11  11  1111  1  111
               12          12  13     112     12  13           112
                           21                 14  21           121
                                                  23           122

Wikipedia, Coprime integers.

Dominated by A355731, firsts A355732, primes A355741, prime-powers A355742.
For weakly increasing instead of coprime we have A355735, primes A355745.
Positions of first appearances are A355738.
For strict instead of coprime we have A355739, zeros A355740.
A000005 counts divisors.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A120383 lists numbers divisible by all of their prime indices.
A289508 gives GCD of prime indices.
A289509 ranks relatively prime partitions, odd A302697, squarefree A302796.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A007359, A051424, A076610, A289507, A296150, A302696, A302698, A355733, A355744, A355748.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Tuples[Divisors/@primeMS[n]],GCD@@#==1&]],{n,100}]

A348551 Heinz numbers of integer partitions whose mean is not an integer.

Original entry on oeis.org

1, 6, 12, 14, 15, 18, 20, 24, 26, 33, 35, 36, 38, 40, 42, 44, 45, 48, 50, 51, 52, 54, 56, 58, 60, 63, 65, 66, 69, 70, 72, 74, 75, 76, 77, 80, 86, 92, 93, 95, 96, 102, 104, 106, 108, 112, 114, 117, 119, 120, 122, 123, 124, 126, 130, 132, 135, 136, 140, 141, 142

Offset: 1

Gus Wiseman, Nov 14 2021

nonn

Equivalently, partitions whose length does not divide their sum.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms and their prime indices begin:
   1: {}
   6: {1,2}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  26: {1,6}
  33: {2,5}
  35: {3,4}
  36: {1,1,2,2}
  38: {1,8}
  40: {1,1,1,3}
  42: {1,2,4}
  44: {1,1,5}
  45: {2,2,3}
  48: {1,1,1,1,2}

A version counting nonempty subsets is A000079 - A051293.
A version counting factorizations is A001055 - A326622.
A version counting compositions is A011782 - A271654.
A version for prime factors is A175352, complement A078175.
A version for distinct prime factors A176587, complement A078174.
The complement is A316413, counted by A067538, strict A102627.
The geometric version is the complement of A326623.
The conjugate version is the complement of A326836.
These partitions are counted by A349156.
A000041 counts partitions.
A001222 counts prime factors with multiplicity.
A018818 counts partitions into divisors, ranked by A326841.
A143773 counts partitions into multiples of the length, ranked by A316428.
A236634 counts unbalanced partitions.
A047993 counts balanced partitions, ranked by A106529.
A056239 adds up prime indices, row sums of A112798.
A326567/A326568 gives the mean of prime indices, conjugate A326839/A326840.
A327472 counts partitions not containing their mean, complement A237984.
Cf. A067539, A096199, A098743, A175397, A175761, A289508, A289509, A290103, A326028, A326645, A326837.

q:= n-> (l-> nops(l)=0 or irem(add(i, i=l), nops(l))>0)(map
        (i-> numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..142])[];  # Alois P. Heinz, Nov 19 2021

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!IntegerQ[Mean[primeMS[#]]]&]

A326674 GCD of the set of positions of 1's in the reversed binary expansion of n.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 2, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 17 2019

a(n) is even if and only if n is in A062880. - Robert Israel, Oct 13 2020

			The reversed binary expansion of 40 is (0,0,0,1,0,1), with positions of 1's being {4,6}, so a(40) = GCD(4,6) = 2.

Robert Israel, Table of n, a(n) for n = 1..10000

Positions of 1's are A291166, and non-1's are A291165.
GCDs of prime indices are A289508.
GCDs of strict partitions encoded by FDH numbers are A319826.
Numbers whose binary positions are pairwise coprime are A326675.
Cf. A000120, A051293, A062880, A070939, A326667, A326668, A326669, A326670, A326672, A326673.

f:= proc(n) local B;
  B:= convert(n,base,2);
  igcd(op(select(t -> B[t]=1, [$1..ilog2(n)+1])))
end proc:
map(f, [$1..100]); # Robert Israel, Oct 13 2020

Table[GCD@@Join@@Position[Reverse[IntegerDigits[n,2]],1],{n,100}]

Trivially, a(n) <= log_2(n). - Charles R Greathouse IV, Nov 15 2022

A298748 Heinz numbers of aperiodic (relatively prime multiplicities) integer partitions with relatively prime parts.

Original entry on oeis.org

2, 6, 10, 12, 14, 15, 18, 20, 22, 24, 26, 28, 30, 33, 34, 35, 38, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 58, 60, 62, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 88, 90, 92, 93, 94, 95, 96, 98, 99, 102, 104, 105, 106, 108, 110, 112, 114

Offset: 1

Gus Wiseman, Mar 01 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of partitions begins: (1), (21), (31), (211), (41), (32), (221), (311), (51), (2111), (61), (411), (321), (52), (71), (43), (81), (3111), (421), (511), (322), (91), (21111), (331), (72), (611), (2221), (53), (4111).

Cf. A000740, A000837, A007916, A052410, A064989, A100953, A289508, A289509, A296302, A300272.

Select[Range[100],With[{t=Transpose[FactorInteger[#]]},And[GCD@@PrimePi/@t[[1]]===1,GCD@@t[[2]]===1]]&] (* Gus Wiseman, Apr 14 2018 *)

A318978 Heinz numbers of integer partitions with a common divisor > 1.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 121, 125, 127, 129, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167, 169

Offset: 1

Gus Wiseman, Sep 06 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Is this the same as A305078 without the leading 2? - R. J. Mathar, Sep 08 2018

			The sequence of all integer partitions with a common divisor begins: (2), (3), (4), (2,2), (5), (6), (7), (8), (4,2), (9), (3,3), (2,2,2), (10), (11), (12), (6,2), (13), (14), (15), (4,4), (16), (8,2), (17), (18), (4,2,2), (6,3), (19), (20), (21), (22), (2,2,2,2), (23), (10,2), (24), (6,4), (25).

Cf. A000837, A018783, A056239, A289508, A289509, A296150, A298748.

Select[Range[100],GCD@@PrimePi/@If[#==1,{},FactorInteger[#]][[All,1]]>1&]

A304709 Number of integer partitions of n whose distinct parts are pairwise coprime.

Original entry on oeis.org

1, 1, 2, 3, 6, 7, 13, 16, 23, 29, 42, 49, 69, 83, 102, 126, 161, 191, 239, 281, 336, 402, 484, 566, 672, 787, 919, 1067, 1251, 1449, 1684, 1934, 2223, 2554, 2920, 3341, 3821, 4344, 4928, 5586, 6334, 7163, 8091, 9100, 10228, 11492, 12902, 14449, 16167, 18058

Offset: 1

Gus Wiseman, May 17 2018

nonn

Two parts are coprime if they have no common divisor greater than 1. For partitions of length 1 note that (1) is coprime but (x) is not coprime for x > 1.

			The a(6) = 7 integer partitions of 6 whose distinct parts are pairwise coprime are (51), (411), (321), (3111), (2211), (21111), (111111).

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000005, A007359, A007360, A018783, A051424, A078374, A101268, A289508, A289509, A302569, A302696, A302698, A302796, A302797, A304711, A304712.

Table[Select[IntegerPartitions[n],CoprimeQ@@Union[#]&]//Length,{n,20}]

lista(nn)={local(Cache=Map());
  my(excl=vector(nn, n, sum(i=1, n-1, if(gcd(i,n)>1, 2^(n-i)))));
  my(c(n, m, b)=
     if(n==0, 1,
        while(m>n || bittest(b,0), m--; b>>=1);
        my(hk=[n, m, b], z);
        if(!mapisdefined(Cache, hk, &z),
          z = if(m, self()(n, m-1, b>>1) + self()(n-m, m, bitor(b, excl[m])), 0);
          mapput(Cache, hk, z)); z));
  my(a(n)=c(n, n, 0) + 1 - numdiv(n));
  for(n=1, nn, print1(a(n), ", "))
} \\ Andrew Howroyd, Nov 02 2019

a(n) = A304712(n) + 1 - A000005(n). - Andrew Howroyd, Nov 02 2019

cp's OEIS Frontend

A034729 a(n) = Sum_{ k, k|n } 2^(k-1).

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A291166 Connected Haar graph numbers.

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A302698 Number of integer partitions of n into relatively prime parts that are all greater than 1.

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A304711 Heinz numbers of integer partitions whose distinct parts are pairwise coprime.

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A355737 Number of ways to choose a sequence of divisors, one of each prime index of n (with multiplicity), such that the result has no common divisor > 1.

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A348551 Heinz numbers of integer partitions whose mean is not an integer.

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A326674 GCD of the set of positions of 1's in the reversed binary expansion of n.

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