A281116 -id:A281116 - cp's OEIS frontend

A318981 Numbers whose prime indices plus 1 are relatively prime.

1, 6, 12, 14, 15, 18, 21, 24, 26, 28, 30, 35, 36, 38, 39, 42, 45, 48, 51, 52, 54, 56, 58, 60, 63, 65, 66, 69, 70, 72, 74, 75, 76, 77, 78, 84, 86, 87, 90, 91, 95, 96, 98, 102, 104, 105, 106, 108, 111, 112, 114, 116, 117, 119, 120, 122, 123, 126, 130, 132, 133

Offset: 1

Gus Wiseman, Sep 06 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			The sequence of integer partitions whose Heinz numbers are in the sequence begins: (), (21), (211), (41), (32), (221), (42), (2111), (61), (411), (321), (43), (2211), (81), (62), (421), (322), (21111).

Cf. A000837, A007359, A018783, A281116, A289508, A289509, A318978, A318980.

Select[Range[100],GCD@@(PrimePi/@FactorInteger[#][[All,1]]+1)==1&]

A316979 Number of strict factorizations of n into factors > 1 with no equivalent primes.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 2, 3, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 7, 1, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 6, 1, 1, 3, 4, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 3, 3, 1, 1, 1, 7, 2, 1, 1, 6, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 18 2018

nonn

In a factorization, two primes are equivalent if each factor has in its prime factorization the same multiplicity of both primes. For example, in 60 = (2*30) the primes {3, 5} are equivalent but {2, 3} and {2, 5} are not.

			The a(24) = 5 factorizations are (2*3*4), (2*12), (3*8), (4*6), (24).
The a(36) = 4 factorizations are (2*3*6), (2*18), (3*12), (4*9).

Cf. A000009, A001055, A007716, A007717, A020555, A045778, A130091, A162247, A281116.

Cf. A316974, A316978, A316980, A316981.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[facs[n],And[UnsameQ@@#,UnsameQ@@dual[primeMS/@#]]&]],{n,100}]

a(prime^n) = A000009(n).

A318980 Number of integer partitions of n whose parts plus 1 are relatively prime.

Original entry on oeis.org

0, 0, 1, 1, 4, 5, 9, 13, 21, 29, 43, 56, 79, 109, 146, 192, 254, 329, 428, 553, 707, 900, 1139, 1434, 1800, 2251, 2799, 3472, 4286, 5275, 6469, 7918, 9655, 11755, 14252, 17248, 20817, 25084, 30134, 36142, 43235, 51644, 61548, 73241, 86961, 103108, 122010

Offset: 1

Gus Wiseman, Sep 06 2018

nonn

			The a(7) = 9 partitions are (61), (43), (421), (4111), (322), (3211), (2221), (22111), (211111).
The a(8) = 13 partitions:
  (62),
  (332), (422), (431), (521), (611),
  (3221), (4211),
  (22211), (32111), (41111),
  (221111),
  (2111111).

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000837, A007359, A018783, A051424, A281116, A289508, A289509, A318978.

Table[Length[Select[IntegerPartitions[n],GCD@@(#+1)==1&]],{n,30}]

seq(n)={Vec(sum(d=1, n+1, moebius(d)*(-1 + 1/prod(k=ceil(2/d), (n+1)\d, 1 - x^(k*d-1) + O(x*x^n)))), -n)} \\ Andrew Howroyd, Oct 17 2019

G.f.: Sum_{d>=1} mu(d)*(-1 + 1/(Prod_{k>=2/d} 1 - x^(k*d - 1))). - Andrew Howroyd, Oct 17 2019

A319237 Positions of nonzero terms in A114592, the list of coefficients in the expansion of Product_{n > 1} (1 - 1/n^s).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 12, 13, 17, 18, 19, 20, 23, 24, 25, 28, 29, 30, 31, 32, 36, 37, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 83, 84, 88, 89, 90, 92, 97, 98, 99, 100, 101, 102

Offset: 1

Gus Wiseman, Sep 15 2018

nonn

Complement of A319238.

Cf. A001055, A045778, A114592, A162247, A190938, A281116, A281118, A303386, A316441, A319239.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Join@@Position[Table[Sum[(-1)^Length[f],{f,Select[facs[n],UnsameQ@@#&]}],{n,100}],_Integer?(Abs[#]>0&)]

A303710 Number of factorizations of numbers that are not perfect powers using only numbers that are not perfect powers.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 1, 3, 1, 3, 2, 2, 1, 4, 2, 3, 1, 5, 1, 2, 2, 2, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 2, 5, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 2, 5, 1, 5, 2, 1, 9, 2, 2, 2, 4, 1, 9, 2, 3, 2, 2, 2, 6, 1, 3, 3

Offset: 1

Gus Wiseman, Apr 29 2018

nonn

Note that a factorization of a number that is not a perfect power (A007916) is always itself aperiodic, meaning the multiplicities of its factors are relatively prime.

			The a(19) = 4 factorizations of 24 are (2*2*2*3), (2*2*6), (2*12), (24).
The a(23) = 5 factorizations of 30 are (2*3*5), (2*15), (3*10), (5*6), (30).

Cf. A001055, A001597, A007716, A007916, A052409, A052410, A281116, A303365, A303386, A303707, A303708, A303709.

radQ[n_] := And[n > 1, GCD@@FactorInteger[n][[All, 2]] === 1]; facsr[n_] := If[n <= 1, {{}}, Join@@Table[Map[Prepend[#, d] &, Select[facsr[n/d], Min@@# >= d &]], {d, Select[Divisors[n], radQ]}]]; Table[Length[facsr[n]], {n, Select[Range[100], radQ]}]

A317449 Regular triangle where T(n,k) is the number of multiset partitions of strongly normal multisets of size n into k blocks, where a multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 5, 21, 16, 5, 7, 52, 72, 32, 7, 11, 141, 306, 216, 65, 11, 15, 327, 1113, 1160, 512, 113, 15, 22, 791, 4033, 6052, 3737, 1154, 199, 22, 30, 1780, 13586, 28749, 24325, 10059, 2317, 323, 30, 42, 4058, 45514, 133642, 151994, 82994, 24854, 4493, 523, 42

Offset: 1

Gus Wiseman, Aug 06 2018

			The T(3,2) = 6 multiset partitions are {{1},{1,1}}, {{1},{1,2}}, {{2},{1,1}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}.
Triangle begins:
    1
    2    2
    3    6    3
    5   21   16    5
    7   52   72   32    7
   11  141  306  216   65   11
   15  327 1113 1160  512  113   15
   ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums are A035310. First and last columns are both A000041.

Cf. A001055, A007716, A045778, A255906, A281116, A317584, A317654, A317755, A317775, A317776.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[Join@@mps/@strnorm[n],Length[#]==k&]],{n,6},{k,n}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=EulerT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n))-1, -n)/prod(i=1, #v, i^v[i]*v[i]!)}
U(m, n)={my(s=0); forpart(p=m, s+=D(p,n)); s}
M(n)={Mat(vector(n,k,(U(k,n)-U(k-1,n))~))}
{ my(A=M(8)); for(n=1, #A~, print(A[n,1..n])) } \\ Andrew Howroyd, Dec 30 2020

Terms a(46) and beyond from Andrew Howroyd, Dec 30 2020

A317624 Number of integer partitions of n where all parts are > 1 and whose LCM is n.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 17, 1, 1, 1, 7, 1, 60, 1, 1, 1, 1, 1, 76, 1, 1, 1, 55, 1, 105, 1, 11, 10, 1, 1, 187, 1, 6, 1, 13, 1, 30, 1, 111, 1, 1, 1, 5043, 1, 1, 15, 1, 1, 230, 1, 17, 1, 242, 1, 4173, 1, 1, 12, 19, 1

Offset: 0

Gus Wiseman, Aug 01 2018

nonn

			The a(20) = 5 partitions are (20), (10,4,4,2), (10,4,2,2,2), (5,5,4,4,2), (5,5,4,2,2,2).
The a(45) = 10 partitions:
  (45),
  (15,15,9,3,3), (15,9,9,9,3),
  (15,9,9,3,3,3,3), (15,9,5,5,5,3,3), (9,9,9,5,5,5,3),
  (15,9,3,3,3,3,3,3,3), (9,9,5,5,5,3,3,3,3), (9,5,5,5,5,5,5,3,3),
  (9,5,5,5,3,3,3,3,3,3,3).
From _David A. Corneth_, Sep 08 2018: (Start)
Let sum(t) denote the sum of elements of a tuple t. The tuples t with distinct divisors of 45 that have lcm(t) = 45 and sum(t) <= 45 are {(45) and (3, 9, 15), (3, 5, 9, 15), (3, 5, 9), (5, 9), (9, 15), (5, 9, 15)}. For each such tuple t, find the number of partitions of 45 - s(t) into distinct parts of t.
For the tuple (45), there is 1 partition of 45 - 45 = 0 into parts with 45. That is: {()}.
For the tuple (3, 9, 15), there are 4 partitions of 45 - (3 + 9 + 15) = 18 into parts with 3, 9 and 15. They are {(3, 15), (9, 9), (3, 3, 3, 9), (3, 3, 3, 3, 3, 3)}.
For the tuple (3, 5, 9), there are 4 partitions of 45 - (3 + 5 + 9) = 28 into parts with 3, 5 and 9; they are {(5, 5, 9, 9), (3, 3, 3, 5, 5, 9), (3, 5, 5, 5, 5, 5), (3, 3, 3, 3, 3, 3, 5, 5)}.
For the tuple (3, 5, 9, 15), there is 1 partition of 45 - (3 + 5 + 9 + 15) = 13 into parts with 3, 5, 9 and 15. That is (3, 5, 5).
The other tuples, (5, 9), (9, 15), and (5, 9, 15); they give no extra tuples. That's because there is no solution to the Diophantine equation for 5x + 9y = 45 - (5 + 9), corresponding to the tuple (5, 9) with nonnegative x, y.
That also excludes (9, 15); if there is a solution for that, there would also be a solution for (5, 9). This could whittle down the number of seeds even further. Similarly, (5, 9, 15) gives no solution.
Therefore a(45) = 1 + 4 + 4 + 1 = 10.
(End)
In general, there are A318670(n) (<= A069626(n)) such seed sets of divisors where to start extending the partition from. (See the second PARI program which uses subroutine toplevel_starting_sets.) - _Antti Karttunen_, Sep 08 2018

Antti Karttunen, Table of n, a(n) for n = 0..359
Index entries for sequences related to lcm's

Cf. A000837, A018818, A066874, A067538, A069626, A074761, A074971, A143773, A259936, A281116, A285572, A290103, A305566, A316429, A316431, A316432, A316433, A318670.

Table[Length[Select[IntegerPartitions[n],And[Min@@#>=2,LCM@@#==n]&]],{n,30}]

strong_divisors_reversed(n) = vecsort(select(x -> (x>1), divisors(n)), , 4);
partitions_into_lcm(orgn,n,parts,from=1,m=1) = if(!n,(m==orgn),my(k = #parts, s=0); for(i=from,k,if(parts[i]<=n, s += partitions_into_lcm(orgn,n-parts[i],parts,i,lcm(m,parts[i])))); (s));
A317624(n) = if(n<=1,0,partitions_into_lcm(n,n,strong_divisors_reversed(n))); \\ Antti Karttunen, Sep 07 2018

strong_divisors_reversed(n) = vecsort(select(x -> (x>1), divisors(n)), , 4);
partitions_into(n,parts,from=1) = if(!n,1, if(#parts==from, (0==(n%parts[from])), my(s=0); for(i=from,#parts,if(parts[i]<=n, s += partitions_into(n-parts[i],parts,i))); (s)));
toplevel_starting_sets(orgn,n,parts,from=1,ss=List([])) = { my(k = #parts, s=0, newss); if(lcm(Vec(ss))==orgn,s += partitions_into(n,ss)); for(i=from,k,if(parts[i]<=n, newss = List(ss); listput(newss,parts[i]); s += toplevel_starting_sets(orgn,n-parts[i],parts,i+1,newss))); (s) };
A317624(n) = if(n<=1,0,toplevel_starting_sets(n,n,strong_divisors_reversed(n))); \\ Antti Karttunen, Sep 08-10 2018

A317751 Number of divisors d of n such that there exists a factorization of n into factors > 1 with GCD d.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 3, 2, 2, 2, 3, 1, 2, 1, 3, 2, 2, 2, 5, 1, 2, 2, 3, 1, 2, 1, 3, 3, 2, 1, 4, 2, 3, 2, 3, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 3, 4, 2, 2, 1, 3, 2, 2, 1, 5, 1, 2, 3, 3, 2, 2, 1, 4, 3, 2, 1, 3, 2, 2, 2, 3, 1, 3, 2, 3, 2, 2, 2, 4, 1, 3, 3, 5, 1, 2, 1, 3, 2

Offset: 1

Gus Wiseman, Aug 06 2018

nonn

Also the number of distinct possible GCDs of factorizations of n into factors > 1.
Also the number of nonzero terms in row n of A317748.
a(prime^n) = A008619(n).
If n is squarefree and composite, a(n) = 2.

			The divisors of 36 that are possible GCDs of factorizations of 36 are {1, 2, 3, 6, 36}, so a(36) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A000837, A001055, A014963, A045778, A050370, A162247, A281116, A289509.

Cf. A317748, A317752, A317755, A317757.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
goc[n_,m_]:=Length[Select[facs[n],And[And@@(Divisible[#,m]&/@#),GCD@@(#/m)==1]&]];
Table[Length[Select[Divisors[n],goc[n,#]!=0&]],{n,100}]

A317751aux(n, m, facs, gcds) = if(1==n, setunion([gcd(Vec(facs))],gcds), my(newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); gcds = setunion(gcds,A317751aux(n/d, d, newfacs, gcds)))); (gcds));
A317751(n) = if(1==n,0,length(A317751aux(n, n, List([]), Set([])))); \\ Antti Karttunen, Sep 08 2018

More terms from Antti Karttunen, Sep 08 2018

A319240 Positions of zeros in A316441, the list of coefficients in the expansion of Product_{n > 1} 1/(1 + 1/n^s).

Original entry on oeis.org

4, 6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 25, 26, 28, 33, 34, 35, 38, 39, 44, 45, 46, 48, 49, 50, 51, 52, 55, 57, 58, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 106, 108, 111, 112, 115, 116, 117, 118, 119, 121, 122

Offset: 1

Gus Wiseman, Sep 15 2018

From Tian Vlasic, Dec 31 2021: (Start)
Numbers that have an equal number of even and odd-length unordered factorizations.
There are infinitely many terms since p^2 is a term for prime p.
Out of all numbers of the form p^k with p prime (listed in A000961), only the numbers of the form p^2 (A001248) are terms.
Out of all numbers of the form p*q^k, p and q prime, only the numbers of the form p*q (A006881), p*q^2 (A054753), p*q^4 (A178739) and p*q^6 (A189987) are terms.
Similar methods can be applied to all prime signatures. (End)

			12 = 2*6 = 3*4 = 2*2*3 has an equal number of even-length factorizations and odd-length factorizations (2). - _Tian Vlasic_, Dec 09 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Complement of A319239.

Cf. A001055, A025487, A045778, A114592, A162247, A190938, A281116, A281118, A303386, A316441, A319238.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Join@@Position[Table[Sum[(-1)^Length[f],{f,facs[n]}],{n,100}],0]

A319766 Number of non-isomorphic strict intersecting multiset partitions (sets of multisets) of weight n whose dual is also a strict intersecting multiset partition.

Original entry on oeis.org

1, 1, 1, 4, 6, 14, 31, 64, 145, 324, 753

Offset: 0

Gus Wiseman, Sep 27 2018

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 14 multiset partitions:
1: {{1}}
2: {{1,1}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1},{1,1}}
   {{2},{1,2}}
4: {{1,1,1,1}}
   {{1,2,2,2}}
   {{1},{1,1,1}}
   {{1},{1,2,2}}
   {{2},{1,2,2}}
   {{1,2},{2,2}}
5: {{1,1,1,1,1}}
   {{1,1,2,2,2}}
   {{1,2,2,2,2}}
   {{1},{1,1,1,1}}
   {{1},{1,2,2,2}}
   {{2},{1,1,2,2}}
   {{2},{1,2,2,2}}
   {{2},{1,2,3,3}}
   {{1,1},{1,1,1}}
   {{1,1},{1,2,2}}
   {{1,2},{1,2,2}}
   {{1,2},{2,2,2}}
   {{2,2},{1,2,2}}
   {{2},{1,2},{2,2}}

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A316983, A317757, A319616.

Cf. A319752, A319765, A319767, A319768, A319769, A319773, A319774.

cp's OEIS Frontend

A318981 Numbers whose prime indices plus 1 are relatively prime.

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A316979 Number of strict factorizations of n into factors > 1 with no equivalent primes.

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A318980 Number of integer partitions of n whose parts plus 1 are relatively prime.

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A319237 Positions of nonzero terms in A114592, the list of coefficients in the expansion of Product_{n > 1} (1 - 1/n^s).

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A303710 Number of factorizations of numbers that are not perfect powers using only numbers that are not perfect powers.

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A317449 Regular triangle where T(n,k) is the number of multiset partitions of strongly normal multisets of size n into k blocks, where a multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.

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Extensions

A317624 Number of integer partitions of n where all parts are > 1 and whose LCM is n.

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PARI

A317751 Number of divisors d of n such that there exists a factorization of n into factors > 1 with GCD d.

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