A317624 -id:A317624 - cp's OEIS frontend

A074761 Number of partitions of n of order n.

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 9, 1, 4, 5, 1, 1, 12, 1, 27, 7, 6, 1, 81, 1, 7, 1, 54, 1, 407, 1, 1, 11, 9, 13, 494, 1, 10, 13, 423, 1, 981, 1, 137, 115, 12, 1, 1309, 1, 59, 17, 193, 1, 240, 21, 1207, 19, 15, 1, 47274, 1, 16, 239, 1, 25, 3284, 1, 333, 23, 3731, 1, 42109, 1, 19

Offset: 1

Vladeta Jovovic, Sep 28 2002

Order of partition is lcm of its parts.

a(n) is the number of conjugacy classes of the symmetric group S_n such that a representative of the class has order n. Here order means the order of an element of a group. Note that a(n) = 1 if and only if n is a prime power. - W. Edwin Clark, Aug 05 2014

			The a(15) = 5 partitions are (15), (5,3,3,3,1), (5,5,3,1,1), (5,3,3,1,1,1,1), (5,3,1,1,1,1,1,1,1). - _Gus Wiseman_, Aug 01 2018

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 1..4000 (first 1025 terms from Joerg Arndt)

Cf. A018818, A074351, A074752.
Main diagonal of A256067, A256554.
Cf. A000837, A074761, A285572, A290103, A305566, A316429, A316431, A316432, A316433, A317624.

A:= proc(n)
      uses numtheory;
      local S;
    S:= add(mobius(n/i)*1/mul(1-x^j,j=divisors(i)),i=divisors(n));
    coeff(series(S,x,n+1),x,n);
end proc:
seq(A(n),n=1..100); # Robert Israel, Aug 06 2014

a[n_] := With[{s = Sum[MoebiusMu[n/i]*1/Product[1-x^j, {j, Divisors[i]}], {i, Divisors[n]}]}, SeriesCoefficient[s, {x, 0, n}]]; Array[a, 80] (* Jean-François Alcover, Feb 29 2016 *)
Table[Length[Select[IntegerPartitions[n],LCM@@#==n&]],{n,50}] (* Gus Wiseman, Aug 01 2018 *)

pr(k, x)={my(t=1); fordiv(k, d, t *= (1-x^d) ); return(t); }
a(n) =
{
    my( x = 'x+O('x^(n+1)) );
    polcoeff( Pol( sumdiv(n, i, moebius(n/i) / pr(i, x) ) ), n );
}
vector(66, n, a(n) )
\\ Joerg Arndt, Aug 06 2014

Coefficient of x^n in expansion of Sum_{i divides n} A008683(n/i)*1/Product_{j divides i} (1-x^j).

A074971 Number of partitions of n into distinct parts of order n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 6, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 32, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 24, 1, 1, 1, 2, 1, 24, 1, 1, 1, 1, 1, 12, 1, 1, 1, 3, 1, 2

Offset: 1

Vladeta Jovovic, Oct 05 2002

nonn

Order of partition is lcm of its parts.

			The a(36) = 6 partitions are (36), (18,12,6), (18,12,4,2), (18,12,3,2,1), (18,9,4,3,2), (12,9,6,4,3,2). - _Gus Wiseman_, Aug 01 2018

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000837, A074761, A285572, A290103, A305566, A316431, A316433, A317624.

Cf. also A033630.

A074971(n) = { my(q=0); fordiv(n,i,my(p=1); fordiv(i,j,p *= (1 + 'x^j)); q += moebius(n/i)*p); polcoeff(q,n); }; \\ Antti Karttunen, Dec 19 2018

Coefficient of x^n in expansion of Sum_{i divides n} mu(n/i)*Product_{j divides i} (1+x^j).

A319333 Heinz numbers of integer partitions whose sum is equal to their LCM.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 198, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263

Offset: 1

Gus Wiseman, Sep 17 2018

nonn

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

			The sequence of partitions whose Heinz numbers are in the sequence begins: (1), (2), (3), (4), (5), (6), (7), (8), (9), (10), (3,2,1), (11), (12).

Cf. A067538, A074761, A143773, A290103, A305566, A316429, A316431, A316432, A316433, A317624, A319315, A319334.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],LCM@@primeMS[#]==Total[primeMS[#]]&]

A318670 Number of subsets of divisors of n whose least common multiple is n and the sum does not exceed n. For n > 1, 1 is excluded from the set of divisors.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 5, 1, 2, 2, 1, 1, 6, 1, 6, 2, 2, 1, 17, 1, 2, 1, 7, 1, 34, 1, 1, 2, 2, 2, 44, 1, 2, 2, 23, 1, 36, 1, 7, 7, 2, 1, 65, 1, 7, 2, 7, 1, 21, 2, 25, 2, 2, 1, 471, 1, 2, 7, 1, 2, 39, 1, 7, 2, 44, 1, 355, 1, 2, 7, 7, 2, 39, 1, 89, 1, 2, 1, 531, 2, 2, 2, 27, 1, 559, 2, 7, 2, 2, 2, 257, 1, 7, 7, 61, 1, 39, 1, 28, 46

Offset: 1

Antti Karttunen and David A. Corneth, Sep 08 2018

nonn

These count the "starter sets" employed by a simple backtracking algorithm that computes A317624. See the PARI program dated Sep 08-10 2018 under that entry.

			For n = 45, there exists the following subsets of its divisors larger than one (3, 5, 9, 15, 45) that satisfy the condition that the least common multiple of the members is 45, and the sum does not exceed 45: (45), (3, 9, 15), (3, 5, 9, 15), (3, 5, 9), (5, 9), (9, 15) and (5, 9, 15), altogether seven subsets, thus a(45) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..1259
Index entries for sequences related to lcm's

Cf. A069626, A317624.

A318670(n) = if(1==n,1,my(ds=(divisors(n)[2..numdiv(n)]), subsets = select(v -> ((vecsum(v)<=n)&&(n==lcm(v))),powerset(ds))); length(subsets)); \\ A memory-hog implementation.
powerset(v) = { my(siz=2^length(v),pv=vector(siz)); for(i=0,siz-1,pv[i+1] = choosebybits(v,i)); pv; };
choosebybits(v,m) = { my(s=vector(hammingweight(m)),i=j=1); while(m>0,if(m%2,s[j] = v[i];j++); i++; m >>= 1); s; };  \\ Antti Karttunen, Sep 08 2018

\\ A better program:
strong_divisors_reversed(n) = vecsort(select(x -> (x>1), divisors(n)), , 4);
A318670aux(orgn,n,parts,from=1,ss=List([])) = { my(k = #parts, s=0, newss); if(lcm(Vec(ss))==orgn,s++); for(i=from,k,if(parts[i]<=n, newss = List(ss); listput(newss,parts[i]); s += A318670aux(orgn,n-parts[i],parts,i+1,newss))); (s) };
A318670(n) = if(1==n,n,A318670aux(n,n,strong_divisors_reversed(n))); \\ Antti Karttunen, Sep 08 2018

a(n) <= A069626(n).
For all n >= 1:
a(A000961(n)) = 1.
a(A006881(n)) = 2.

A319334 Nonprime Heinz numbers of integer partitions whose sum is equal to their LCM.

Original entry on oeis.org

30, 198, 264, 273, 364, 490, 525, 630, 700, 840, 918, 1120, 1224, 1495, 1632, 1794, 2392, 2420, 2750, 3105, 3450, 3726, 4140, 4263, 4400, 4466, 4921, 4968, 5481, 5520, 5684, 6327, 6624, 7030, 7040, 7308, 8436, 8832, 9744, 11248, 12992, 14079, 14450, 14993

Offset: 1

Gus Wiseman, Sep 17 2018

nonn

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

			The sequence of all non-singleton integer partitions whose sum is equal to their LCM begins: (321), (5221), (52111), (642), (6411), (4431), (4332), (43221), (43311), (432111), (72221), (4311111), (722111), (963), (7211111), (9621).

Cf. A067538, A074761, A143773, A290103, A305566, A316429, A316431, A316432, A316433, A317624, A319315, A319333.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,1000],And[!PrimeQ[#],LCM@@primeMS[#]==Total[primeMS[#]]]&]

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A074761 Number of partitions of n of order n.

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A074971 Number of partitions of n into distinct parts of order n.

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A319333 Heinz numbers of integer partitions whose sum is equal to their LCM.

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A318670 Number of subsets of divisors of n whose least common multiple is n and the sum does not exceed n. For n > 1, 1 is excluded from the set of divisors.

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A319334 Nonprime Heinz numbers of integer partitions whose sum is equal to their LCM.

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