A316432 -id:A316432 - 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).

A316430 Heinz numbers of integer partitions whose length is equal to the GCD of all the parts.

Original entry on oeis.org

1, 2, 9, 21, 39, 57, 87, 91, 111, 125, 129, 159, 183, 203, 213, 237, 247, 267, 301, 303, 321, 325, 339, 377, 393, 417, 427, 453, 489, 519, 543, 551, 553, 559, 575, 579, 597, 669, 687, 689, 707, 717, 753, 789, 791, 813, 817, 843, 845, 879, 923, 925, 933, 951, 973

Offset: 1

Gus Wiseman, Jul 02 2018

nonn

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

2 is the only even term in the sequence. 3k is in the sequence if and only if k is in A031215. 5k is in the sequence if and only if k = pq with p and q in A031336.

			Sequence of integer partitions whose length is equal to their GCD begins: (), (1), (2,2), (4,2), (6,2), (8,2), (10,2), (6,4), (12,2), (3,3,3), (14,2), (16,2), (18,2), (10,4), (20,2), (22,2), (8,6), (24,2), (14,4), (26,2), (28,2), (6,3,3).

Subsequence of A004280.

Cf. A056239, A067538, A074761, A143773, A289508, A289509, A296150, A316413, A316431, A316432, A316433, A031215, A031336.

Select[Range[200],PrimeOmega[#]==GCD@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]]&]

is(n,f=factor(n))=gcd(apply(primepi,f[,1]))==vecsum(f[,2]) \\ Charles R Greathouse IV, Jul 25 2024

a(n) << n log^2 n, can this be improved? - Charles R Greathouse IV, Jul 25 2024

A316433 Number of integer partitions of n whose length is equal to the LCM of all parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 2, 1, 4, 3, 4, 4, 8, 5, 7, 8, 10, 8, 13, 13, 20, 18, 25, 25, 36, 34, 48, 52, 64, 64, 85, 85, 108, 106, 129, 133, 160, 158, 189, 194, 229, 228, 276, 279, 332, 336, 394, 402, 476, 489, 572, 599, 699, 728, 845, 889, 1032, 1094, 1251, 1332, 1523

Offset: 1

Gus Wiseman, Jul 02 2018

nonn

			The a(13) = 8 partitions are (4441), (55111), (322222), (332221), (333211), (622111), (631111), (7111111).

Cf. A067538, A074761, A143773, A285572, A289508, A290103, A305566, A316429, A316430, A316431, A316432.

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

a(n) = {my(nb = 0); forpart(p=n, if (lcm(Vec(p))==#p, nb++);); nb;} \\ Michel Marcus, Jul 03 2018

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[#]]&]

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

A316437 Take the integer partition with Heinz number n, divide all parts by the GCD of the parts, then take the Heinz number of the resulting partition.

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 4, 10, 2, 12, 2, 14, 15, 16, 2, 18, 2, 20, 6, 22, 2, 24, 4, 26, 8, 28, 2, 30, 2, 32, 33, 34, 35, 36, 2, 38, 10, 40, 2, 42, 2, 44, 45, 46, 2, 48, 4, 50, 51, 52, 2, 54, 55, 56, 14, 58, 2, 60, 2, 62, 12, 64, 6, 66, 2, 68, 69, 70, 2, 72, 2, 74, 75, 76, 77, 78, 2, 80, 16, 82, 2, 84, 85, 86, 22, 88, 2, 90, 15

Offset: 1

Gus Wiseman, Jul 03 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
This sequence is idempotent, meaning a(a(n)) = a(n) for all n.
All terms belong to A289509.

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

Cf. A000720, A056239, A289508, A289509, A290103, A296150, A316430, A316431, A316432, A316438.

f[n_]:=If[n==1,1,With[{pms=Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]},Times@@Prime/@(pms/GCD@@pms)]];
Table[f[n],{n,100}]

A316437(n) = if(1==n,1,my(f = factor(n), pis = apply(p -> primepi(p), f[, 1]~), es = f[, 2]~, g = gcd(pis)); factorback(vector(#f~, k, prime(pis[k]/g)^es[k]))); \\ Antti Karttunen, Aug 06 2018

More terms from Antti Karttunen, Aug 06 2018

A316436 Sum divided by GCD of the integer partition with Heinz number n > 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 4, 1, 5, 5, 4, 1, 5, 1, 5, 3, 6, 1, 5, 2, 7, 3, 6, 1, 6, 1, 5, 7, 8, 7, 6, 1, 9, 4, 6, 1, 7, 1, 7, 7, 10, 1, 6, 2, 7, 9, 8, 1, 7, 8, 7, 5, 11, 1, 7, 1, 12, 4, 6, 3, 8, 1, 9, 11, 8, 1, 7, 1, 13, 8, 10, 9, 9, 1, 7, 4, 14, 1, 8, 10, 15, 6, 8, 1, 8, 5, 11, 13, 16, 11, 7, 1, 9, 9, 8, 1, 10, 1, 9, 9

Offset: 2

Gus Wiseman, Jul 03 2018

nonn

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

Cf. A056239, A289508, A289509, A290103, A290104, A296150, A316430, A316431, A316432, A316437.

a:= n-> (l-> add(i, i=l)/igcd(l[]))(map(i->
      numtheory[pi](i[1])$i[2], ifactors(n)[2])):
seq(a(n), n=2..100);  # Alois P. Heinz, Jul 03 2018

Table[With[{pms=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]},Total[pms]/GCD@@pms],{n,2,100}]

A316436(n) = { my(f = factor(n), pis = apply(p -> primepi(p), f[, 1]~), es = f[, 2]~, g = gcd(pis)); sum(i=1, #f~, pis[i]*es[i])/g; }; \\ Antti Karttunen, Sep 10 2018

More terms from Antti Karttunen, Sep 10 2018

A319329 Heinz numbers of integer partitions, whose length is equal to the GCD of the parts and whose sum is equal to the LCM of the parts, in increasing order.

Original entry on oeis.org

2, 1495, 179417, 231133, 727531, 1378583, 1787387, 3744103, 4556993, 7566167, 18977519, 29629391, 30870587, 34174939, 39973571, 53508983, 70946617, 110779141, 138820187, 139681069, 170583017, 225817751, 409219217, 441317981, 493580417, 539462099, 544392433, 712797613, 802903541

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 corresponding sequence of partitions, whose length is equal to their GCD and whose sum is equal to their LCM: (1), (9,6,3), (20,8,8,4), (24,16,4,4), (16,16,12,4).

Subsequence of A316430.

Cf. A056239, A067538, A074761, A143773, A289508, A289509, A290103, A290104, A316431, A316432, A319328, A319330, A319333.

Select[Range[2,10000],With[{m=If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]},And[LCM@@m==Total[m],GCD@@m==Length[m]]]&]

More terms from Max Alekseyev, Jul 25 2024

A319330 Number of integer partitions of n whose length is equal to the GCD of the parts and whose sum is equal to the LCM of the parts.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3, 0, 0, 0, 0, 2

Offset: 1

Gus Wiseman, Sep 17 2018

nonn

			A list of all such partitions begins (1), (9,6,3), (20,8,8,4), (24,16,4,4), (16,16,12,4), (16,12,12,8), (25,10,5,5,5), (20,15,15,5,5), (20,15,10,10,5).

Cf. A067538, A074761, A143773, A316432, A319328, A319329, A319333.

Table[Length[Select[IntegerPartitions[n],And[LCM@@#==Total[#],GCD@@#==Length[#]]&]],{n,30}]

a(71)-a(105) from Alois P. Heinz, Sep 18 2018

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[#]]]&]

cp's OEIS Frontend

A074761 Number of partitions of n of order n.

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A316430 Heinz numbers of integer partitions whose length is equal to the GCD of all the parts.

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A316433 Number of integer partitions of n whose length is equal to the LCM of all parts.

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

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

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A316437 Take the integer partition with Heinz number n, divide all parts by the GCD of the parts, then take the Heinz number of the resulting partition.

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A316436 Sum divided by GCD of the integer partition with Heinz number n > 1.

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A319329 Heinz numbers of integer partitions, whose length is equal to the GCD of the parts and whose sum is equal to the LCM of the parts, in increasing order.

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