A074761 -id:A074761 - cp's OEIS frontend

A316413 Heinz numbers of integer partitions whose length divides their sum.

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 21, 22, 23, 25, 27, 28, 29, 30, 31, 32, 34, 37, 39, 41, 43, 46, 47, 49, 53, 55, 57, 59, 61, 62, 64, 67, 68, 71, 73, 78, 79, 81, 82, 83, 84, 85, 87, 88, 89, 90, 91, 94, 97, 98, 99, 100, 101, 103, 105, 107, 109, 110

Offset: 1

Gus Wiseman, Jul 02 2018

nonn

In other words, partitions whose average is an integer.

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

			Sequence of partitions whose length divides their sum begins (1), (2), (11), (3), (4), (111), (22), (31), (5), (6), (1111), (7), (8), (42), (51), (9), (33), (222), (411).

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1327 terms from R. J. Mathar)

Cf. A056239, A067538, A074761, A143773, A237984, A289508, A289509, A290103, A296150, A298423, A316428, A316431.

isA326413 := proc(n)
    psigsu := A056239(n) ;
    psigle := numtheory[bigomega](n) ;
    if modp(psigsu,psigle) = 0 then
        true;
    else
        false;
    end if;
end proc:
n := 1:
for i from 2 to 3000 do
    if isA326413(i) then
        printf("%d %d\n",n,i);
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Aug 09 2019
# second Maple program:
q:= n-> (l-> nops(l)>0 and irem(add(i, i=l), nops(l))=0)(map
        (i-> numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..110])[];  # Alois P. Heinz, Nov 19 2021

Select[Range[2,100],Divisible[Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]],PrimeOmega[#]]&]

A349156 Number of integer partitions of n whose mean is not an integer.

Original entry on oeis.org

1, 0, 0, 1, 1, 5, 3, 13, 11, 21, 28, 54, 31, 99, 111, 125, 165, 295, 259, 488, 425, 648, 933, 1253, 943, 1764, 2320, 2629, 2962, 4563, 3897, 6840, 6932, 9187, 11994, 12840, 12682, 21635, 25504, 28892, 28187, 44581, 42896, 63259, 66766, 74463, 104278, 124752

Offset: 0

Gus Wiseman, Nov 14 2021

nonn

Equivalently, partitions whose length does not divide their sum.

By conjugation, also the number of integer partitions of n with greatest part not dividing n.

			The a(3) = 1 through a(8) = 11 partitions:
  (21)  (211)  (32)    (2211)   (43)      (332)
               (41)    (3111)   (52)      (422)
               (221)   (21111)  (61)      (431)
               (311)            (322)     (521)
               (2111)           (331)     (611)
                                (421)     (22211)
                                (511)     (32111)
                                (2221)    (41111)
                                (3211)    (221111)
                                (4111)    (311111)
                                (22111)   (2111111)
                                (31111)
                                (211111)

Below, "!" means either enumerative or set theoretical complement.
The version for nonempty subsets is !A051293.
The complement is counted by A067538, ranked by A316413.
The geometric version is !A067539, strict !A326625, ranked by !A326623.
The strict case is !A102627.
The version for prime factors is A175352, complement A078175.
The version for distinct prime factors is A176587, complement A078174.
The ordered version (compositions) is !A271654, ranked by !A096199.
The multiplicative version (factorizations) is !A326622, geometric !A326028.
The conjugate is ranked by !A326836.
The conjugate strict version is !A326850.
These partitions are ranked by A348551.
A000041 counts integer partitions.
A326567/A326568 give the mean of prime indices, conjugate A326839/A326840.
A236634 counts unbalanced partitions, complement of A047993.
A327472 counts partitions not containing their mean, complement of A237984.
Cf. A001700, A074761, A098743, A143773, A175397, A175761, A298423, A326027, A326641, A326842, A326849, A327778.

Table[Length[Select[IntegerPartitions[n],!IntegerQ[Mean[#]]&]],{n,0,30}]

a(n > 0) = A000041(n) - A067538(n).

A316428 Heinz numbers of integer partitions such that every part is divisible by the number of parts.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 29, 31, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 67, 71, 73, 79, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 125, 127, 129, 131, 133, 137, 139, 149, 151, 157, 159, 163, 167, 169, 173, 179, 181, 183, 191, 193, 197

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).

			93499 is the Heinz number of (12,8,8,4) and belongs to the sequence because each part is divisible by 4.
Sequence of partitions such that every part is divisible by the number of parts begins (1), (2), (3), (4), (2,2), (5), (6), (7), (8), (4,2), (9).

Cf. A056239, A067538, A074761, A143773, A237984, A289509, A296150, A298423, A316413.

Select[Range[200],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>Divisible[PrimePi[p],PrimeOmega[#]]]&]

A316431 Least common multiple divided by greatest common divisor of the integer partition with Heinz number n > 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 4, 6, 1, 1, 2, 1, 3, 2, 5, 1, 2, 1, 6, 1, 4, 1, 6, 1, 1, 10, 7, 12, 2, 1, 8, 3, 3, 1, 4, 1, 5, 6, 9, 1, 2, 1, 3, 14, 6, 1, 2, 15, 4, 4, 10, 1, 6, 1, 11, 2, 1, 2, 10, 1, 7, 18, 12, 1, 2, 1, 12, 6, 8, 20, 6, 1, 3, 1, 13, 1, 4, 21, 14, 5, 5, 1, 6, 6, 9, 22, 15, 24, 2, 1, 4, 10, 3, 1, 14, 1, 6, 12

Offset: 2

Gus Wiseman, Jul 02 2018

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

			63 is the Heinz number of (4,2,2), which has LCM 4 and GCD 2, so a(63) = 4/2 = 2.
91 is the Heinz number of (6,4), which has LCM 12 and GCD 2, so a(91) = 12/2 = 6.

Cf. A005117, A056239, A074761, A289508, A289509, A290103, A296150, A316429, A316430, A316437.

Table[With[{pms=Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]},LCM@@pms/GCD@@pms],{n,2,100}]

A316431(n) = if(1==n,1,my(pis = apply(p -> primepi(p), factor(n)[, 1]~)); lcm(pis)/gcd(pis)); \\ Antti Karttunen, Sep 06 2018

a(n) = A290103(n)/A289508(n).

a(n) = a(A005117(n)). - David A. Corneth, Sep 06 2018

More terms from Antti Karttunen, Sep 06 2018

A353426 Number of integer partitions of n that are empty or a singleton or whose multiplicities are a sub-multiset that is already counted.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 3, 3, 5, 4, 6, 5, 6, 6, 7, 8, 10, 12, 12, 14, 13, 13, 18, 15, 16, 19, 20, 20, 32, 37, 53, 74, 105

Offset: 0

Gus Wiseman, May 16 2022

a(n) is number of integer partitions of n whose Heinz number belongs to A353393, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(n) partitions for selected n (A..M = 10..22):
  n=1: n=4:  n=14:     n=16:     n=17:     n=18:        n=22:
------------------------------------------------------------------
  (1)  (4)   (E)       (G)       (H)       (I)          (M)
       (22)  (5522)    (4444)    (652211)  (7722)       (9922)
             (532211)  (6622)    (742211)  (752211)     (972211)
                       (642211)  (832211)  (842211)     (A62211)
                       (732211)            (932211)     (B52211)
                                           (333222111)  (C42211)
                                                        (D32211)

The non-recursive version is A325702, ranked by A325755.
The version for compositions is A353391, non-recursive A353390.
These partitions are ranked by A353393, nonprime A353389.
A047966 counts uniform partitions, compositions A329738.
A239455 counts Look-and-Say partitions, ranked by A351294.
Cf. A000041, A002033, A074761, A181819, A181821, A323014, A325534, A333755.

oosQ[y_]:=Length[y]<=1||MemberQ[Subsets[Sort[y],{Length[Union[y]]}],Sort[Length/@Split[y]]]&&oosQ[Sort[Length/@Split[y]]];
Table[Length[Select[IntegerPartitions[n],oosQ]],{n,0,30}]

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).

A316429 Heinz numbers of integer partitions whose length is equal to their LCM.

Original entry on oeis.org

2, 6, 9, 20, 50, 56, 84, 125, 126, 176, 189, 196, 240, 294, 360, 416, 441, 540, 600, 624, 686, 810, 900, 936, 968, 1029, 1040, 1088, 1215, 1350, 1404, 1500, 1560, 2025, 2106, 2250, 2340, 2401, 2432, 2600, 2704, 3159, 3375, 3510, 3648, 3750, 3900, 4056, 5265

Offset: 1

Gus Wiseman, Jul 02 2018

A110295 is a subsequence.

			3750 is the Heinz number of (3,3,3,3,2,1), whose length and lcm are both 6.

David A. Corneth, Table of n, a(n) for n = 1..10822 (Terms <= 5 * 10^11)

Cf. A056239, A074761, A110295, A143773, A237984, A289508, A289509, A290103, A296150, A316413, A316428, A316430, A316431.

Select[Range[2,200],PrimeOmega[#]==LCM@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]]&]

heinz(n) = my(f=factor(n), pr=f[,1]~,exps=f[,2], res=vector(vecsum(exps)), t=0); for(i = 1, #pr, pr[i] = primepi(pr[i]); for(j=1, exps[i],t++; res[t] = pr[i])); res
is(n) = my(h = heinz(n)); lcm(h)==#h \\ David A. Corneth, Jul 05 2018

A256067 Irregular table T(n,k): the number of partitions of n where the least common multiple of all parts equals k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 0, 0, 1, 0, 1, 1, 4, 2, 4, 1, 5, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 4, 3, 4, 1, 7, 1, 1, 1, 2, 0, 2, 0, 1, 1, 0, 0, 0, 0, 1, 1, 5, 3, 6, 2, 9, 1, 2, 1, 3, 0, 4, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 3, 6, 2

Offset: 0

R. J. Mathar, Mar 18 2015

			The 5 partitions of n=4 are 1+1+1+1 (lcm=1), 1+1+2 (lcm=2), 2+2 (lcm=2), 1+3 (lcm=3) and 4 (lcm=4). So k=1, 3 and 4 appear once, k=2 appears twice.
The triangle starts:
  1 ;
  1 ;
  1  1;
  1  1  1;
  1  2  1  1;
  1  2  1  1  1  1;
  1  3  2  2  1  2;
  1  3  2  2  1  3  1  0  0  1  0  1;
  ...

Alois P. Heinz, Rows n = 0..30, flattened

Cf. A000041 (row sums), A000793 (row lengths), A213952, A074761 (diagonal), A074752 (6th column), A008642 (4th column), A002266 (5th column), A002264 (3rd column), A132270 (7th column), A008643 (8th column), A008649 (9th column), A258470 (10th column).

Cf. A009490 (number of nonzero terms of rows), A074064 (last elements of rows), A168532 (the same for gcd), A181844 (Sum k*T(n,k)).

A256067 := proc(n,k)
        local a,p ;
        a := 0 ;
        for p in combinat[partition](n) do
                ilcm(op(p)) ;
                if % = k then
                        a := a+1 ;
                end if;
        end do:
        a;
end proc:
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x,
      b(n, i-1)+(p-> add(coeff(p, x, t)*x^ilcm(t, i),
      t=1..degree(p)))(add(b(n-i*j, i-1), j=1..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 27 2015

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x, b[n, i-1] + Function[{p}, Sum[ Coefficient[p, x, t]*x^LCM[t, i], {t, 1, Exponent[p, x]}]][Sum[b[n-i*j, i-1], {j, 1, n/i}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jun 22 2015, after Alois P. Heinz *)

T(0,1)=1 prepended by Alois P. Heinz, Mar 27 2015

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

A316432 Number of integer partitions of n whose length is equal to the GCD of all parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 2, 0, 3, 2, 3, 0, 5, 0, 3, 4, 5, 0, 8, 1, 6, 6, 6, 0, 11, 0, 8, 10, 8, 2, 18, 0, 9, 14, 15, 0, 19, 0, 16, 21, 11, 0, 34, 1, 16, 24, 24, 0, 30, 10, 27, 30, 14, 0, 71, 0, 15, 34, 38, 18, 47, 0, 47, 44, 36, 0, 88, 0, 18, 79, 63, 5

Offset: 1

Gus Wiseman, Jul 02 2018

nonn

			The a(24) = 8 partitions:
(14,10), (22,2),
(9,9,6), (12,9,3), (15,6,3), (18,3,3),
(8,8,4,4), (12,4,4,4).

Cf. A000837, A067538, A074761, A289508, A289509, A290103, A316429, A316430, A316431, A316433.

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

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

cp's OEIS Frontend

A316413 Heinz numbers of integer partitions whose length divides their sum.

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A349156 Number of integer partitions of n whose mean is not an integer.

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A316428 Heinz numbers of integer partitions such that every part is divisible by the number of parts.

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Mathematica

A316431 Least common multiple divided by greatest common divisor of the integer partition with Heinz number n > 1.

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A353426 Number of integer partitions of n that are empty or a singleton or whose multiplicities are a sub-multiset that is already counted.

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Mathematica

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

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A316429 Heinz numbers of integer partitions whose length is equal to their LCM.

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Mathematica

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A256067 Irregular table T(n,k): the number of partitions of n where the least common multiple of all parts equals k.

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