A112798 -id:A112798 - cp's OEIS frontend

A355744 Number of multisets that can be obtained by choosing a prime factor of each prime index of n.

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

Offset: 1

Gus Wiseman, Jul 18 2022

nonn

First differs from A355741 at a(169) = 3, A355741(169) = 4.

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(169) = 3 multisets are: {2,2}, {2,3}, {3,3}.
The a(507) = 3 multisets are: {2,2,2}, {2,2,3}, {2,3,3}.

Wikipedia, Axiom of choice.

Choosing from all divisors gives A355733, firsts A355734.
Counting sequences instead of multisets gives A355741.
Choosing weakly increasing sequences of divisors gives A355745.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A324850 lists numbers divisible by the product of their prime indices.
A344606 counts alternating permutations of prime indices.
Cf. A000720, A076610, A120383, A289509, A335433, A340852, A355731, A355735, A355737, A355739, A355740.

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

A048768 Numbers n such that A048767(n) = n.

Original entry on oeis.org

1, 2, 9, 12, 18, 40, 112, 125, 250, 352, 360, 675, 832, 1008, 1125, 1350, 1500, 2176, 2250, 2401, 3168, 3969, 4802, 4864, 7488, 7938, 11776, 14000, 19584, 21609, 28812, 29403, 29696, 43218, 43776, 44000, 58806, 63488, 75600, 96040, 104000, 105984, 123201, 126000

Offset: 1

Naohiro Nomoto

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions that are fixed points under the map described in A217605 (which interchanges the parts with their multiplicities). The enumeration of these partitions by sum is given by A217605. - Gus Wiseman, May 04 2019

			12 = (2^2)*(3^1) = (2nd prime)^pi(2) * (first prime)^pi(3).
From _Gus Wiseman_, May 04 2019: (Start)
The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     9: {2,2}
    12: {1,1,2}
    18: {1,2,2}
    40: {1,1,1,3}
   112: {1,1,1,1,4}
   125: {3,3,3}
   250: {1,3,3,3}
   352: {1,1,1,1,1,5}
   360: {1,1,1,2,2,3}
   675: {2,2,2,3,3}
   832: {1,1,1,1,1,1,6}
  1008: {1,1,1,1,2,2,4}
  1125: {2,2,3,3,3}
  1350: {1,2,2,2,3,3}
  1500: {1,1,2,3,3,3}
  2176: {1,1,1,1,1,1,1,7}
  2250: {1,2,2,3,3,3}
  2401: {4,4,4,4}
(End)

Amiram Eldar, Table of n, a(n) for n = 1..128

A subsequence of A130091.

Cf. A008478, A048767, A048769, A056239, A098859, A109297, A109298, A112798, A118914, A217605, A325326, A325368.

wt[n_]:=Times@@Cases[FactorInteger[n],{p_,k_}:>Prime[k]^PrimePi[p]];
Select[Range[1000],wt[#]==#&] (* Gus Wiseman, May 04 2019 *)

is(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); #Set(e) == #e && prod(i = 1, #e, prime(e[i])^primepi(p[i])) == n;} \\ Amiram Eldar, Oct 20 2023

a(1) inserted and more terms added by Amiram Eldar, Oct 20 2023

A241909 Self-inverse permutation of natural numbers: a(1)=1, a(p_i) = 2^i, and if n = p_i1 * p_i2 * p_i3 * ... * p_{ik-1} * p_ik, where p's are primes, with their indexes are sorted into nondescending order: i1 <= i2 <= i3 <= ... <= i_{k-1} <= ik, then a(n) = 2^(i1-1) * 3^(i2-i1) * 5^(i3-i2) * ... * p_k^(1+(ik-i_{k-1})). Here k = A001222(n) and ik = A061395(n).

Original entry on oeis.org

1, 2, 4, 3, 8, 9, 16, 5, 6, 27, 32, 25, 64, 81, 18, 7, 128, 15, 256, 125, 54, 243, 512, 49, 12, 729, 10, 625, 1024, 75, 2048, 11, 162, 2187, 36, 35, 4096, 6561, 486, 343, 8192, 375, 16384, 3125, 50, 19683, 32768, 121, 24, 45, 1458, 15625, 65536, 21, 108, 2401

Offset: 1

Antti Karttunen, May 03 2014, partly inspired by Marc LeBrun's Jan 11 2006 message on SeqFan mailing list

nonn

This permutation maps between the partitions as ordered in A112798 and A241918 (the original motivation for this sequence).
For all n > 2, A007814(a(n)) = A055396(n)-1, which implies that this self-inverse permutation maps between primes (A000040) and the powers of two larger than one (A000079(n>=1)), and apart from a(1) & a(2), this also maps each even number to some odd number, and vice versa, which means there are no fixed points after 2.
A122111 commutes with this one, that is, a(n) = A122111(a(A122111(n))).
Conjugates between A243051 and A242424 and other rows of A243060 and A243070.

			For n = 12 = 2 * 2 * 3 = p_1 * p_1 * p_2, we obtain by the first formula 2^(1-1) * 3^(1-1) * 5^(1+(2-1)) = 5^2 = 25. By the second formula, as n = 2^2 * 3^1, we obtain the same result, p_{1+2} * p_{2+1} = p_3 * p_3 = 25, thus a(12) = 25.
Using the product formula over the terms of row n of table A241918, we see, because 9450 = 2*3*3*3*5*5*7 = p_1^1 * p_2^3 * p_3^2 * p_4^1, that the corresponding row in A241918 is {2,5,7,7}, and multiplying p_2 * p_5 * p_7^2 yields 3 * 11 * 17 * 17 = 9537, thus a(9450) = 9537.
Similarly, for 9537, the corresponding row in A241918 is {1,2,2,2,3,3,4}, and multiplying p_1^1 * p_2^3 * p_3^2 * p_4^1, we obtain 9450 back.

Antti Karttunen, Table of n, a(n) for n = 1..1024
A. Karttunen, A few notes on A122111, A241909 & A241916.
Index entries for sequences that are permutations of the natural numbers

Cf. A203623, A241918, A242378, A007814, A064989, A064216, A001222, A061395, A125976, A243060, A243070.
Cf. also A278220 (= A046523(a(n))), A331280 (its rgs_transform), A331299 (= min(n,a(n))).
{A000027, A122111, A241909, A241916} form a 4-group.
Cf. A049084, A027746.

a241909 1 = 1
a241909 n = product $ zipWith (^) a000040_list $ zipWith (-) is (1 : is)
            where is = reverse ((j + 1) : js)
                  (j:js) = reverse $ map a049084 $ a027746_row n
-- Reinhard Zumkeller, Aug 04 2014

Array[If[# == 1, 1, Function[t, Times @@ Prime@ Accumulate[If[Length@ t < 2, {0}, Join[{1}, ConstantArray[0, Length@ t - 2], {-1}]] + ReplacePart[t, Map[#1 -> #2 & @@ # &, #]]]]@ ConstantArray[0, Transpose[#][[1, -1]]] &[FactorInteger[#] /. {p_, e_} /; p > 0 :> {PrimePi@ p, e}]] &, 56] (* Michael De Vlieger, Jan 23 2020 *)

A241909(n) = if(1==n||isprime(n),2^primepi(n),my(f=factor(n),h=1,i,m=1,p=1,k=1); while(k<=#f~, p = nextprime(1+p); i = primepi(f[k,1]); m *= p^(i-h); h = i; if(f[k,2]>1, f[k,2]--, k++)); (p*m)); \\ Antti Karttunen, Jan 17 2020

If n is a prime with index i (p_i), then a(n) = 2^i, otherwise when n = p_i1 * p_i2 * p_i3 * ... p_ik, where p_i1, p_i2, p_i3, ..., p_ik are the primes present (not necessarily all distinct) in the prime factorization of n, sorted into nondescending order, a(n) = 2^(i1-1) * 3^(i2-i1) * 5^(i3-i2) * ... * p_k^(1+(ik-i_{k-1})).
Equally, if n = 2^k, then a(n) = p_k, otherwise, when n = 2^e1 * 3^e2 * 5^e3 * ... * p_k^{e_k}, i.e., where e1 ... e_k are the exponents (some of them possibly zero, except the last) of the primes 2, 3, 5, ... in the prime factorization of n, a(n) = p_{1+e1} * p_{1+e1+e2} * p_{1+e1+e2+e3} * ... * p_{e1+e2+e3+...+e_k}.
From the equivalence of the above two formulas (which are inverses of each other) it follows that a(a(n)) = n, i.e., that this permutation is an involution. For a proof, please see the attached notes.
The first formula corresponds to this recurrence:
a(1) = 1, a(p_k) = 2^k for primes with index k, otherwise a(n) = (A000040(A001222(n))^(A241917(n)+1)) * A052126(a(A052126(n))).
And the latter formula with this recurrence:
a(1) = 1, and for n>1, if n = 2^k, a(n) = A000040(k), otherwise a(n) = A000040(A001511(n)) * A242378(A007814(n), a(A064989(n))).
[Here A242378(k,n) changes each prime p(i) in the prime factorization of n to p(i+k), i.e., it's the result of A003961 iterated k times starting from n.]
We also have:
a(1)=1, and for n>1, a(n) = Product_{i=A203623(n-1)+2..A203623(n)+1} A000040(A241918(i)).
For all n >= 1, A001222(a(n)) = A061395(n), and vice versa, A061395(a(n)) = A001222(n).
For all n > 1, a(2n-1) = 2*a(A064216(n)).

Typos in the name corrected by Antti Karttunen, May 31 2014

A324695 Lexicographically earliest sequence of positive integers whose prime indices are not already in the sequence.

Original entry on oeis.org

1, 3, 7, 9, 11, 13, 19, 21, 27, 29, 33, 37, 39, 43, 47, 49, 53, 57, 59, 61, 63, 71, 77, 79, 81, 83, 87, 89, 91, 97, 99, 101, 107, 111, 113, 117, 121, 127, 129, 131, 133, 139, 141, 143, 147, 149, 151, 159, 163, 169, 171, 173, 177, 179, 181, 183, 189, 193, 197

Offset: 1

Gus Wiseman, Mar 10 2019

nonn

A self-describing sequence, similar to A304360.

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 sequence of terms together with their prime indices begins:
   1: {}
   3: {2}
   7: {4}
   9: {2,2}
  11: {5}
  13: {6}
  19: {8}
  21: {2,4}
  27: {2,2,2}
  29: {10}
  33: {2,5}
  37: {12}
  39: {2,6}
  43: {14}
  47: {15}
  49: {4,4}
  53: {16}
  57: {2,8}
  59: {17}
  61: {18}
  63: {2,2,4}

Complement of A324694. Prime indices are A304360.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098.
Cf. A324696, A324697, A324698, A324699, A324700, A324701, A324702, A324703, A324704, A324705.

aQ[n_]:=And@@Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>!aQ[PrimePi[p]]];
Select[Range[100],aQ]

A326149 Numbers whose product of prime indices is divisible by their sum of prime indices.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 43, 47, 49, 53, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 108, 109, 113, 125, 127, 131, 137, 139, 149, 150, 151, 154, 157, 163, 165, 167, 169, 173, 179, 181, 190, 191, 193, 197

Offset: 1

Gus Wiseman, Jun 09 2019

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 Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose product of parts is divisible by their sum of parts. The enumeration of these partitions by sum is given by A057568.

			The sequence of terms together with their prime indices begins:
    2: {1}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   23: {9}
   29: {10}
   30: {1,2,3}
   31: {11}
   37: {12}
   41: {13}
   43: {14}
   47: {15}
   49: {4,4}
   53: {16}
   59: {17}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Satisfies A056239(a(n))|A003963(a(n)).
The nonprime case is A326150, with squarefree case A326158.
Cf. A000720, A001222, A057567, A057568, A112798, A301987.
Cf. A325037, A325042, A325044, A326151, A326153/A326154, A326155, A326156.

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

A026791 Triangle in which n-th row lists juxtaposed lexicographically ordered partitions of n; e.g., the partitions of 3 (1+1+1,1+2,3) appear as 1,1,1,1,2,3 in row 3.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

Differs from A080576 in a(18): Here, (...,1+3,2+2,4), there (...,2+2,1+3,4).
The representation of the partitions (for fixed n) is as (weakly) increasing lists of parts, the order between individual partitions (for the same n) is lexicographic (see example). - Joerg Arndt, Sep 03 2013
The equivalent sequence for compositions (ordered partitions) is A228369. - Omar E. Pol, Oct 19 2019

			First six rows are:
[[1]];
[[1, 1], [2]];
[[1, 1, 1], [1, 2], [3]];
[[1, 1, 1, 1], [1, 1, 2], [1, 3], [2, 2], [4]];
[[1, 1, 1, 1, 1], [1, 1, 1, 2], [1, 1, 3], [1, 2, 2], [1, 4], [2, 3], [5]];
[[1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 2], [1, 1, 1, 3], [1, 1, 2, 2], [1, 1, 4], [1, 2, 3], [1, 5], [2, 2, 2], [2, 4], [3, 3], [6]];
...
From _Omar E. Pol_, Sep 03 2013: (Start)
Illustration of initial terms:
----------------------------------
.                     Ordered
n  j      Diagram     partition j
----------------------------------
.               _
1  1           |_|    1;
.             _ _
2  1         | |_|    1, 1,
2  2         |_ _|    2;
.           _ _ _
3  1       | | |_|    1, 1, 1,
3  2       | |_ _|    1, 2,
3  3       |_ _ _|    3;
.         _ _ _ _
4  1     | | | |_|    1, 1, 1, 1,
4  2     | | |_ _|    1, 1, 2,
4  3     | |_ _ _|    1, 3,
4  4     |   |_ _|    2, 2,
4  5     |_ _ _ _|    4;
...
(End)

Alois P. Heinz, Rows n = 1..19, flattened
Wikiversity, Lexicographic and colexicographic order

Row lengths are given in A006128.
Partition lengths are in A193173.
Other partition orderings: A026792, A036037, A080577, A125106, A139100, A181087, A181317, A182937, A228100, A240837, A242628.
Row lengths are A000041.
Partition sums are A036042.
Partition minima are A196931.
Partition maxima are A194546.
The reflected version is A211992.
The length-sensitive version (sum/length/lex) is A036036.
The colexicographic version (sum/colex) is A080576.
The version for non-reversed partitions is A193073.
Compositions under the same ordering (sum/lex) are A228369.
The reverse-lexicographic version (sum/revlex) is A228531.
The Heinz numbers of these partitions are A334437.
Cf. A049085, A103921, A112798, A115623, A129129, A331581, A334435, A334439, A334442.

T:= proc(n) local b, ll;
      b:= proc(n,l)
            if n=0 then ll:= ll, l[]
          else seq(b(n-i, [l[], i]), i=`if`(l=[],1,l[-1])..n)
            fi
          end;
      ll:= NULL; b(n, []); ll
    end:
seq(T(n), n=1..8);  # Alois P. Heinz, Jul 16 2011

T[n0_] := Module[{b, ll}, b[n_, l_] := If[n == 0, ll = Join[ll, l], Table[ b[n - i, Append[l, i]], {i, If[l == {}, 1, l[[-1]]], n}]]; ll = {}; b[n0, {}]; ll]; Table[T[n], {n, 1, 8}] // Flatten (* Jean-François Alcover, Aug 05 2015, after Alois P. Heinz *)
Table[DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions[n]], x_ /; x == 0, 2], {n, 7}] // Flatten (* Robert Price, May 18 2020 *)

t = [[[]]]
for n in range(1, 10):
    p = []
    for minp in range(1, n):
        p += [[minp] + pp for pp in t[n-minp] if min(pp) >= minp]
    t.append(p + [[n]])
print(t)
# Andrey Zabolotskiy, Oct 18 2019

A299202 Moebius function of the multiorder of integer partitions indexed by their Heinz numbers.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 05 2018

sign

By convention, mu() = 0.

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

			Heinz number of (2,1,1) is 12, so mu(2,1,1) = a(12) = 2.

Gus Wiseman, Comcategories and Multiorders

Cf. A000041, A063834, A112798, A196545, A273873, A281145, A289501, A290261, A296150, A299200, A299201, A299203.

nn=120;
ptns=Table[If[n===1,{},Join@@Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]],{n,nn}];
tris=Join@@Map[Tuples[IntegerPartitions/@#]&,ptns];
mu[y_]:=mu[y]=If[Length[y]===1,1,-Sum[Times@@mu/@t,{t,Select[tris,And[Length[#]>1,Sort[Join@@#,Greater]===y]&]}]];
mu/@ptns

mu(y) = Sum_{g(t)=y} (-1)^d(t), where the sum is over all enriched p-trees (A289501, A299203) whose multiset of leaves is the integer partition y, and d(t) is the number of non-leaf nodes in t.

A305936 Irregular triangle whose n-th row is the multiset spanning an initial interval of positive integers with multiplicities equal to the n-th row of A296150 (the prime indices of n in weakly decreasing order).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 2, 2, 1

Offset: 1

Gus Wiseman, Aug 23 2018

			Row 90 is {1,1,1,2,2,3,3,4} because 90 = prime(3)*prime(2)*prime(2)*prime(1).
Triangle begins:
   1:
   2:  1
   3:  1  1
   4:  1  2
   5:  1  1  1
   6:  1  1  2
   7:  1  1  1  1
   8:  1  2  3
   9:  1  1  2  2
  10:  1  1  1  2
  11:  1  1  1  1  1
  12:  1  1  2  3
  13:  1  1  1  1  1  1

Row lengths are A056239. Number of distinct elements in row n is A001222(n). Number of distinct multiplicities in row n is A001221(n).
Cf. A000041, A000720, A112798, A181821, A182850, A255906, A296150, A304464.
Cf. A318283, A318284, A318285, A318286, A318287, A318360, A318361, A318362, A318371.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Array[nrmptn,30]

A325038 Heinz numbers of integer partitions whose sum of parts is greater than their product.

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 32, 34, 36, 38, 40, 44, 46, 48, 52, 56, 58, 60, 62, 64, 68, 72, 74, 76, 80, 82, 86, 88, 92, 94, 96, 104, 106, 112, 116, 118, 120, 122, 124, 128, 134, 136, 142, 144, 146, 148, 152, 158, 160, 164, 166, 168, 172

Offset: 1

Gus Wiseman, Mar 25 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose product of prime indices (A003963) is less than their sum of prime indices (A056239).

The enumeration of these partitions by sum is given by A096276 shifted once to the right.

			The sequence of terms together with their prime indices begins:
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  16: {1,1,1,1}
  18: {1,2,2}
  20: {1,1,3}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}
  32: {1,1,1,1,1}
  34: {1,7}
  36: {1,1,2,2}
  38: {1,8}
  40: {1,1,1,3}
  44: {1,1,5}
  46: {1,9}
  48: {1,1,1,1,2}

Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.

Cf. A325032, A325033, A325036, A325037, A325041, A325042, A325044.

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

A003963(a(n)) < A056239(a(n)).

a(n) = 2 * A325044(n).

A327476 Heinz numbers of integer partitions whose mean A326567/A326568 is not a part.

Original entry on oeis.org

1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 106

Offset: 1

Gus Wiseman, Sep 13 2019

nonn

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

			The sequence of terms together with their prime indices begins:
    1: {}
    6: {1,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   39: {2,6}
   40: {1,1,1,3}

Complement of A327473.
The enumeration of these partitions by sum is given by A327472.
Subsets whose mean is not an element are A327471.
Cf. A056239, A067538, A112798, A114639, A237984, A240851, A316413, A324756, A324758, A326567/A326568, A327477.

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

cp's OEIS Frontend

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A048768 Numbers n such that A048767(n) = n.

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A324695 Lexicographically earliest sequence of positive integers whose prime indices are not already in the sequence.

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A326149 Numbers whose product of prime indices is divisible by their sum of prime indices.

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A026791 Triangle in which n-th row lists juxtaposed lexicographically ordered partitions of n; e.g., the partitions of 3 (1+1+1,1+2,3) appear as 1,1,1,1,2,3 in row 3.

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A299202 Moebius function of the multiorder of integer partitions indexed by their Heinz numbers.

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A305936 Irregular triangle whose n-th row is the multiset spanning an initial interval of positive integers with multiplicities equal to the n-th row of A296150 (the prime indices of n in weakly decreasing order).

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