A279784 -id:A279784 - cp's OEIS frontend

A381635 Number of ways to partition the prime indices of n into constant blocks with distinct sums.

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

Offset: 1

Gus Wiseman, Mar 09 2025

nonn

First differs from A381716 at a(1728) = 5, A381716(1728) = 4.

Also the number of factorizations on n into prime powers > 1 with distinct sums of prime indices (A056239).

			The a(432) = 3 multiset partitions:
  {{2,2,2},{1,1,1,1}}
  {{1},{1,1,1},{2,2,2}}
  {{1},{2},{2,2},{1,1,1}}
Note {{2},{2,2},{1,1,1,1}} is not included, as it does not have distinct block-sums.

Without distinct block-sums we have A000688, after sums A381455 (upper), A381453 (lower).
For distinct blocks instead of sums we have A050361, after sums A381715.
For strict instead of constant we have A381633 (zeros A381806), after sums A381634.
Positions of 0 are A381636.
Taking block-sums (and sorting) gives A381716.
Other multiset partitions of prime indices:
More on multiset partitions into constant blocks: A006171, A279784, A295935.
A001055 counts multiset partitions, see A317141 (upper), A300383 (lower).
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A001222, A002846, A005117, A213242, A213385, A293511, A299202, A300385, A317142, A381870.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Table[Length[Select[pfacs[n],UnsameQ@@hwt/@#&]],{n,100}]

A295935 Number of twice-factorizations of n where the latter factorizations are constant, i.e., type (P,P,R).

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 1, 5, 3, 2, 1, 5, 1, 2, 2, 12, 1, 5, 1, 5, 2, 2, 1, 10, 3, 2, 5, 5, 1, 5, 1, 18, 2, 2, 2, 15, 1, 2, 2, 10, 1, 5, 1, 5, 5, 2, 1, 22, 3, 5, 2, 5, 1, 10, 2, 10, 2, 2, 1, 13, 1, 2, 5, 40, 2, 5, 1, 5, 2, 5, 1, 28, 1, 2, 5, 5, 2, 5, 1, 22, 12, 2, 1

Offset: 1

Gus Wiseman, Nov 29 2017

nonn

a(n) is also the number of ways to choose a perfect divisor of each factor in a factorization of n.

			The a(24) = 10 twice-factorizations are:
(2)*(2)*(2)*(3), (2)*(3)*(2*2), (3)*(2*2*2)
(2)*(2)*(6), (2*2)*(6),
(2)*(3)*(4),
(2)*(12),
(3)*(8),
(4)*(6),
(24).

Cf. A000005, A001055, A052409, A052410, A089723, A279784, A281113, A284639, A295923, A295924, A295931.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Product[Length[Divisors[GCD@@FactorInteger[d][[All,2]]]],{d,f}],{f,facs[n]}],{n,100}]

Dirichlet g.f.: 1/Product_{n > 1}(1 - A089723(n)/n^s).

A381716 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into constant blocks with distinct sums.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 10 2025

nonn

First differs from A381635 at a(1728) = 4, A381635(1728) = 5.

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 prime indices of 1728 are {1,1,1,1,1,1,2,2,2}, with multiset partitions into constant multisets with distinct sums:
  {{1,1,1,1,1,1},{2,2},{2}}
  {{1,1,1,1,1},{1},{2,2,2}}
  {{1,1,1,1,1},{1},{2,2},{2}}
  {{1,1,1,1},{1,1},{2,2,2}}
  {{1,1,1},{1,1},{1},{2,2,2}}
with block-sums: {1,5,6}, {2,4,6}, {1,2,3,6}, {1,2,4,5}, so a(1728) = 4.

Without distinct sums we have A000688, after sums A381455 (upper), A381453 (lower).
More on multiset partitions into constant blocks: A006171, A279784, A295935.
For strict instead of constant we have A381633, before sums A381634.
Before taking sums we had A381635.
Positions of 0 are A381636.
For distinct blocks instead of sums we have A381715.
A001055 counts multiset partitions, see A317141 (upper), A300383 (lower).
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A001222, A002846, A005117, A050361, A213242, A265947, A293511, A299202, A300385, A362421.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mce[y_]:=Table[ConstantArray[y[[1]],#]&/@ptn,{ptn,IntegerPartitions[Length[y]]}];
Table[Length[Union[Sort[Total/@#]&/@Select[Join@@@Tuples[mce/@Split[prix[n]]],UnsameQ@@Total/@#&]]],{n,100}]

A323583 Number of ways to split an integer partition of n into consecutive subsequences.

Original entry on oeis.org

1, 1, 3, 7, 17, 37, 83, 175, 373, 773, 1603, 3275, 6693, 13557, 27447, 55315, 111397, 223769, 449287, 900795, 1805465, 3615929, 7240327, 14491623, 29001625, 58027017, 116093259, 232237583, 464558201, 929224589, 1858623819, 3717475031, 7435314013, 14871103069

Offset: 0

Gus Wiseman, Jan 19 2019

nonn

			The a(3) = 7 ways to split an integer partition of 3 into consecutive subsequences are (3), (21), (2)(1), (111), (11)(1), (1)(11), (1)(1)(1).

Cf. A006951, A070933, A100883, A279784, A279786, A323433, A323582.

b:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, 2*b(n-i, i))))
    end:
a:= n-> ceil(b(n$2)):
seq(a(n), n=0..33);  # Alois P. Heinz, Jan 01 2023

Table[Sum[2^(Length[ptn]-1),{ptn,IntegerPartitions[n]}],{n,40}]
(* Second program: *)
(1/2) CoefficientList[1 - 1/QPochhammer[2, x] + O[x]^100 , x] (* Jean-François Alcover, Jan 02 2022, after Vladimir Reshetnikov in A070933 *)

a(n) = A070933(n)/2.
O.g.f.: (1/2)*Product_{n >= 1} 1/(1 - 2*x^n).
G.f.: 1 + Sum_{k>=1} 2^(k - 1) * x^k / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Jan 28 2020

A355743 Numbers whose prime indices are all prime-powers.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25, 27, 31, 33, 35, 41, 45, 49, 51, 53, 55, 57, 59, 63, 67, 69, 75, 77, 81, 83, 85, 93, 95, 97, 99, 103, 105, 109, 115, 119, 121, 123, 125, 127, 131, 133, 135, 147, 153, 155, 157, 159, 161, 165, 171, 175, 177, 179, 187

Offset: 1

Gus Wiseman, Jul 24 2022

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.

Also MM-numbers of multiset partitions into constant multisets, where the multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The terms together with their prime indices begin:
   1: {}
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  15: {2,3}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  31: {11}
  33: {2,5}
  35: {3,4}
  41: {13}
  45: {2,2,3}

Robert Price, Table of n, a(n) for n = 1..1410

The multiplicative version is A000688, strict A050361, coprime A354911.
The case of only primes (not all prime-powers) is A076610, strict A302590.
Allowing prime index 1 gives A302492.
These are the products of elements of A302493.
Requiring n to be a prime-power gives A302601.
These are the positions of 1's in A355741.
The squarefree case is A356065.
The complement is A356066.
A001222 counts prime-power divisors.
A023894 counts ptns into prime-powers, strict A054685, with 1's A023893.
A034699 gives maximal prime-power divisor.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A355742 chooses a prime-power divisor of each prime index.
Cf. A085970, A106244, A279784, A295935, A355731, A356064.

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

A355742 Number of ways to choose a sequence of prime-power divisors, one of each prime index of n. Product of bigomega over the prime indices of n, with multiplicity.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 20 2022

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 prime indices of 49 are {4,4}, and the a(49) = 4 choices are: (2,2), (2,4), (4,2), (4,4).
The prime indices of 777 are {2,4,12}, and the a(777) = 6 choices are: (2,2,2), (2,2,3), (2,2,4), (2,4,2), (2,4,3), (2,4,4).

Wikipedia, Cartesian product.

The unordered version is A001970, row-sums of A061260.
Positions of 1's are A076610, just primes A355743.
Positions of 0's are A299174.
Allowing all divisors (not just primes) gives A355731, firsts A355732.
Choosing only prime factors (not prime-powers) gives A355741.
Counting multisets of primes gives A355744.
The case of weakly increasing primes A355745, all divisors A355735.
A000688 counts factorizations into prime powers.
A001414 adds up distinct prime factors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Cf. A000720, A120383, A279784, A289509, A324850, A355733, A355739, A355746.

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

Totally multiplicative with a(prime(k)) = A001222(k).

A381453 Number of multisets that can be obtained by choosing a constant integer partition of each prime index of n and taking the multiset union.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 08 2025

nonn

First differs from A355733 and A355735 at a(21) = 6, A355733(21) = A355735(21) = 5.
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.
A multiset partition can be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
Multisets of constant multisets are generally not transitive. For example, we have arrows: {{1,1},{2}}: {1,1,2} -> {2,2} and {{2,2}}: {2,2} -> {4}, but there is no multiset of constant multisets {1,1,2} -> {4}.

			The a(21) = 6 multisets are: {2,4}, {1,1,4}, {2,2,2}, {1,1,2,2}, {2,1,1,1,1}, {1,1,1,1,1,1}.
The a(n) partitions for n = 1, 3, 7, 13, 53, 21 (G = 16):
  ()  (2)   (4)     (6)       (G)                 (42)
      (11)  (22)    (33)      (88)                (411)
            (1111)  (222)     (4444)              (222)
                    (111111)  (22222222)          (2211)
                              (1111111111111111)  (21111)
                                                  (111111)

Robert Price, Table of n, a(n) for n = 1..300

Positions of 1 are A000079.
The strict case is A008966.
Before sorting we had A355731.
Choosing divisors instead of constant multisets gives A355733.
The upper version is A381455, before taking sums A000688.
Multiset partitions of prime indices:
- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
- For strict multiset partitions (A045778) see A381452.
- For set multipartitions (A050320) see A381078 (upper), A381454 (lower).
- For set systems (A050326, zeros A293243) see A381441 (upper).
- For sets of constant multisets (A050361) see A381715.
- For strict multiset partitions with distinct sums (A321469) see A381637.
- For set systems with distinct sums (A381633, zeros A381806) see A381634.
- For sets of constant multisets with distinct sums (A381635, zeros A381636) see A381716.
More on multiset partitions into constant blocks: A006171, A279784, A295935.
A000041 counts integer partitions, strict A000009.
A000040 lists the primes.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A000961, A001222, A002577, A018818, A213242, A213385, A213427, A275870, A299200, A300273, A300385.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@prix[n]]]],{n,nn}]

a(A002110(n)) = A381807(n).

A381455 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into a multiset of constant multisets.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Mar 06 2025

nonn

First differs from A000688 at a(144) = 9, A000688(144) = 10.
First differs from A295879 at a(128) = 15, A295879(128) = 13.
Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a factorization of n into prime powers > 1.
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.
A multiset partition can be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
Multisets of constant multisets are generally not transitive. For example, we have arrows: {{1,1},{2}}: {1,1,2} -> {2,2} and {{2,2}}: {2,2} -> {4}, but there is no multiset of constant multisets {1,1,2} -> {4}.

			The prime indices of 36 are {1,1,2,2}, with the following 4 partitions into a multiset of constant multisets:
  {{1,1},{2,2}}
  {{1},{1},{2,2}}
  {{2},{2},{1,1}}
  {{1},{1},{2},{2}}
with block-sums: {2,4}, {1,1,4}, {2,2,2}, {1,1,2,2}, which are all different, so a(36) = 4.
The prime indices of 144 are {1,1,1,1,2,2}, with the following 10 partitions into a multiset of constant multisets:
  {{2,2},{1,1,1,1}}
  {{1},{2,2},{1,1,1}}
  {{2},{2},{1,1,1,1}}
  {{1,1},{1,1},{2,2}}
  {{1},{1},{1,1},{2,2}}
  {{1},{2},{2},{1,1,1}}
  {{2},{2},{1,1},{1,1}}
  {{1},{1},{1},{1},{2,2}}
  {{1},{1},{2},{2},{1,1}}
  {{1},{1},{1},{1},{2},{2}}
with block-sums: {4,4}, {1,3,4}, {2,2,4}, {2,2,4}, {1,1,2,4}, {1,2,2,3}, {2,2,2,2}, {1,1,1,1,4}, {1,1,2,2,2}, {1,1,1,1,2,2}, of which 9 are distinct, so a(144) = 9.
The a(n) partitions for n = 4, 8, 16, 32, 36, 64, 72, 128:
  (2)   (3)    (4)     (5)      (42)    (6)       (43)     (7)
  (11)  (21)   (22)    (32)     (222)   (33)      (322)    (43)
        (111)  (31)    (41)     (411)   (42)      (421)    (52)
               (211)   (221)    (2211)  (51)      (2221)   (61)
               (1111)  (311)            (222)     (4111)   (322)
                       (2111)           (321)     (22111)  (331)
                       (11111)          (411)              (421)
                                        (2211)             (511)
                                        (3111)             (2221)
                                        (21111)            (3211)
                                        (111111)           (4111)
                                                           (22111)
                                                           (31111)
                                                           (211111)
                                                           (1111111)

Robert Price, Table of n, a(n) for n = 1..1000

Before taking sums we had A000688.
Positions of 1 are A005117.
There is a chain from the prime indices of n to a singleton iff n belongs to A300273.
The lower version is A381453.
For distinct blocks we have A381715, before sum A050361.
For distinct block-sums we have A381716, before sums A381635 (zeros A381636).
Other multiset partitions of prime indices:
- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
- For strict multiset partitions (A045778) see A381452.
- For set multipartitions (A050320) see A381078 (upper), A381454 (lower).
- For set systems (A050326) see A381441 (upper).
- For strict multiset partitions with distinct sums (A321469) see A381637.
- For set systems with distinct sums (A381633) see A381634, A293243.
More on multiset partitions into constant blocks: A006171, A279784, A295935.
A000041 counts integer partitions, strict A000009.
A000040 lists the primes.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A001222, A002577, A002846, A018818, A213242, A213427, A275870, A289078, A299200, A299202, A300385.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[sqfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Table[Length[Union[Sort[hwt/@#]&/@sqfacs[n]]],{n,100}]

a(s) = 1 for any squarefree number s.

a(p^k) = A000041(k) for any prime p.

A381871 Numbers whose prime indices cannot be partitioned into constant blocks having a common sum.

Original entry on oeis.org

6, 10, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 38, 39, 42, 44, 45, 46, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 110

Offset: 1

Gus Wiseman, Mar 13 2025

nonn

First differs from A383100 in lacking 108.
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, sum A056239.
Also numbers that cannot be written as a product of prime powers with equal sums of prime indices.
Partitions of this type are counted by A381993.

			The terms together with their prime indices begin:
    6: {1,2}
   10: {1,3}
   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}
   30: {1,2,3}

Constant blocks: A000688, A006171, A279784, A295935, A381453 (lower), A381455 (upper).
Constant blocks with distinct sums: A381635, A381716.
For distinct instead of equal sums we have A381636, counted by A381717.
Partitions of this type are counted by A381993, complement A383093.
These are the positions of 0 in A381995.
A001055 counts multiset partitions of prime indices, strict A045778.
A050361 counts multiset partitions into distinct constant blocks.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A317141 counts coarsenings of prime indices, refinements A300383.
Cf. A000720, A000961, A001222, A265947, A321469, A381633, A381715, A381719, A381806.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Select[Range[100],Select[mps[prix[#]],SameQ@@Total/@#&&And@@SameQ@@@#&]=={}&]

A302492 Products of any power of 2 with prime numbers of prime-power index, i.e., prime numbers p of the form p = prime(q^k), for q prime, k >= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 30, 31, 32, 33, 34, 35, 36, 38, 40, 41, 42, 44, 45, 46, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 62, 63, 64, 66, 67, 68, 69, 70, 72, 75, 76, 77, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Apr 08 2018

nonn

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

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of multiset multisystems.
01: {}
02: {{}}
03: {{1}}
04: {{},{}}
05: {{2}}
06: {{},{1}}
07: {{1,1}}
08: {{},{},{}}
09: {{1},{1}}
10: {{},{2}}
11: {{3}}
12: {{},{},{1}}
14: {{},{1,1}}
15: {{1},{2}}
16: {{},{},{},{}}
17: {{4}}
18: {{},{1},{1}}
19: {{1,1,1}}
20: {{},{},{2}}
21: {{1},{1,1}}
22: {{},{3}}
23: {{2,2}}
24: {{},{},{},{1}}

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

Cf. A000961, A001222, A003963, A005117, A007716, A050361, A056239, A076610, A270995, A275024, A279784, A281113, A295935, A301762, A302242, A302243, A302493.

Select[Range[100],Or[#===1,And@@PrimePowerQ/@PrimePi/@DeleteCases[FactorInteger[#][[All,1]],2]]&]

ok(n)={!#select(p->p<>2&&!isprimepower(primepi(p)), factor(n)[,1])} \\ Andrew Howroyd, Aug 26 2018

cp's OEIS Frontend

A381635 Number of ways to partition the prime indices of n into constant blocks with distinct sums.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A295935 Number of twice-factorizations of n where the latter factorizations are constant, i.e., type (P,P,R).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A381716 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into constant blocks with distinct sums.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A323583 Number of ways to split an integer partition of n into consecutive subsequences.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A355743 Numbers whose prime indices are all prime-powers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A355742 Number of ways to choose a sequence of prime-power divisors, one of each prime index of n. Product of bigomega over the prime indices of n, with multiplicity.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A381453 Number of multisets that can be obtained by choosing a constant integer partition of each prime index of n and taking the multiset union.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A381455 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into a multiset of constant multisets.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs