cp's OEIS Frontend

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.

%I A381455 #19 Apr 01 2025 12:16:27
%S A381455 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,
%T A381455 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,
%U A381455 1,1,1,6,1,1,2,2,1,1,1,5,5,1,1,2,1,1,1
%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.
%C A381455 First differs from A000688 at a(144) = 9, A000688(144) = 10.
%C A381455 First differs from A295879 at a(128) = 15, A295879(128) = 13.
%C A381455 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.
%C A381455 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.
%C A381455 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).
%C A381455 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}.
%H A381455 Robert Price, <a href="/A381455/b381455.txt">Table of n, a(n) for n = 1..1000</a>
%F A381455 a(s) = 1 for any squarefree number s.
%F A381455 a(p^k) = A000041(k) for any prime p.
%e A381455 The prime indices of 36 are {1,1,2,2}, with the following 4 partitions into a multiset of constant multisets:
%e A381455   {{1,1},{2,2}}
%e A381455   {{1},{1},{2,2}}
%e A381455   {{2},{2},{1,1}}
%e A381455   {{1},{1},{2},{2}}
%e A381455 with block-sums: {2,4}, {1,1,4}, {2,2,2}, {1,1,2,2}, which are all different, so a(36) = 4.
%e A381455 The prime indices of 144 are {1,1,1,1,2,2}, with the following 10 partitions into a multiset of constant multisets:
%e A381455   {{2,2},{1,1,1,1}}
%e A381455   {{1},{2,2},{1,1,1}}
%e A381455   {{2},{2},{1,1,1,1}}
%e A381455   {{1,1},{1,1},{2,2}}
%e A381455   {{1},{1},{1,1},{2,2}}
%e A381455   {{1},{2},{2},{1,1,1}}
%e A381455   {{2},{2},{1,1},{1,1}}
%e A381455   {{1},{1},{1},{1},{2,2}}
%e A381455   {{1},{1},{2},{2},{1,1}}
%e A381455   {{1},{1},{1},{1},{2},{2}}
%e A381455 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.
%e A381455 The a(n) partitions for n = 4, 8, 16, 32, 36, 64, 72, 128:
%e A381455   (2)   (3)    (4)     (5)      (42)    (6)       (43)     (7)
%e A381455   (11)  (21)   (22)    (32)     (222)   (33)      (322)    (43)
%e A381455         (111)  (31)    (41)     (411)   (42)      (421)    (52)
%e A381455                (211)   (221)    (2211)  (51)      (2221)   (61)
%e A381455                (1111)  (311)            (222)     (4111)   (322)
%e A381455                        (2111)           (321)     (22111)  (331)
%e A381455                        (11111)          (411)              (421)
%e A381455                                         (2211)             (511)
%e A381455                                         (3111)             (2221)
%e A381455                                         (21111)            (3211)
%e A381455                                         (111111)           (4111)
%e A381455                                                            (22111)
%e A381455                                                            (31111)
%e A381455                                                            (211111)
%e A381455                                                            (1111111)
%t A381455 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t A381455 sqfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[sqfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
%t A381455 Table[Length[Union[Sort[hwt/@#]&/@sqfacs[n]]],{n,100}]
%Y A381455 Before taking sums we had A000688.
%Y A381455 Positions of 1 are A005117.
%Y A381455 There is a chain from the prime indices of n to a singleton iff n belongs to A300273.
%Y A381455 The lower version is A381453.
%Y A381455 For distinct blocks we have A381715, before sum A050361.
%Y A381455 For distinct block-sums we have A381716, before sums A381635 (zeros A381636).
%Y A381455 Other multiset partitions of prime indices:
%Y A381455 - For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
%Y A381455 - For strict multiset partitions (A045778) see A381452.
%Y A381455 - For set multipartitions (A050320) see A381078 (upper), A381454 (lower).
%Y A381455 - For set systems (A050326) see A381441 (upper).
%Y A381455 - For strict multiset partitions with distinct sums (A321469) see A381637.
%Y A381455 - For set systems with distinct sums (A381633) see A381634, A293243.
%Y A381455 More on multiset partitions into constant blocks: A006171, A279784, A295935.
%Y A381455 A000041 counts integer partitions, strict A000009.
%Y A381455 A000040 lists the primes.
%Y A381455 A003963 gives product of prime indices.
%Y A381455 A055396 gives least prime index, greatest A061395.
%Y A381455 A056239 adds up prime indices, row sums of A112798.
%Y A381455 A122111 represents conjugation in terms of Heinz numbers.
%Y A381455 A265947 counts refinement-ordered pairs of integer partitions.
%Y A381455 Cf. A000720, A001222, A002577, A002846, A018818, A213242, A213427, A275870, A289078, A299200, A299202, A300385.
%K A381455 nonn
%O A381455 1,4
%A A381455 _Gus Wiseman_, Mar 06 2025