cp's OEIS Frontend

A381433 Heinz numbers of non section-sum partitions. Complement of A381431.

%I A381433 #6 Feb 28 2025 10:35:48
%S A381433 6,12,18,21,24,30,36,42,48,54,60,63,66,70,72,78,84,90,96,102,105,108,
%T A381433 110,114,120,126,132,138,140,144,147,150,154,156,162,165,168,174,180,
%U A381433 186,189,192,198,204,210,216,220,222,228,231,234,238,240,246,252,258
%N A381433 Heinz numbers of non section-sum partitions. Complement of A381431.
%C A381433 First differs from A364348, A364537, A350845 in not containing 65.
%C A381433 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%C A381433 The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
%C A381433 Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
%C A381433 The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.
%e A381433 The terms together with their prime indices begin:
%e A381433     6: {1,2}
%e A381433    12: {1,1,2}
%e A381433    18: {1,2,2}
%e A381433    21: {2,4}
%e A381433    24: {1,1,1,2}
%e A381433    30: {1,2,3}
%e A381433    36: {1,1,2,2}
%e A381433    42: {1,2,4}
%e A381433    48: {1,1,1,1,2}
%e A381433    54: {1,2,2,2}
%e A381433    60: {1,1,2,3}
%e A381433    63: {2,2,4}
%e A381433    66: {1,2,5}
%e A381433    70: {1,3,4}
%e A381433    72: {1,1,1,2,2}
%e A381433    78: {1,2,6}
%e A381433    84: {1,1,2,4}
%e A381433    90: {1,2,2,3}
%e A381433    96: {1,1,1,1,1,2}
%e A381433   102: {1,2,7}
%e A381433   105: {2,3,4}
%e A381433   108: {1,1,2,2,2}
%t A381433 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A381433 egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
%t A381433 Select[Range[100],!MemberQ[Times@@Prime/@#&/@egs/@IntegerPartitions[Total[prix[#]]],#]&]
%Y A381433 Partitions of this type are counted by A351293, complement A239455.
%Y A381433 The conjugate is A351295, union of A048767 (parts A381440, fixed A048768, A217605).
%Y A381433 The complement is A381432, union of A381431 (conjugate A351294, parts A381436).
%Y A381433 A000040 lists the primes, differences A001223.
%Y A381433 A055396 gives least prime index, greatest A061395.
%Y A381433 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A381433 A122111 represents conjugation in terms of Heinz numbers.
%Y A381433 Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
%Y A381433 Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
%Y A381433 Cf. A000720, A003557, A051903, A066328, A116861, A130091, A181819, A238745, A239964, A317081, A381437.
%K A381433 nonn
%O A381433 1,1
%A A381433 _Gus Wiseman_, Feb 27 2025