cp's OEIS Frontend

A381434 Numbers appearing only once in A381431 (section-sum partition of prime indices).

%I A381434 #14 Mar 02 2025 22:34:48
%S A381434 1,2,3,4,8,9,10,14,15,16,20,22,27,28,32,33,35,40,44,45,50,55,56,64,75,
%T A381434 77,80,81,88,98,99,100,112,128,130,135,160,170,175,176,182,190,195,
%U A381434 196,200
%N A381434 Numbers appearing only once in A381431 (section-sum partition of prime indices).
%C A381434 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 A381434 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 A381434 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 A381434 The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.
%F A381434 The complement is A381433 U A381435.
%e A381434 The terms together with their prime indices begin:
%e A381434     1: {}
%e A381434     2: {1}
%e A381434     3: {2}
%e A381434     4: {1,1}
%e A381434     8: {1,1,1}
%e A381434     9: {2,2}
%e A381434    10: {1,3}
%e A381434    14: {1,4}
%e A381434    15: {2,3}
%e A381434    16: {1,1,1,1}
%e A381434    20: {1,1,3}
%e A381434    22: {1,5}
%e A381434    27: {2,2,2}
%e A381434    28: {1,1,4}
%e A381434    32: {1,1,1,1,1}
%t A381434 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A381434 egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
%t A381434 Select[Range[100],Count[Times@@Prime/@#&/@egs/@IntegerPartitions[Total[prix[#]]],#]==1&]
%Y A381434 In A381431:
%Y A381434 - fixed points are A000961, A000005
%Y A381434 - conjugate is A048767, fixed points A048768, A217605
%Y A381434 - all numbers present are A381432, conjugate A351294
%Y A381434 - numbers missing are A381433, conjugate A351295
%Y A381434 - numbers appearing only once are A381434 (this), conjugate A381540
%Y A381434 - numbers appearing more than once are A381435, conjugate A381541
%Y A381434 A000040 lists the primes, differences A001223.
%Y A381434 A055396 gives least prime index, greatest A061395.
%Y A381434 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A381434 A122111 represents conjugation in terms of Heinz numbers.
%Y A381434 A239455 counts section-sum partitions, complement A351293.
%Y A381434 A381436 lists section-sum partition of prime indices, conjugate A381440.
%Y A381434 Set multipartitions: A050320, A089259, A116540, A296119, A318360, A318361.
%Y A381434 Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
%Y A381434 Cf. A000720, A003557, A051903, A066328, A116861, A130091, A212166, A239964, A317081, A381437.
%K A381434 nonn
%O A381434 1,2
%A A381434 _Gus Wiseman_, Feb 27 2025