A381434 - OEIS

%I #11 Feb 28 2025 07:45:17

%S 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 77,80,81,88,98,99,100,112,128,130,135,160,170,175,176,182,190,195,

%U 196,200

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

%C 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 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 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 The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

%F The complement is A381433 U A381435.

%e The terms together with their prime indices begin:

%e     1: {}

%e     2: {1}

%e     3: {2}

%e     4: {1,1}

%e     8: {1,1,1}

%e     9: {2,2}

%e    10: {1,3}

%e    14: {1,4}

%e    15: {2,3}

%e    16: {1,1,1,1}

%e    20: {1,1,3}

%e    22: {1,5}

%e    27: {2,2,2}

%e    28: {1,1,4}

%e    32: {1,1,1,1,1}

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];

%t Select[Range[100],Count[Times@@Prime/@#&/@egs/@IntegerPartitions[Total[prix[#]]],#]==1&]

%Y In A381431:

%Y - fixed points are A000961, A000005

%Y - conjugate is A048767, fixed points A048768, A217605

%Y - all numbers present are A381432, conjugate A351294

%Y - numbers missing are A381433, conjugate A351295

%Y - numbers appearing only once are A381434 (this), conjugate A381540

%Y - numbers appearing more than once are A381435, conjugate A381541

%Y A000040 lists the primes, differences A001223.

%Y A055396 gives least prime index, greatest A061395.

%Y A056239 adds up prime indices, row sums of A112798, counted by A001222.

%Y A122111 represents conjugation in terms of Heinz numbers.

%Y A239455 counts section-sum partitions, complement A351293.

%Y A381436 lists section-sum partition of prime indices, complement A381440.

%Y Set multipartitions: A050320, A089259, A116540, A296119, A318360, A318361.

%Y Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.

%Y Cf. A000720, A003557, A051903, A066328, A116861, A130091, A212166, A239964, A317081, A381437.

%K nonn,new

%O 1,2

%A _Gus Wiseman_, Feb 27 2025