cp's OEIS Frontend

A381435 Numbers appearing more than once in A381431 (section-sum partition of prime indices).

%I A381435 #17 May 21 2025 10:43:15
%S A381435 5,7,11,13,17,19,23,25,26,29,31,34,37,38,39,41,43,46,47,49,51,52,53,
%T A381435 57,58,59,61,62,65,67,68,69,71,73,74,76,79,82,83,85,86,87,89,91,92,93,
%U A381435 94,95,97,101,103,104,106,107,109,111,113,115,116,117,118,119
%N A381435 Numbers appearing more than once in A381431 (section-sum partition of prime indices).
%C A381435 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 A381435 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 A381435 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 A381435 The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.
%F A381435 The complement is A381434 U A381433.
%e A381435 The terms together with their prime indices begin:
%e A381435    5: {3}
%e A381435    7: {4}
%e A381435   11: {5}
%e A381435   13: {6}
%e A381435   17: {7}
%e A381435   19: {8}
%e A381435   23: {9}
%e A381435   25: {3,3}
%e A381435   26: {1,6}
%e A381435   29: {10}
%e A381435   31: {11}
%e A381435   34: {1,7}
%e A381435   37: {12}
%e A381435   38: {1,8}
%e A381435   39: {2,6}
%e A381435   41: {13}
%e A381435   43: {14}
%e A381435   46: {1,9}
%e A381435   47: {15}
%e A381435   49: {4,4}
%e A381435   51: {2,7}
%e A381435   52: {1,1,6}
%t A381435 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A381435 egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
%t A381435 Select[Range[100],Count[Times@@Prime/@#&/@egs/@IntegerPartitions[Total[prix[#]]],#]>1&]
%Y A381435 In A381431:
%Y A381435 - fixed points are A000961, A000005
%Y A381435 - conjugate is A048767, fixed points A048768, A217605
%Y A381435 - all numbers present are A381432, conjugate A351294
%Y A381435 - numbers missing are A381433, conjugate A351295
%Y A381435 - numbers appearing only once are A381434, conjugate A381540
%Y A381435 - numbers appearing more than once are A381435 (this), conjugate A381541
%Y A381435 A000040 lists the primes, differences A001223.
%Y A381435 A055396 gives least prime index, greatest A061395.
%Y A381435 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A381435 A122111 represents conjugation in terms of Heinz numbers.
%Y A381435 A239455 counts section-sum partitions, complement A351293.
%Y A381435 A381436 lists section-sum partition of prime indices, conjugate A381440.
%Y A381435 Set multipartitions: A050320, A089259, A116540, A296119, A318360, A318361.
%Y A381435 Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
%Y A381435 Cf. A000720, A003557, A051903, A066328, A116861, A130091, A212166, A239964, A317081, A381437.
%K A381435 nonn
%O A381435 1,1
%A A381435 _Gus Wiseman_, Feb 27 2025