cp's OEIS Frontend

A381540 Numbers appearing only once in A048767 (Look-and-Say partition of prime indices).

%I A381540 #7 Mar 02 2025 22:34:30
%S A381540 1,2,3,4,5,7,9,11,12,13,17,18,19,20,23,24,25,28,29,31,37,40,41,43,44,
%T A381540 45,47,48,49,50,52,53,54,56,59,61,63,67,68,71,72,73,75,76,79,80,83,88,
%U A381540 89,92,97,98,99,101,103,104,107,108,109,112,113,116,117,121
%N A381540 Numbers appearing only once in A048767 (Look-and-Say partition of prime indices).
%C A381540 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 A381540 The Look-and-Say partition of a multiset or partition y is obtained by interchanging parts with multiplicities. For example, starting with (3,2,2,1,1) we get (2,2,2,1,1,1), the multiset union of ((1,1,1),(2,2),(2)).
%C A381540 The conjugate of a Look-and-Say partition is a section-sum partition; see A381431, union A381432, count A239455.
%e A381540 The terms together with their prime indices begin:
%e A381540    1: {}
%e A381540    2: {1}
%e A381540    3: {2}
%e A381540    4: {1,1}
%e A381540    5: {3}
%e A381540    7: {4}
%e A381540    9: {2,2}
%e A381540   11: {5}
%e A381540   12: {1,1,2}
%e A381540   13: {6}
%e A381540   17: {7}
%e A381540   18: {1,2,2}
%e A381540   19: {8}
%e A381540   20: {1,1,3}
%e A381540   23: {9}
%e A381540   24: {1,1,1,2}
%t A381540 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A381540 hls[y_]:=Product[Prime[Count[y,x]]^x,{x,Union[y]}];
%t A381540 Select[Range[100],Count[hls/@IntegerPartitions[Total[prix[#]]],#]==1&]
%Y A381540 In A048767:
%Y A381540 - fixed points are A048768, A217605
%Y A381540 - conjugate is A381431, fixed points A000961, A000005
%Y A381540 - all numbers present are A351294, conjugate A381432
%Y A381540 - numbers missing are A351295, conjugate A381433
%Y A381540 - numbers appearing only once are A381540 (this), conjugate A381434
%Y A381540 - numbers appearing more than once are A381541, conjugate A381435
%Y A381540 A000040 lists the primes.
%Y A381540 A055396 gives least prime index, greatest A061395.
%Y A381540 A056239 adds up prime indices, row sums of A112798.
%Y A381540 A122111 represents conjugation in terms of Heinz numbers.
%Y A381540 A239455 counts Look-and-Say partitions, complement A351293.
%Y A381540 A381440 lists Look-and-Say partition of prime indices, conjugate A381436.
%Y A381540 Cf. A000720, A001222, A003557, A047966, A051903, A051904, A066328, A071178, A116861, A130091, A239964.
%K A381540 nonn
%O A381540 1,2
%A A381540 _Gus Wiseman_, Mar 02 2025