cp's OEIS Frontend

A381541 Numbers appearing more than once in A048767 (Look-and-Say partition of prime indices).

%I A381541 #7 Mar 02 2025 22:45:57
%S A381541 8,16,27,32,64,81,96,125,128,144,160,192,216,224,243,256,288
%N A381541 Numbers appearing more than once in A048767 (Look-and-Say partition of prime indices).
%C A381541 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 A381541 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 A381541 The conjugate of a Look-and-Say partition is a section-sum partition; see A381431, union A381432, count A239455.
%e A381541 The terms together with their prime indices begin:
%e A381541     8: {1,1,1}
%e A381541    16: {1,1,1,1}
%e A381541    27: {2,2,2}
%e A381541    32: {1,1,1,1,1}
%e A381541    64: {1,1,1,1,1,1}
%e A381541    81: {2,2,2,2}
%e A381541    96: {1,1,1,1,1,2}
%e A381541   125: {3,3,3}
%e A381541   128: {1,1,1,1,1,1,1}
%e A381541   144: {1,1,1,1,2,2}
%e A381541   160: {1,1,1,1,1,3}
%e A381541   192: {1,1,1,1,1,1,2}
%e A381541   216: {1,1,1,2,2,2}
%e A381541   224: {1,1,1,1,1,4}
%e A381541   243: {2,2,2,2,2}
%e A381541   256: {1,1,1,1,1,1,1,1}
%e A381541   288: {1,1,1,1,1,2,2}
%e A381541 For example, the term 96 appears in A048767 at positions 44 and 60, with prime indices:
%e A381541   44: {1,1,5}
%e A381541   60: {1,1,2,3}
%t A381541 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A381541 hls[y_]:=Product[Prime[Count[y,x]]^x,{x,Union[y]}];
%t A381541 Select[Range[100],Count[hls/@IntegerPartitions[Total[prix[#]]],#]>1&]
%Y A381541 In A048767:
%Y A381541 - fixed points are A048768, A217605
%Y A381541 - conjugate is A381431, fixed points A000961, A000005
%Y A381541 - all numbers present are A351294, conjugate A381432
%Y A381541 - numbers missing are A351295, conjugate A381433
%Y A381541 - numbers appearing only once are A381540, conjugate A381434
%Y A381541 - numbers appearing more than once are A381541 (this), conjugate A381435
%Y A381541 A000040 lists the primes.
%Y A381541 A055396 gives least prime index, greatest A061395.
%Y A381541 A056239 adds up prime indices, row sums of A112798.
%Y A381541 A122111 represents conjugation in terms of Heinz numbers.
%Y A381541 A239455 counts Look-and-Say partitions, complement A351293.
%Y A381541 A381440 lists Look-and-Say partitions of prime indices, conjugate A381436.
%Y A381541 Cf. A000720, A001222, A003557, A047966, A051903, A051904, A066328, A071178, A116861, A130091, A239964.
%K A381541 nonn,more
%O A381541 1,1
%A A381541 _Gus Wiseman_, Mar 02 2025