cp's OEIS Frontend

A367401 Numbers m such that bigomega(m) is not the sum of prime indices of any semiprime divisor of m.

%I A367401 #6 Nov 21 2023 10:38:34
%S A367401 1,2,3,5,6,7,8,9,10,11,13,14,15,16,17,19,20,21,22,23,24,25,26,27,28,
%T A367401 29,31,32,33,34,35,37,38,39,41,43,44,45,46,47,48,49,50,51,52,53,55,56,
%U A367401 57,58,59,61,62,63,64,65,67,68,69,70,71,72,73,74,75,76,77
%N A367401 Numbers m such that bigomega(m) is not the sum of prime indices of any semiprime divisor of m.
%C A367401 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 A367401 These are the Heinz numbers of the partitions counted by A367398.
%e A367401 60 has semiprime divisor 10 with prime indices {1,3} summing to 4 = bigomega(60), so 60 is not in the sequence.
%e A367401 The terms together with their prime indices begin:
%e A367401    1: {}
%e A367401    2: {1}
%e A367401    3: {2}
%e A367401    5: {3}
%e A367401    6: {1,2}
%e A367401    7: {4}
%e A367401    8: {1,1,1}
%e A367401    9: {2,2}
%e A367401   10: {1,3}
%e A367401   11: {5}
%e A367401   13: {6}
%e A367401   14: {1,4}
%e A367401   15: {2,3}
%e A367401   16: {1,1,1,1}
%e A367401   17: {7}
%e A367401   19: {8}
%e A367401   20: {1,1,3}
%t A367401 prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A367401 Select[Range[100], FreeQ[Total/@Subsets[prix[#],{2}], PrimeOmega[#]]&]
%Y A367401 The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
%Y A367401              sum-full  sum-free  comb-full comb-free semi-full semi-free
%Y A367401             -----------------------------------------------------------
%Y A367401 partitions:  A367212   A367213   A367218   A367219   A367394   A367398
%Y A367401 strict:      A367214   A367215   A367220   A367221   A367395   A367399
%Y A367401 subsets:     A367216   A367217   A367222   A367223   A367396   A367400
%Y A367401 ranks:       A367224   A367225   A367226   A367227   A367397   A367401*
%Y A367401 A002865 counts partitions w/ length, complement A229816, ranks A325761.
%Y A367401 A088809 and A093971 count subsets containing semi-sums.
%Y A367401 A236912 counts partitions with no semi-sum of the parts, ranks A364461.
%Y A367401 A237113 counts partitions with a semi-sum of the parts, ranks A364462.
%Y A367401 A366738 counts semi-sums of partitions, strict A366741.
%Y A367401 Triangles:
%Y A367401 A365381 counts subsets with a subset summing to k, complement A366320.
%Y A367401 A365541 counts subsets with a semi-sum k.
%Y A367401 A367404 counts partitions with a semi-sum k, strict A367405.
%Y A367401 Cf. A000700, A237667, A238628, A304792, A363225, A364272, A365543, A365658, A365918, A366740.
%K A367401 nonn
%O A367401 1,2
%A A367401 _Gus Wiseman_, Nov 21 2023