%I #5 Nov 15 2023 08:10:30

%S 3,5,7,10,11,13,14,17,19,22,23,25,26,27,28,29,31,34,35,37,38,41,43,44,

%T 46,47,49,52,53,55,58,59,61,62,63,65,67,68,71,73,74,76,77,79,82,83,85,

%U 86,88,89,91,92,94,95,97,98,99,101,103,104,106,107,109,113

%N Numbers m without a divisor whose prime indices sum to bigomega(m).

%C Also numbers m whose prime indices do not have a submultiset summing to bigomega(m).

%C 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 These are the Heinz numbers of the partitions counted by A367213.

%e The prime indices of 24 are {1,1,1,2} with submultiset {1,1,2} summing to 4, so 24 is not in the sequence.

%e The terms together with their prime indices begin:

%e 3: {2} 29: {10} 58: {1,10}

%e 5: {3} 31: {11} 59: {17}

%e 7: {4} 34: {1,7} 61: {18}

%e 10: {1,3} 35: {3,4} 62: {1,11}

%e 11: {5} 37: {12} 63: {2,2,4}

%e 13: {6} 38: {1,8} 65: {3,6}

%e 14: {1,4} 41: {13} 67: {19}

%e 17: {7} 43: {14} 68: {1,1,7}

%e 19: {8} 44: {1,1,5} 71: {20}

%e 22: {1,5} 46: {1,9} 73: {21}

%e 23: {9} 47: {15} 74: {1,12}

%e 25: {3,3} 49: {4,4} 76: {1,1,8}

%e 26: {1,6} 52: {1,1,6} 77: {4,5}

%e 27: {2,2,2} 53: {16} 79: {22}

%e 28: {1,1,4} 55: {3,5} 82: {1,13}

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

%t Select[Range[100], FreeQ[Total/@prix/@Divisors[#], PrimeOmega[#]]&]

%Y The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.

%Y sum-full sum-free comb-full comb-free

%Y -------------------------------------------

%Y partitions: A367212 A367213 A367218 A367219

%Y strict: A367214 A367215 A367220 A367221

%Y subsets: A367216 A367217 A367222 A367223

%Y ranks: A367224 A367225* A367226 A367227

%Y A000700 counts self-conjugate partitions, ranks A088902.

%Y A108917 counts knapsack partitions, ranks A299702, strict A275972.

%Y A112798 lists prime indices, reverse A296150, length A001222, sum A056239.

%Y A229816 counts partitions whose length is not a part, ranks A367107.

%Y A237667 counts sum-free partitions, ranks A364531.

%Y A365924 counts incomplete partitions, ranks A365830.

%Y Triangles:

%Y A046663 counts partitions of n without a subset-sum k, strict A365663.

%Y A365543 counts partitions of n with a subset-sum k, strict A365661.

%Y A365658 counts partitions by number of subset-sums, strict A365832.

%Y Cf. A000720, A055396, A061395, A106529, A288728, A304792, A325761, A325781, A364345, A364347.

%K nonn

%O 1,1

%A _Gus Wiseman_, Nov 15 2023