A367227 - OEIS

%I #6 Nov 15 2023 08:06:12

%S 3,5,7,11,13,17,19,23,25,27,29,31,35,37,41,43,47,49,53,55,59,61,63,65,

%T 67,71,73,77,79,83,85,89,91,95,97,99,101,103,107,109,113,115,117,119,

%U 121,127,131,133,137,139,143,145,147,149,151,153,155,157,161,163

%N Numbers m whose prime indices have no nonnegative linear combination equal 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 A367219.

%e The prime indices of 24 are {1,1,1,2} with (1+1+1+1) = 4 or (1+1)+(2) = 4 or (2+2) = 4, so 24 is not in the sequence.

%e The terms together with their prime indices begin:

%e      3: {2}        43: {14}        85: {3,7}

%e      5: {3}        47: {15}        89: {24}

%e      7: {4}        49: {4,4}       91: {4,6}

%e     11: {5}        53: {16}        95: {3,8}

%e     13: {6}        55: {3,5}       97: {25}

%e     17: {7}        59: {17}        99: {2,2,5}

%e     19: {8}        61: {18}       101: {26}

%e     23: {9}        63: {2,2,4}    103: {27}

%e     25: {3,3}      65: {3,6}      107: {28}

%e     27: {2,2,2}    67: {19}       109: {29}

%e     29: {10}       71: {20}       113: {30}

%e     31: {11}       73: {21}       115: {3,9}

%e     35: {3,4}      77: {4,5}      117: {2,2,6}

%e     37: {12}       79: {22}       119: {4,7}

%e     41: {13}       83: {23}       121: {5,5}

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

%t combs[n_,y_]:=With[{s=Table[{k,i}, {k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];

%t Select[Range[100], combs[PrimeOmega[#], Union[prix[#]]]=={}&]

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

%Y A124506 appears to count combination-free subsets, differences of A326083.

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

%Y A304792 counts subset-sums of partitions, strict A365925.

%Y A365046 counts combination-full subsets, differences of A364914.

%Y Cf. A000720, A046663, A088314, A106529, A116861, A236912, A364345, A364346, A364347, A364350, A365073, A365312.

%K nonn

%O 1,1

%A _Gus Wiseman_, Nov 15 2023