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A344291 Numbers whose sum of prime indices is at least twice their number of prime indices (counted with multiplicity).

%I A344291 #14 May 26 2021 02:30:17
%S A344291 1,3,5,7,9,10,11,13,14,15,17,19,21,22,23,25,26,27,28,29,30,31,33,34,
%T A344291 35,37,38,39,41,42,43,44,45,46,47,49,50,51,52,53,55,57,58,59,61,62,63,
%U A344291 65,66,67,68,69,70,71,73,74,75,76,77,78,79,81,82,83,84,85
%N A344291 Numbers whose sum of prime indices is at least twice their number of prime indices (counted with multiplicity).
%C A344291 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.
%F A344291 A056239(a(n)) >= 2*A001222(a(n)).
%e A344291 The sequence of terms together with their prime indices begins:
%e A344291      1: {}       25: {3,3}      43: {14}       62: {1,11}
%e A344291      3: {2}      26: {1,6}      44: {1,1,5}    63: {2,2,4}
%e A344291      5: {3}      27: {2,2,2}    45: {2,2,3}    65: {3,6}
%e A344291      7: {4}      28: {1,1,4}    46: {1,9}      66: {1,2,5}
%e A344291      9: {2,2}    29: {10}       47: {15}       67: {19}
%e A344291     10: {1,3}    30: {1,2,3}    49: {4,4}      68: {1,1,7}
%e A344291     11: {5}      31: {11}       50: {1,3,3}    69: {2,9}
%e A344291     13: {6}      33: {2,5}      51: {2,7}      70: {1,3,4}
%e A344291     14: {1,4}    34: {1,7}      52: {1,1,6}    71: {20}
%e A344291     15: {2,3}    35: {3,4}      53: {16}       73: {21}
%e A344291     17: {7}      37: {12}       55: {3,5}      74: {1,12}
%e A344291     19: {8}      38: {1,8}      57: {2,8}      75: {2,3,3}
%e A344291     21: {2,4}    39: {2,6}      58: {1,10}     76: {1,1,8}
%e A344291     22: {1,5}    41: {13}       59: {17}       77: {4,5}
%e A344291     23: {9}      42: {1,2,4}    61: {18}       78: {1,2,6}
%e A344291 For example, the prime indices of 45 are {2,2,3} with sum 7 >= 2*3, so 45 is in the sequence.
%t A344291 Select[Range[100],PrimeOmega[#]<=Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/2&]
%Y A344291 The partitions with these Heinz numbers are counted by A110618.
%Y A344291 The conjugate version is A322109.
%Y A344291 The case of equality is A340387, counted by A035363.
%Y A344291 The 5-smooth case is A344293, with non-3-smooth case A344294.
%Y A344291 The opposite version is A344296.
%Y A344291 The conjugate opposite version is A344414.
%Y A344291 The conjugate case of equality is A344415.
%Y A344291 A001221 counts distinct prime indices.
%Y A344291 A001222 counts prime indices with multiplicity.
%Y A344291 A056239 adds up prime indices, row sums of A112798.
%Y A344291 Cf. A001358, A025065, A084127, A244990, A344292.
%K A344291 nonn
%O A344291 1,2
%A A344291 _Gus Wiseman_, May 15 2021