cp's OEIS Frontend

A344413 Numbers n whose sum of prime indices A056239(n) is even and is at least twice the number of prime factors A001222(n).

%I A344413 #20 Oct 10 2024 15:20:32
%S A344413 1,3,7,9,10,13,19,21,22,25,27,28,29,30,34,37,39,43,46,49,52,53,55,57,
%T A344413 61,62,63,66,70,71,75,76,79,81,82,84,85,87,88,89,90,91,94,100,101,102,
%U A344413 107,111,113,115,116,117,118,121,129,130,131,133,134,136,138
%N A344413 Numbers n whose sum of prime indices A056239(n) is even and is at least twice the number of prime factors A001222(n).
%C A344413 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 A344413 Also Heinz numbers of integer partitions of even numbers m with at most m/2 parts, counted by A209816 riffled with zeros, or A110618 with odd positions zeroed out.
%H A344413 Robert Israel, <a href="/A344413/b344413.txt">Table of n, a(n) for n = 1..10000</a>
%F A344413 Members m of A300061 such that A056239(m) >= 2*A001222(m).
%e A344413 The sequence of terms together with their prime indices begins:
%e A344413       1: {}          37: {12}        75: {2,3,3}
%e A344413       3: {2}         39: {2,6}       76: {1,1,8}
%e A344413       7: {4}         43: {14}        79: {22}
%e A344413       9: {2,2}       46: {1,9}       81: {2,2,2,2}
%e A344413      10: {1,3}       49: {4,4}       82: {1,13}
%e A344413      13: {6}         52: {1,1,6}     84: {1,1,2,4}
%e A344413      19: {8}         53: {16}        85: {3,7}
%e A344413      21: {2,4}       55: {3,5}       87: {2,10}
%e A344413      22: {1,5}       57: {2,8}       88: {1,1,1,5}
%e A344413      25: {3,3}       61: {18}        89: {24}
%e A344413      27: {2,2,2}     62: {1,11}      90: {1,2,2,3}
%e A344413      28: {1,1,4}     63: {2,2,4}     91: {4,6}
%e A344413      29: {10}        66: {1,2,5}     94: {1,15}
%e A344413      30: {1,2,3}     70: {1,3,4}    100: {1,1,3,3}
%e A344413      34: {1,7}       71: {20}       101: {26}
%e A344413 For example, 75 has 3 prime indices {2,3,3} with sum 8 >= 2*3, so 75 is in the sequence.
%p A344413 filter:= proc(n) local F,a,t;
%p A344413   F:= ifactors(n)[2];
%p A344413   a:= add((numtheory:-pi(t[1])-2)*t[2],t=F);
%p A344413   a::even and a >= 0
%p A344413 end proc:
%p A344413 select(filter, [$1..300]); # _Robert Israel_, Oct 10 2024
%t A344413 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A344413 Select[Range[100],EvenQ[Total[primeMS[#]]]&&PrimeOmega[#]<=Total[primeMS[#]]/2&]
%Y A344413 These are the Heinz numbers of partitions counted by A209816 and A110618.
%Y A344413 A subset of A300061 (sum of prime indices is even).
%Y A344413 The conjugate version appears to be A320924 (allowing odd weights: A322109).
%Y A344413 The case of equality is A340387.
%Y A344413 Allowing odd weights gives A344291.
%Y A344413 The 5-smooth case is A344295, or A344293 allowing odd weights.
%Y A344413 The opposite version allowing odd weights is A344296.
%Y A344413 The conjugate opposite version allowing odd weights is A344414.
%Y A344413 The case of equality in the conjugate case is A344415.
%Y A344413 The conjugate opposite version is A344416, counted by A000070.
%Y A344413 A001222 counts prime factors with multiplicity.
%Y A344413 A027187 counts partitions of even length, ranked by A028260.
%Y A344413 A056239 adds up prime indices, row sums of A112798.
%Y A344413 A058696 counts partitions of even numbers, ranked by A300061.
%Y A344413 A301987 lists numbers whose sum of prime indices equals their product.
%Y A344413 A330950 counts partitions of n with Heinz number divisible by n.
%Y A344413 A334201 adds up all prime indices except the greatest.
%Y A344413 Cf. A001414, A025065, A067538, A301988, A316413, A316428, A325037, A325038, A325044, A338914, A344294, A344297.
%K A344413 nonn
%O A344413 1,2
%A A344413 _Gus Wiseman_, May 19 2021