cp's OEIS Frontend

A340610 Numbers whose number of prime factors (A001222) divides their greatest prime index (A061395).

%I A340610 #10 Feb 09 2021 02:45:59
%S A340610 2,3,5,6,7,9,11,13,14,17,19,20,21,23,26,29,30,31,35,37,38,39,41,43,45,
%T A340610 47,49,50,52,53,56,57,58,59,61,65,67,71,73,74,75,78,79,83,84,86,87,89,
%U A340610 91,92,95,97,101,103,106,107,109,111,113,117,122,125,126,127
%N A340610 Numbers whose number of prime factors (A001222) divides their greatest prime index (A061395).
%C A340610 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.
%H A340610 Robert Israel, <a href="/A340610/b340610.txt">Table of n, a(n) for n = 1..10000</a>
%F A340610 A001222(a(n)) divides A061395(a(n)).
%e A340610 The sequence of terms together with their prime indices begins:
%e A340610      2: {1}        29: {10}       56: {1,1,1,4}
%e A340610      3: {2}        30: {1,2,3}    57: {2,8}
%e A340610      5: {3}        31: {11}       58: {1,10}
%e A340610      6: {1,2}      35: {3,4}      59: {17}
%e A340610      7: {4}        37: {12}       61: {18}
%e A340610      9: {2,2}      38: {1,8}      65: {3,6}
%e A340610     11: {5}        39: {2,6}      67: {19}
%e A340610     13: {6}        41: {13}       71: {20}
%e A340610     14: {1,4}      43: {14}       73: {21}
%e A340610     17: {7}        45: {2,2,3}    74: {1,12}
%e A340610     19: {8}        47: {15}       75: {2,3,3}
%e A340610     20: {1,1,3}    49: {4,4}      78: {1,2,6}
%e A340610     21: {2,4}      50: {1,3,3}    79: {22}
%e A340610     23: {9}        52: {1,1,6}    83: {23}
%e A340610     26: {1,6}      53: {16}       84: {1,1,2,4}
%p A340610 filter:= proc(n) local F,m,g,t;
%p A340610   F:= ifactors(n)[2];
%p A340610   m:= add(t[2],t=F);
%p A340610   g:= numtheory:-pi(max(seq(t[1],t=F)));
%p A340610   g mod m = 0;
%p A340610 end proc:
%p A340610 select(filter, [$2..1000]); # _Robert Israel_, Feb 08 2021
%t A340610 Select[Range[2,100],Divisible[PrimePi[FactorInteger[#][[-1,1]]],PrimeOmega[#]]&]
%Y A340610 Note: Heinz numbers are given in parentheses below.
%Y A340610 The case of equality is A047993 (A106529).
%Y A340610 The case where all parts are multiples, not just the maximum part, is A143773 (A316428), with strict case A340830, while the case of factorizations is A340853.
%Y A340610 These are the Heinz numbers of certain partitions counted by A168659.
%Y A340610 The reciprocal version is A340609.
%Y A340610 The squarefree case is A340828 (A340856).
%Y A340610 A001222 counts prime factors.
%Y A340610 A006141 counts partitions whose length equals their minimum (A324522).
%Y A340610 A056239 adds up prime indices.
%Y A340610 A061395 selects the maximum prime index.
%Y A340610 A067538 counts partitions whose length divides their sum (A316413).
%Y A340610 A067538 counts partitions whose maximum divides their sum (A326836).
%Y A340610 A112798 lists the prime indices of each positive integer.
%Y A340610 A200750 counts partitions with length coprime to maximum (A340608).
%Y A340610 A325134 = A001222 + A061395.
%Y A340610 A326845 = A056239 * A061395.
%Y A340610 Cf. A244990/A244991, A326837, A326849 (A326848), A340653, A340691, A340693 (A340606), A340787/A340788.
%K A340610 nonn
%O A340610 1,1
%A A340610 _Gus Wiseman_, Jan 27 2021