cp's OEIS Frontend

A340609 Numbers whose number of prime factors (A001222) is divisible by their greatest prime index (A061395).

%I A340609 #11 Feb 09 2021 02:45:50
%S A340609 2,4,6,8,9,16,20,24,30,32,36,45,50,54,56,64,75,81,84,96,125,126,128,
%T A340609 140,144,160,176,189,196,210,216,240,256,264,294,315,324,350,360,384,
%U A340609 396,400,416,440,441,486,490,512,525,540,576,594,600,616,624,660,686
%N A340609 Numbers whose number of prime factors (A001222) is divisible by their greatest prime index (A061395).
%C A340609 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 A340609 If n is a term, then so is n^k for k > 1. - _Robert Israel_, Feb 08 2021
%H A340609 Robert Israel, <a href="/A340609/b340609.txt">Table of n, a(n) for n = 1..10000</a>
%F A340609 A061395(a(n)) divides A001222(a(n)).
%e A340609 The sequence of terms together with their prime indices begins:
%e A340609       2: {1}             64: {1,1,1,1,1,1}      216: {1,1,1,2,2,2}
%e A340609       4: {1,1}           75: {2,3,3}            240: {1,1,1,1,2,3}
%e A340609       6: {1,2}           81: {2,2,2,2}          256: {1,1,1,1,1,1,1,1}
%e A340609       8: {1,1,1}         84: {1,1,2,4}          264: {1,1,1,2,5}
%e A340609       9: {2,2}           96: {1,1,1,1,1,2}      294: {1,2,4,4}
%e A340609      16: {1,1,1,1}      125: {3,3,3}            315: {2,2,3,4}
%e A340609      20: {1,1,3}        126: {1,2,2,4}          324: {1,1,2,2,2,2}
%e A340609      24: {1,1,1,2}      128: {1,1,1,1,1,1,1}    350: {1,3,3,4}
%e A340609      30: {1,2,3}        140: {1,1,3,4}          360: {1,1,1,2,2,3}
%e A340609      32: {1,1,1,1,1}    144: {1,1,1,1,2,2}      384: {1,1,1,1,1,1,1,2}
%e A340609      36: {1,1,2,2}      160: {1,1,1,1,1,3}      396: {1,1,2,2,5}
%e A340609      45: {2,2,3}        176: {1,1,1,1,5}        400: {1,1,1,1,3,3}
%e A340609      50: {1,3,3}        189: {2,2,2,4}          416: {1,1,1,1,1,6}
%e A340609      54: {1,2,2,2}      196: {1,1,4,4}          440: {1,1,1,3,5}
%e A340609      56: {1,1,1,4}      210: {1,2,3,4}          441: {2,2,4,4}
%p A340609 filter:= proc(n) local F,m,g,t;
%p A340609   F:= ifactors(n)[2];
%p A340609   m:= add(t[2],t=F);
%p A340609   g:= numtheory:-pi(max(seq(t[1],t=F)));
%p A340609   m mod g = 0;
%p A340609 end proc:
%p A340609 seelect(filter, [$2..1000]); # _Robert Israel_, Feb 08 2021
%t A340609 Select[Range[2,100],Divisible[PrimeOmega[#],PrimePi[FactorInteger[#][[-1,1]]]]&]
%Y A340609 Note: Heinz numbers are given in parentheses below.
%Y A340609 The case of equality is A047993 (A106529).
%Y A340609 These are the Heinz numbers of certain partitions counted by A168659.
%Y A340609 The reciprocal version is A340610, with strict case A340828 (A340856).
%Y A340609 If all parts (not just the greatest) are divisors we get A340693 (A340606).
%Y A340609 A001222 counts prime factors.
%Y A340609 A006141 counts partitions whose length equals their minimum (A324522).
%Y A340609 A056239 adds up prime indices.
%Y A340609 A061395 selects the maximum prime index.
%Y A340609 A067538 counts partitions whose length divides their sum (A316413).
%Y A340609 A067538 counts partitions whose maximum divides their sum (A326836).
%Y A340609 A112798 lists the prime indices of each positive integer.
%Y A340609 A200750 counts partitions with length coprime to maximum (A340608).
%Y A340609 A325134 = A001222 + A061395.
%Y A340609 A326845 = A056239 * A061395.
%Y A340609 Cf. A039900, A064174, A143773, A244990/A244991, A326837, A326849 (A326848), A340653, A340787/A340788.
%K A340609 nonn
%O A340609 1,1
%A A340609 _Gus Wiseman_, Jan 27 2021