A340610 - OEIS

%I #10 Feb 09 2021 02:45:59

%S 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 47,49,50,52,53,56,57,58,59,61,65,67,71,73,74,75,78,79,83,84,86,87,89,

%U 91,92,95,97,101,103,106,107,109,111,113,117,122,125,126,127

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

%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.

%H Robert Israel, <a href="/A340610/b340610.txt">Table of n, a(n) for n = 1..10000</a>

%F A001222(a(n)) divides A061395(a(n)).

%e The sequence of terms together with their prime indices begins:

%e      2: {1}        29: {10}       56: {1,1,1,4}

%e      3: {2}        30: {1,2,3}    57: {2,8}

%e      5: {3}        31: {11}       58: {1,10}

%e      6: {1,2}      35: {3,4}      59: {17}

%e      7: {4}        37: {12}       61: {18}

%e      9: {2,2}      38: {1,8}      65: {3,6}

%e     11: {5}        39: {2,6}      67: {19}

%e     13: {6}        41: {13}       71: {20}

%e     14: {1,4}      43: {14}       73: {21}

%e     17: {7}        45: {2,2,3}    74: {1,12}

%e     19: {8}        47: {15}       75: {2,3,3}

%e     20: {1,1,3}    49: {4,4}      78: {1,2,6}

%e     21: {2,4}      50: {1,3,3}    79: {22}

%e     23: {9}        52: {1,1,6}    83: {23}

%e     26: {1,6}      53: {16}       84: {1,1,2,4}

%p filter:= proc(n) local F,m,g,t;

%p   F:= ifactors(n)[2];

%p   m:= add(t[2],t=F);

%p   g:= numtheory:-pi(max(seq(t[1],t=F)));

%p   g mod m = 0;

%p end proc:

%p select(filter, [$2..1000]); # _Robert Israel_, Feb 08 2021

%t Select[Range[2,100],Divisible[PrimePi[FactorInteger[#][[-1,1]]],PrimeOmega[#]]&]

%Y Note: Heinz numbers are given in parentheses below.

%Y The case of equality is A047993 (A106529).

%Y 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 These are the Heinz numbers of certain partitions counted by A168659.

%Y The reciprocal version is A340609.

%Y The squarefree case is A340828 (A340856).

%Y A001222 counts prime factors.

%Y A006141 counts partitions whose length equals their minimum (A324522).

%Y A056239 adds up prime indices.

%Y A061395 selects the maximum prime index.

%Y A067538 counts partitions whose length divides their sum (A316413).

%Y A067538 counts partitions whose maximum divides their sum (A326836).

%Y A112798 lists the prime indices of each positive integer.

%Y A200750 counts partitions with length coprime to maximum (A340608).

%Y A325134 = A001222 + A061395.

%Y A326845 = A056239 * A061395.

%Y Cf. A244990/A244991, A326837, A326849 (A326848), A340653, A340691, A340693 (A340606), A340787/A340788.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jan 27 2021