cp's OEIS Frontend

A381542 Numbers > 1 whose greatest prime index equals their greatest prime multiplicity.

%I A381542 #6 Mar 24 2025 22:35:24
%S A381542 2,9,12,18,36,40,112,120,125,135,200,250,270,336,352,360,375,500,540,
%T A381542 560,567,600,675,750,784,832,1000,1008,1056,1080,1125,1134,1350,1500,
%U A381542 1680,1760,1800,2176,2250,2268,2352,2401,2464,2496,2673,2700,2800,2835,3000
%N A381542 Numbers > 1 whose greatest prime index equals their greatest prime multiplicity.
%C A381542 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, sum A056239.
%F A381542 A061395(a(n)) = A051903(a(n)).
%e A381542 The terms together with their prime indices begin:
%e A381542      2: {1}
%e A381542      9: {2,2}
%e A381542     12: {1,1,2}
%e A381542     18: {1,2,2}
%e A381542     36: {1,1,2,2}
%e A381542     40: {1,1,1,3}
%e A381542    112: {1,1,1,1,4}
%e A381542    120: {1,1,1,2,3}
%e A381542    125: {3,3,3}
%e A381542    135: {2,2,2,3}
%e A381542    200: {1,1,1,3,3}
%e A381542    250: {1,3,3,3}
%e A381542    270: {1,2,2,2,3}
%e A381542    336: {1,1,1,1,2,4}
%e A381542    352: {1,1,1,1,1,5}
%e A381542    360: {1,1,1,2,2,3}
%t A381542 Select[Range[2,1000],PrimePi[FactorInteger[#][[-1,1]]]==Max@@FactorInteger[#][[All,2]]&]
%Y A381542 Counting partitions by the LHS gives A008284, rank statistic A061395.
%Y A381542 Counting partitions by the RHS gives A091602, rank statistic A051903.
%Y A381542 For length instead of maximum we have A106529, counted by A047993 (balanced partitions).
%Y A381542 For number of distinct factors instead of max index we have A212166, counted by A239964.
%Y A381542 Partitions of this type are counted by A240312.
%Y A381542 Including number of distinct parts gives A381543, counted by A382302.
%Y A381542 A000005 counts divisors.
%Y A381542 A000040 lists the primes, differences A001223.
%Y A381542 A001222 counts prime factors, distinct A001221.
%Y A381542 A051903 gives greatest prime exponent, least A051904.
%Y A381542 A055396 gives least prime index, greatest A061395.
%Y A381542 A056239 adds up prime indices, row sums of A112798.
%Y A381542 A122111 represents partition conjugation in terms of Heinz numbers.
%Y A381542 A381544 counts partitions without more ones than any other part, ranks A381439.
%Y A381542 Cf. A000720, A047966, A048767, A130091, A317090, A381632.
%K A381542 nonn
%O A381542 1,1
%A A381542 _Gus Wiseman_, Mar 24 2025