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

2, 9, 12, 18, 36, 40, 112, 120, 125, 135, 200, 250, 270, 336, 352, 360, 375, 500, 540, 560, 567, 600, 675, 750, 784, 832, 1000, 1008, 1056, 1080, 1125, 1134, 1350, 1500, 1680, 1760, 1800, 2176, 2250, 2268, 2352, 2401, 2464, 2496, 2673, 2700, 2800, 2835, 3000

Offset: 1

Author

Gus Wiseman, Mar 24 2025

Keywords

• nonn

Comments

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.

Examples

			The terms together with their prime indices begin:
     2: {1}
     9: {2,2}
    12: {1,1,2}
    18: {1,2,2}
    36: {1,1,2,2}
    40: {1,1,1,3}
   112: {1,1,1,1,4}
   120: {1,1,1,2,3}
   125: {3,3,3}
   135: {2,2,2,3}
   200: {1,1,1,3,3}
   250: {1,3,3,3}
   270: {1,2,2,2,3}
   336: {1,1,1,1,2,4}
   352: {1,1,1,1,1,5}
   360: {1,1,1,2,2,3}
		

Crossrefs

Counting partitions by the LHS gives A008284, rank statistic A061395.

Counting partitions by the RHS gives A091602, rank statistic A051903.

For length instead of maximum we have A106529, counted by A047993 (balanced partitions).

For number of distinct factors instead of max index we have A212166, counted by A239964.

Partitions of this type are counted by A240312.

Including number of distinct parts gives A381543, counted by A382302.

A000005 counts divisors.

A000040 lists the primes, differences A001223.

A001222 counts prime factors, distinct A001221.

A051903 gives greatest prime exponent, least A051904.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A122111 represents partition conjugation in terms of Heinz numbers.

A381544 counts partitions without more ones than any other part, ranks A381439.

Cf. A000720, A047966, A048767, A130091, A317090, A381632.

Programs

Mathematica

Select[Range[2,1000],PrimePi[FactorInteger[#][[-1,1]]]==Max@@FactorInteger[#][[All,2]]&]

Formula

A061395(a(n)) = A051903(a(n)).