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

2, 4, 6, 8, 9, 16, 20, 24, 30, 32, 36, 45, 50, 54, 56, 64, 75, 81, 84, 96, 125, 126, 128, 140, 144, 160, 176, 189, 196, 210, 216, 240, 256, 264, 294, 315, 324, 350, 360, 384, 396, 400, 416, 440, 441, 486, 490, 512, 525, 540, 576, 594, 600, 616, 624, 660, 686

Offset: 1

Author

Gus Wiseman, Jan 27 2021

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.

If n is a term, then so is n^k for k > 1. - Robert Israel, Feb 08 2021

Examples

			The sequence of terms together with their prime indices begins:
      2: {1}             64: {1,1,1,1,1,1}      216: {1,1,1,2,2,2}
      4: {1,1}           75: {2,3,3}            240: {1,1,1,1,2,3}
      6: {1,2}           81: {2,2,2,2}          256: {1,1,1,1,1,1,1,1}
      8: {1,1,1}         84: {1,1,2,4}          264: {1,1,1,2,5}
      9: {2,2}           96: {1,1,1,1,1,2}      294: {1,2,4,4}
     16: {1,1,1,1}      125: {3,3,3}            315: {2,2,3,4}
     20: {1,1,3}        126: {1,2,2,4}          324: {1,1,2,2,2,2}
     24: {1,1,1,2}      128: {1,1,1,1,1,1,1}    350: {1,3,3,4}
     30: {1,2,3}        140: {1,1,3,4}          360: {1,1,1,2,2,3}
     32: {1,1,1,1,1}    144: {1,1,1,1,2,2}      384: {1,1,1,1,1,1,1,2}
     36: {1,1,2,2}      160: {1,1,1,1,1,3}      396: {1,1,2,2,5}
     45: {2,2,3}        176: {1,1,1,1,5}        400: {1,1,1,1,3,3}
     50: {1,3,3}        189: {2,2,2,4}          416: {1,1,1,1,1,6}
     54: {1,2,2,2}      196: {1,1,4,4}          440: {1,1,1,3,5}
     56: {1,1,1,4}      210: {1,2,3,4}          441: {2,2,4,4}
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Note: Heinz numbers are given in parentheses below.

The case of equality is A047993 (A106529).

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

The reciprocal version is A340610, with strict case A340828 (A340856).

If all parts (not just the greatest) are divisors we get A340693 (A340606).

A001222 counts prime factors.

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

A056239 adds up prime indices.

A061395 selects the maximum prime index.

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

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

A112798 lists the prime indices of each positive integer.

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

A325134 = A001222 + A061395.

A326845 = A056239 * A061395.

Cf. A039900, A064174, A143773, A244990/A244991, A326837, A326849 (A326848), A340653, A340787/A340788.

Programs

Maple

filter:= proc(n) local F,m,g,t;
  F:= ifactors(n)[2];
  m:= add(t[2],t=F);
  g:= numtheory:-pi(max(seq(t[1],t=F)));
  m mod g = 0;
end proc:
seelect(filter, [$2..1000]); # Robert Israel, Feb 08 2021

Mathematica

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

Formula

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