A361855 Numbers > 1 whose prime indices satisfy (maximum) * (length) = 2*(sum).

28, 40, 78, 84, 171, 190, 198, 220, 240, 252, 280, 351, 364, 390, 406, 435, 714, 748, 756, 765, 777, 784, 814, 840, 850, 925, 988, 1118, 1197, 1254, 1330, 1352, 1419, 1425, 1440, 1505, 1564, 1600, 1638, 1716, 1755, 1794, 1802, 1820, 1950, 2067, 2204, 2254

Offset

1,1

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.

Also positive integers whose prime indices satisfy (maximum) = 2*(mean).

Also Heinz numbers of partitions of the same size as their complement (see example).

Links

Table of n, a(n) for n=1..48.

Formula

A061395(a(n)) * A001222(a(n)) = 2*A056239(a(n)).

Example

The terms together with their prime indices begin:
   28: {1,1,4}
   40: {1,1,1,3}
   78: {1,2,6}
   84: {1,1,2,4}
  171: {2,2,8}
  190: {1,3,8}
  198: {1,2,2,5}
  220: {1,1,3,5}
  240: {1,1,1,1,2,3}
  252: {1,1,2,2,4}
  280: {1,1,1,3,4}
The prime indices of 84 are {1,1,2,4}, with maximum 4, length 4, and sum 8, and 4*4 = 2*8, so 84 is in the sequence.
The prime indices of 120 are {1,1,1,2,3}, with maximum 3, length 5, and sum 8, and 3*5 != 2*8, so 120 is not in the sequence.
The prime indices of 252 are {1,1,2,2,4}, with maximum 4, length 5, and sum 10, and 4*5 = 2*10, so 252 is in the sequence.
The partition (5,2,2,1) with Heinz number 198 has diagram:
  o o o o o
  o o . . .
  o o . . .
  o . . . .
Since the partition and its complement (shown in dots) both have size 10, 198 is in the sequence.

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[2, 100], Max@@prix[#]*PrimeOmega[#]==2*Total[prix[#]]&]

Crossrefs

These partitions are counted by A361853, strict A361854.

For median instead of mean we have A361856, counted by A361849.

For minimum instead of mean we have A361908, counted by A118096.

For length instead of mean we have A361909, counted by A237753.

A001222 (bigomega) counts prime factors, distinct A001221 (omega).

A061395 gives greatest prime index.

A112798 lists prime indices, sum A056239.

A326567/A326568 gives mean of prime indices.

Cf. A067801, A237824, A316413, A324521, A326844, A361205, A361851, A361906.

Sequence in context: A084807 A367975 A184032 * A047630 A135789 A219882

Adjacent sequences: A361852 A361853 A361854 * A361856 A361857 A361858

Keyword

nonn

Author

Gus Wiseman, Mar 29 2023

Status

approved