A255201 -id:A255201

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A361909

Positive integers > 1 whose prime indices satisfy: (maximum) = 2*(length).

+10
6

3, 14, 21, 35, 49, 52, 78, 117, 130, 152, 182, 195, 228, 273, 286, 325, 338, 342, 380, 429, 455, 464, 507, 513, 532, 570, 637, 696, 715, 798, 836, 845, 855, 950, 988, 1001, 1044, 1160, 1183, 1184, 1197, 1254, 1292, 1330, 1425, 1444, 1482, 1566, 1573, 1624

(list; graph; refs; listen; history; text; internal format)

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.

LINKS

Table of n, a(n) for n=1..50.

EXAMPLE

The terms together with their prime indices begin:

3: {2}

14: {1,4}

21: {2,4}

35: {3,4}

49: {4,4}

52: {1,1,6}

78: {1,2,6}

117: {2,2,6}

130: {1,3,6}

152: {1,1,1,8}

182: {1,4,6}

195: {2,3,6}

228: {1,1,2,8}

273: {2,4,6}

286: {1,5,6}

325: {3,3,6}

338: {1,6,6}

342: {1,2,2,8}

MATHEMATICA

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

CROSSREFS

The LHS is A061395 (greatest prime index), least A055396.

Without multiplying by 2 in the RHS, we have A106529.

For omega instead of bigomega we have A111907, counted by A239959.

Partitions of this type are counted by A237753.

The RHS is A255201 (twice bigomega).

For mean instead of length we have A361855, counted by A361853.

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

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

A001221 (omega) counts distinct prime factors.

A001222 (bigomega) counts prime factors with multiplicity.

A112798 lists prime indices, sum A056239.

A316413 ranks partitions with integer mean, counted by A067538.

A326567/A326568 gives mean of prime indices.

Cf. A067801, A324521, A237752, A237820, A360005, A361205, A361907.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 05 2023

STATUS

approved

A362050

Numbers whose prime indices satisfy: (length) = 2*(median).

+10
5

4, 54, 81, 90, 100, 126, 135, 140, 189, 198, 220, 234, 260, 297, 306, 340, 342, 351, 380, 414, 459, 460, 513, 522, 558, 580, 620, 621, 666, 738, 740, 774, 783, 820, 837, 846, 860, 940, 954, 999, 1060, 1062, 1098, 1107, 1161, 1180, 1206, 1220, 1269, 1278, 1314

(list; graph; refs; listen; history; text; internal format)

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.

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

All terms are squarefree.

LINKS

Table of n, a(n) for n=1..51.

EXAMPLE

The terms together with their prime indices begin:

4: {1,1}

54: {1,2,2,2}

81: {2,2,2,2}

90: {1,2,2,3}

100: {1,1,3,3}

126: {1,2,2,4}

135: {2,2,2,3}

140: {1,1,3,4}

189: {2,2,2,4}

198: {1,2,2,5}

MATHEMATICA

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Select[Range[100], PrimeOmega[#]==2*Median[prix[#]]&]

CROSSREFS

The LHS is A001222 (bigomega).

The RHS is A360005 (twice median).

Before multiplying the median by 2, A361800 counts partitions of this type.

For maximum instead of length we have A361856, counted by A361849.

Partitions of this type are counted by A362049.

A061395 gives greatest prime index, least A055396.