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A363134
Positive integers whose multiset of prime indices satisfies: (length) = 2*(minimum).
3
4, 6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 81, 82, 86, 94, 106, 118, 122, 134, 135, 142, 146, 158, 166, 178, 189, 194, 202, 206, 214, 218, 225, 226, 254, 262, 274, 278, 297, 298, 302, 314, 315, 326, 334, 346, 351, 358, 362, 375, 382, 386, 394, 398, 422, 441
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.
FORMULA
A001222(a(n)) = 2*A055396(a(n)).
EXAMPLE
The terms together with their prime indices begin:
4: {1,1} 94: {1,15} 214: {1,28}
6: {1,2} 106: {1,16} 218: {1,29}
10: {1,3} 118: {1,17} 225: {2,2,3,3}
14: {1,4} 122: {1,18} 226: {1,30}
22: {1,5} 134: {1,19} 254: {1,31}
26: {1,6} 135: {2,2,2,3} 262: {1,32}
34: {1,7} 142: {1,20} 274: {1,33}
38: {1,8} 146: {1,21} 278: {1,34}
46: {1,9} 158: {1,22} 297: {2,2,2,5}
58: {1,10} 166: {1,23} 298: {1,35}
62: {1,11} 178: {1,24} 302: {1,36}
74: {1,12} 189: {2,2,2,4} 314: {1,37}
81: {2,2,2,2} 194: {1,25} 315: {2,2,3,4}
82: {1,13} 202: {1,26} 326: {1,38}
86: {1,14} 206: {1,27} 334: {1,39}
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Length[prix[#]]==2*Min[prix[#]]&]
CROSSREFS
Partitions of this type are counted by A237757.
Removing the factor 2 gives A324522.
For maximum instead of length we have A361908, counted by A118096.
For mean instead of length we have A363133, counted by A363132.
For maximum instead of minimum we have A363218, counted by A237753.
A055396 gives minimum prime index, maximum A061395.
A112798 lists prime indices, length A001222, sum A056239.
A360005 gives twice median of prime indices.
Sequence in context: A184335 A243428 A091376 * A100484 A076924 A103801
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 05 2023
STATUS
approved