A363134 Positive integers whose multiset of prime indices satisfies: (length) = 2*(minimum).

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

Author

Gus Wiseman, Jun 05 2023

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.

Examples

			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}
		

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.

Cf. A000961, A006141, A046660, A051293, A106529, A111907, A237755, A237824, A327482, A361860, A361861, A362050.

Programs

Mathematica

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

Formula

A001222(a(n)) = 2*A055396(a(n)).