A338909 Numbers of the form prime(x) * prime(y) where x and y have a common divisor > 1.

9, 21, 25, 39, 49, 57, 65, 87, 91, 111, 115, 121, 129, 133, 159, 169, 183, 185, 203, 213, 235, 237, 247, 259, 267, 289, 299, 301, 303, 305, 319, 321, 339, 361, 365, 371, 377, 393, 417, 427, 445, 453, 481, 489, 497, 515, 517, 519, 529, 543, 551, 553, 559, 565

Offset: 1

Author

Gus Wiseman, Nov 20 2020

Keywords

• nonn

Examples

			The sequence of terms together with their prime indices begins:
      9: {2,2}     169: {6,6}     319: {5,10}
     21: {2,4}     183: {2,18}    321: {2,28}
     25: {3,3}     185: {3,12}    339: {2,30}
     39: {2,6}     203: {4,10}    361: {8,8}
     49: {4,4}     213: {2,20}    365: {3,21}
     57: {2,8}     235: {3,15}    371: {4,16}
     65: {3,6}     237: {2,22}    377: {6,10}
     87: {2,10}    247: {6,8}     393: {2,32}
     91: {4,6}     259: {4,12}    417: {2,34}
    111: {2,12}    267: {2,24}    427: {4,18}
    115: {3,9}     289: {7,7}     445: {3,24}
    121: {5,5}     299: {6,9}     453: {2,36}
    129: {2,14}    301: {4,14}    481: {6,12}
    133: {4,8}     303: {2,26}    489: {2,38}
    159: {2,16}    305: {3,18}    497: {4,20}
		

Crossrefs

A082023 counts partitions with these as Heinz numbers, complement A023022.

A300912 is the complement in A001358.

A339002 is the squarefree case.

A001221 counts distinct prime indices.

A001222 counts prime indices.

A001358 lists semiprimes, with odds A046315 and evens A100484.

A004526 counts 2-part partitions, with strict case A140106 (shifted left).

A006881 lists squarefree semiprimes, with odds A046388 and evens A100484.

A176504/A176506/A087794 give sum/difference/product of semiprime indices.

A318990 lists semiprimes with divisible indices.

A320655 counts factorizations into semiprimes.

A338898, A338912, and A338913 give semiprime indices.

A338899, A270650, and A270652 give squarefree semiprime indices.

A338910 lists semiprimes with odd indices.

A338911 lists semiprimes with even indices.

Cf. A005117, A037143, A055684, A056239, A065516, A112798, A115392, A128301, A289182, A338900, A338904.

Programs

Mathematica

Select[Range[100],PrimeOmega[#]==2&&GCD@@PrimePi/@First/@FactorInteger[#]>1&]

Formula

Equals A001358 \ A300912.

Equals A339002 \/ (A001248 \ {4}).