A338899 Concatenated sequence of prime indices of squarefree semiprimes (A006881).

1, 2, 1, 3, 1, 4, 2, 3, 2, 4, 1, 5, 1, 6, 2, 5, 1, 7, 3, 4, 1, 8, 2, 6, 1, 9, 2, 7, 3, 5, 2, 8, 1, 10, 1, 11, 3, 6, 2, 9, 1, 12, 4, 5, 1, 13, 3, 7, 1, 14, 2, 10, 4, 6, 2, 11, 1, 15, 3, 8, 1, 16, 2, 12, 3, 9, 1, 17, 4, 7, 1, 18, 2, 13, 2, 14, 4, 8, 1, 19, 2, 15

Offset

1,2

Comments

This is a triangle with two columns and strictly increasing rows, namely {A270650(n), A270652(n)}.

A squarefree semiprime is a product of any two distinct prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798.

Links

Table of n, a(n) for n=1..82.

Example

The sequence of terms together with their prime indices begins:
      6: {1,2}     57: {2,8}     106: {1,16}    155: {3,11}
     10: {1,3}     58: {1,10}    111: {2,12}    158: {1,22}
     14: {1,4}     62: {1,11}    115: {3,9}     159: {2,16}
     15: {2,3}     65: {3,6}     118: {1,17}    161: {4,9}
     21: {2,4}     69: {2,9}     119: {4,7}     166: {1,23}
     22: {1,5}     74: {1,12}    122: {1,18}    177: {2,17}
     26: {1,6}     77: {4,5}     123: {2,13}    178: {1,24}
     33: {2,5}     82: {1,13}    129: {2,14}    183: {2,18}
     34: {1,7}     85: {3,7}     133: {4,8}     185: {3,12}
     35: {3,4}     86: {1,14}    134: {1,19}    187: {5,7}
     38: {1,8}     87: {2,10}    141: {2,15}    194: {1,25}
     39: {2,6}     91: {4,6}     142: {1,20}    201: {2,19}
     46: {1,9}     93: {2,11}    143: {5,6}     202: {1,26}
     51: {2,7}     94: {1,15}    145: {3,10}    203: {4,10}
     55: {3,5}     95: {3,8}     146: {1,21}    205: {3,13}

Mathematica

Join@@Cases[Select[Range[100], SquareFreeQ[#]&&PrimeOmega[#]==2&], k_:>PrimePi/@First/@FactorInteger[k]]

Crossrefs

A270650 is the first column.

A270652 is the second column.

A320656 counts multiset partitions using these rows, or factorizations into squarefree semiprimes.

A338898 is the version including squares, with columns A338912 and A338913.

A338900 gives row differences.

A338901 gives the row numbers for first appearances.

A001221 and A001222 count distinct/all prime indices.

A001358 lists semiprimes.

A004526 counts 2-part partitions, with strict case shifted right once.

A005117 lists squarefree numbers.

A006881 lists squarefree semiprimes.

A046315 and A100484 list odd and even semiprimes.

A046388 lists odd squarefree semiprimes.

A166237 gives first differences of squarefree semiprimes.

Cf. A030229, A056239, A065516, A112798, A115392, A167171, A176506, A320893, A338904, A338906, A338907, A338910, A338911.

Sequence in context: A211110 A325828 A200780 * A362071 A194943 A087145

Adjacent sequences: A338896 A338897 A338898 * A338900 A338901 A338902

Keyword

nonn,hear,tabf

Author

Gus Wiseman, Nov 16 2020

Status

approved