A338907 Semiprimes whose prime indices sum to an odd number.

6, 14, 15, 26, 33, 35, 38, 51, 58, 65, 69, 74, 77, 86, 93, 95, 106, 119, 122, 123, 141, 142, 143, 145, 158, 161, 177, 178, 185, 201, 202, 209, 214, 215, 217, 219, 221, 226, 249, 262, 265, 278, 287, 291, 299, 302, 305, 309, 319, 323, 326, 327, 329, 346, 355

Offset

1,1

Comments

All terms are squarefree (A005117).

A semiprime is a product of any two 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.

The semiprimes in A300063; the semiprimes in A332820. - Peter Munn, Dec 25 2020

Links

Table of n, a(n) for n=1..55.

Formula

A338906 \/ A338907 = A001358.

Example

The sequence of terms together with their prime indices begins:
      6: {1,2}      95: {3,8}     202: {1,26}
     14: {1,4}     106: {1,16}    209: {5,8}
     15: {2,3}     119: {4,7}     214: {1,28}
     26: {1,6}     122: {1,18}    215: {3,14}
     33: {2,5}     123: {2,13}    217: {4,11}
     35: {3,4}     141: {2,15}    219: {2,21}
     38: {1,8}     142: {1,20}    221: {6,7}
     51: {2,7}     143: {5,6}     226: {1,30}
     58: {1,10}    145: {3,10}    249: {2,23}
     65: {3,6}     158: {1,22}    262: {1,32}
     69: {2,9}     161: {4,9}     265: {3,16}
     74: {1,12}    177: {2,17}    278: {1,34}
     77: {4,5}     178: {1,24}    287: {4,13}
     86: {1,14}    185: {3,12}    291: {2,25}
     93: {2,11}    201: {2,19}    299: {6,9}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], PrimeOmega[#]==2&&OddQ[Total[primeMS[#]]]&]

Crossrefs

A031368 looks at primes instead of semiprimes.

A098350 has this as union of odd-indexed antidiagonals.

A300063 looks at all numbers (not just semiprimes).

A338904 has this as union of odd-indexed rows.

A338906 is the even version.

A001358 lists semiprimes, with odd/even terms A046315/A100484.

A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.

A056239 gives the sum of prime indices (Heinz weight).

A084126 and A084127 give the prime factors of semiprimes.

A087112 groups semiprimes by greater factor.

A289182/A115392 list the positions of odd/even terms in A001358.

A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.

A338899, A270650, and A270652 give the prime indices of squarefree semiprimes, with difference A338900.

A338908 lists squarefree semiprimes of even weight.

A339114/A339115 give the least/greatest semiprime of weight n.

Cf. A000040, A001222, A014342, A024697, A062198, A112798, A300061, A319242, A320655, A338910, A339003.

Subsequence of A332820.

Sequence in context: A355941 A359912 A325698 * A218005 A337384 A063600

Adjacent sequences: A338904 A338905 A338906 * A338908 A338909 A338910

Keyword

nonn

Author

Gus Wiseman, Nov 28 2020

Status

approved