A302568 Odd numbers that are either prime or whose prime indices are pairwise coprime.

3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 55, 59, 61, 67, 69, 71, 73, 77, 79, 83, 85, 89, 93, 95, 97, 101, 103, 107, 109, 113, 119, 123, 127, 131, 137, 139, 141, 143, 145, 149, 151, 155, 157, 161, 163, 165, 167, 173, 177, 179

Offset

1,1

Comments

Also Heinz numbers of partitions with pairwise coprime parts all greater than 1 (A007359), where singletons are considered coprime. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

Links

Table of n, a(n) for n=1..59.

Formula

Equals A065091 \/ A337984.

Equals A302569 /\ A005408.

Example

The sequence of terms together with their prime indices begins:
      3: {2}       43: {14}      89: {24}      141: {2,15}
      5: {3}       47: {15}      93: {2,11}    143: {5,6}
      7: {4}       51: {2,7}     95: {3,8}     145: {3,10}
     11: {5}       53: {16}      97: {25}      149: {35}
     13: {6}       55: {3,5}    101: {26}      151: {36}
     15: {2,3}     59: {17}     103: {27}      155: {3,11}
     17: {7}       61: {18}     107: {28}      157: {37}
     19: {8}       67: {19}     109: {29}      161: {4,9}
     23: {9}       69: {2,9}    113: {30}      163: {38}
     29: {10}      71: {20}     119: {4,7}     165: {2,3,5}
     31: {11}      73: {21}     123: {2,13}    167: {39}
     33: {2,5}     77: {4,5}    127: {31}      173: {40}
     35: {3,4}     79: {22}     131: {32}      177: {2,17}
     37: {12}      83: {23}     137: {33}      179: {41}
     41: {13}      85: {3,7}    139: {34}      181: {42}
Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of multiset systems.
03: {{1}}
05: {{2}}
07: {{1,1}}
11: {{3}}
13: {{1,2}}
15: {{1},{2}}
17: {{4}}
19: {{1,1,1}}
23: {{2,2}}
29: {{1,3}}
31: {{5}}
33: {{1},{3}}
35: {{2},{1,1}}
37: {{1,1,2}}
41: {{6}}
43: {{1,4}}
47: {{2,3}}
51: {{1},{4}}
53: {{1,1,1,1}}

Mathematica

primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[1, 400, 2], Or[PrimeQ[#], CoprimeQ@@primeMS[#]]&]

Crossrefs

A005117 is a superset.

A007359 counts partitions with these Heinz numbers.

A302569 allows evens, with squarefree version A302798.

A337694 is the pairwise non-coprime instead of pairwise coprime version.

A337984 does not include the primes.

A305713 counts pairwise coprime strict partitions.

A327516 counts pairwise coprime partitions, ranked by A302696.

A337462 counts pairwise coprime compositions, ranked by A333227.

A337561 counts pairwise coprime strict compositions.

A337667 counts pairwise non-coprime compositions, ranked by A337666.

A337697 counts pairwise coprime compositions with no 1's.

Cf. A005408, A051424, A056239, A087087, A112798, A200976, A302797, A303282, A304711, A335235, A338468.

Sequence in context: A359080 A103796 A368100 * A359260 A294674 A340077

Adjacent sequences: A302565 A302566 A302567 * A302569 A302570 A302571

Keyword

nonn

Author

Gus Wiseman, Apr 10 2018

Extensions

Extended by Gus Wiseman, Oct 29 2020

Status

approved