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.
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.
A337694 is the pairwise non-coprime instead of pairwise coprime version.
A337984 does not include the primes.
A305713 counts pairwise coprime strict partitions.
A337561 counts pairwise coprime strict compositions.
A337697 counts pairwise coprime compositions with no 1's.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 10 2018
EXTENSIONS
Extended by Gus Wiseman, Oct 29 2020
STATUS
approved