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%I #12 Nov 20 2020 17:14:46
%S 3,5,7,11,13,15,17,19,23,29,31,33,35,37,41,43,47,51,53,55,59,61,67,69,
%T 71,73,77,79,83,85,89,93,95,97,101,103,107,109,113,119,123,127,131,
%U 137,139,141,143,145,149,151,155,157,161,163,165,167,173,177,179
%N Odd numbers that are either prime or whose prime indices are pairwise coprime.
%C 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.
%F Equals A065091 \/ A337984.
%F Equals A302569 /\ A005408.
%e The sequence of terms together with their prime indices begins:
%e 3: {2} 43: {14} 89: {24} 141: {2,15}
%e 5: {3} 47: {15} 93: {2,11} 143: {5,6}
%e 7: {4} 51: {2,7} 95: {3,8} 145: {3,10}
%e 11: {5} 53: {16} 97: {25} 149: {35}
%e 13: {6} 55: {3,5} 101: {26} 151: {36}
%e 15: {2,3} 59: {17} 103: {27} 155: {3,11}
%e 17: {7} 61: {18} 107: {28} 157: {37}
%e 19: {8} 67: {19} 109: {29} 161: {4,9}
%e 23: {9} 69: {2,9} 113: {30} 163: {38}
%e 29: {10} 71: {20} 119: {4,7} 165: {2,3,5}
%e 31: {11} 73: {21} 123: {2,13} 167: {39}
%e 33: {2,5} 77: {4,5} 127: {31} 173: {40}
%e 35: {3,4} 79: {22} 131: {32} 177: {2,17}
%e 37: {12} 83: {23} 137: {33} 179: {41}
%e 41: {13} 85: {3,7} 139: {34} 181: {42}
%e Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of multiset systems.
%e 03: {{1}}
%e 05: {{2}}
%e 07: {{1,1}}
%e 11: {{3}}
%e 13: {{1,2}}
%e 15: {{1},{2}}
%e 17: {{4}}
%e 19: {{1,1,1}}
%e 23: {{2,2}}
%e 29: {{1,3}}
%e 31: {{5}}
%e 33: {{1},{3}}
%e 35: {{2},{1,1}}
%e 37: {{1,1,2}}
%e 41: {{6}}
%e 43: {{1,4}}
%e 47: {{2,3}}
%e 51: {{1},{4}}
%e 53: {{1,1,1,1}}
%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[1,400,2],Or[PrimeQ[#],CoprimeQ@@primeMS[#]]&]
%Y A005117 is a superset.
%Y A007359 counts partitions with these Heinz numbers.
%Y A302569 allows evens, with squarefree version A302798.
%Y A337694 is the pairwise non-coprime instead of pairwise coprime version.
%Y A337984 does not include the primes.
%Y A305713 counts pairwise coprime strict partitions.
%Y A327516 counts pairwise coprime partitions, ranked by A302696.
%Y A337462 counts pairwise coprime compositions, ranked by A333227.
%Y A337561 counts pairwise coprime strict compositions.
%Y A337667 counts pairwise non-coprime compositions, ranked by A337666.
%Y A337697 counts pairwise coprime compositions with no 1's.
%Y Cf. A005408, A051424, A056239, A087087, A112798, A200976, A302797, A303282, A304711, A335235, A338468.
%K nonn
%O 1,1
%A _Gus Wiseman_, Apr 10 2018
%E Extended by _Gus Wiseman_, Oct 29 2020
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