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%I #11 Nov 05 2020 22:55:54
%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,22,23,24,25,26,27,
%T 28,29,30,31,32,33,34,35,36,37,38,40,41,43,44,45,46,47,48,49,50,51,52,
%U 53,54,55,56,58,59,60,61,62,64,66,67,68,69,70,71,72,73
%N Numbers whose set of distinct prime indices (A304038) is pairwise coprime, where a singleton is always considered coprime.
%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C Also Heinz numbers of partitions whose set of distinct parts is a singleton or pairwise 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 A304711 \/ A000961.
%e The sequence of terms together with their prime indices begins:
%e 1: {} 16: {1,1,1,1} 32: {1,1,1,1,1}
%e 2: {1} 17: {7} 33: {2,5}
%e 3: {2} 18: {1,2,2} 34: {1,7}
%e 4: {1,1} 19: {8} 35: {3,4}
%e 5: {3} 20: {1,1,3} 36: {1,1,2,2}
%e 6: {1,2} 22: {1,5} 37: {12}
%e 7: {4} 23: {9} 38: {1,8}
%e 8: {1,1,1} 24: {1,1,1,2} 40: {1,1,1,3}
%e 9: {2,2} 25: {3,3} 41: {13}
%e 10: {1,3} 26: {1,6} 43: {14}
%e 11: {5} 27: {2,2,2} 44: {1,1,5}
%e 12: {1,1,2} 28: {1,1,4} 45: {2,2,3}
%e 13: {6} 29: {10} 46: {1,9}
%e 14: {1,4} 30: {1,2,3} 47: {15}
%e 15: {2,3} 31: {11} 48: {1,1,1,1,2}
%t Select[Range[100],#==1||PrimePowerQ[#]||CoprimeQ@@PrimePi/@First/@FactorInteger[#]&]
%Y A302798 is the squarefree case.
%Y A304709 counts partitions with pairwise coprime distinct parts, with ordered version A337665 and Heinz numbers A304711.
%Y A304711 does not consider singletons relatively prime, except for (1).
%Y A304712 counts the partitions with these Heinz numbers.
%Y A316476 is the version for indivisibility instead of relative primality.
%Y A328867 is the pairwise non-coprime instead of pairwise coprime version.
%Y A337600 counts triples of this type, with ordered version A337602.
%Y A338330 is the complement.
%Y A000961 lists powers of primes.
%Y A051424 counts pairwise coprime or singleton partitions.
%Y A304038 gives the distinct prime indices of each positive integer.
%Y A327516 counts pairwise coprime partitions.
%Y A333228 ranks compositions whose distinct parts are pairwise coprime.
%Y Cf. A000837, A047968, A056239, A112798, A289509, A302797, A305148, A318716, A318719, A337664, A337695.
%K nonn
%O 1,2
%A _Gus Wiseman_, Oct 31 2020
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