cp's OEIS Frontend

A338331 Numbers whose set of distinct prime indices (A304038) is pairwise coprime, where a singleton is always considered coprime.

%I A338331 #11 Nov 05 2020 22:55:54
%S A338331 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 A338331 28,29,30,31,32,33,34,35,36,37,38,40,41,43,44,45,46,47,48,49,50,51,52,
%U A338331 53,54,55,56,58,59,60,61,62,64,66,67,68,69,70,71,72,73
%N A338331 Numbers whose set of distinct prime indices (A304038) is pairwise coprime, where a singleton is always considered coprime.
%C A338331 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 A338331 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 A338331 Equals A304711 \/ A000961.
%e A338331 The sequence of terms together with their prime indices begins:
%e A338331       1: {}          16: {1,1,1,1}     32: {1,1,1,1,1}
%e A338331       2: {1}         17: {7}           33: {2,5}
%e A338331       3: {2}         18: {1,2,2}       34: {1,7}
%e A338331       4: {1,1}       19: {8}           35: {3,4}
%e A338331       5: {3}         20: {1,1,3}       36: {1,1,2,2}
%e A338331       6: {1,2}       22: {1,5}         37: {12}
%e A338331       7: {4}         23: {9}           38: {1,8}
%e A338331       8: {1,1,1}     24: {1,1,1,2}     40: {1,1,1,3}
%e A338331       9: {2,2}       25: {3,3}         41: {13}
%e A338331      10: {1,3}       26: {1,6}         43: {14}
%e A338331      11: {5}         27: {2,2,2}       44: {1,1,5}
%e A338331      12: {1,1,2}     28: {1,1,4}       45: {2,2,3}
%e A338331      13: {6}         29: {10}          46: {1,9}
%e A338331      14: {1,4}       30: {1,2,3}       47: {15}
%e A338331      15: {2,3}       31: {11}          48: {1,1,1,1,2}
%t A338331 Select[Range[100],#==1||PrimePowerQ[#]||CoprimeQ@@PrimePi/@First/@FactorInteger[#]&]
%Y A338331 A302798 is the squarefree case.
%Y A338331 A304709 counts partitions with pairwise coprime distinct parts, with ordered version A337665 and Heinz numbers A304711.
%Y A338331 A304711 does not consider singletons relatively prime, except for (1).
%Y A338331 A304712 counts the partitions with these Heinz numbers.
%Y A338331 A316476 is the version for indivisibility instead of relative primality.
%Y A338331 A328867 is the pairwise non-coprime instead of pairwise coprime version.
%Y A338331 A337600 counts triples of this type, with ordered version A337602.
%Y A338331 A338330 is the complement.
%Y A338331 A000961 lists powers of primes.
%Y A338331 A051424 counts pairwise coprime or singleton partitions.
%Y A338331 A304038 gives the distinct prime indices of each positive integer.
%Y A338331 A327516 counts pairwise coprime partitions.
%Y A338331 A333228 ranks compositions whose distinct parts are pairwise coprime.
%Y A338331 Cf. A000837, A047968, A056239, A112798, A289509, A302797, A305148, A318716, A318719, A337664, A337695.
%K A338331 nonn
%O A338331 1,2
%A A338331 _Gus Wiseman_, Oct 31 2020