cp's OEIS Frontend

A338316 Odd numbers whose distinct prime indices are pairwise coprime, where a singleton is always considered coprime.

%I A338316 #11 Nov 01 2020 01:47:17
%S A338316 1,3,5,7,9,11,13,15,17,19,23,25,27,29,31,33,35,37,41,43,45,47,49,51,
%T A338316 53,55,59,61,67,69,71,73,75,77,79,81,83,85,89,93,95,97,99,101,103,107,
%U A338316 109,113,119,121,123,125,127,131,135,137,139,141,143,145,149,151
%N A338316 Odd numbers whose distinct prime indices are pairwise coprime, where a singleton is always considered coprime.
%C A338316 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 A338316 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions. a(n) gives the n-th Heinz number of an integer partition with no 1's and pairwise coprime distinct parts, where a singleton is always considered coprime (A338317).
%e A338316 The sequence of terms together with their prime indices begins:
%e A338316       1: {}          33: {2,5}       71: {20}
%e A338316       3: {2}         35: {3,4}       73: {21}
%e A338316       5: {3}         37: {12}        75: {2,3,3}
%e A338316       7: {4}         41: {13}        77: {4,5}
%e A338316       9: {2,2}       43: {14}        79: {22}
%e A338316      11: {5}         45: {2,2,3}     81: {2,2,2,2}
%e A338316      13: {6}         47: {15}        83: {23}
%e A338316      15: {2,3}       49: {4,4}       85: {3,7}
%e A338316      17: {7}         51: {2,7}       89: {24}
%e A338316      19: {8}         53: {16}        93: {2,11}
%e A338316      23: {9}         55: {3,5}       95: {3,8}
%e A338316      25: {3,3}       59: {17}        97: {25}
%e A338316      27: {2,2,2}     61: {18}        99: {2,2,5}
%e A338316      29: {10}        67: {19}       101: {26}
%e A338316      31: {11}        69: {2,9}      103: {27}
%t A338316 Select[Range[1,100,2],#==1||PrimePowerQ[#]||CoprimeQ@@Union[PrimePi/@First/@FactorInteger[#]]&]
%Y A338316 A338315 does not consider singletons coprime, with Heinz numbers A337987.
%Y A338316 A338317 counts the partitions with these Heinz numbers.
%Y A338316 A337694 is a pairwise non-coprime instead of pairwise coprime version.
%Y A338316 A007359 counts singleton or pairwise coprime partitions with no 1's, with Heinz numbers A302568.
%Y A338316 A101268 counts pairwise coprime or singleton compositions, ranked by A335235.
%Y A338316 A302797 lists squarefree numbers whose distinct parts are pairwise coprime.
%Y A338316 A304709 counts partitions whose distinct parts are pairwise coprime, with Heinz numbers A304711.
%Y A338316 A327516 counts pairwise coprime partitions, ranked by A302696.
%Y A338316 A337485 counts pairwise coprime partitions with no 1's, with Heinz numbers A337984.
%Y A338316 A337561 counts pairwise coprime strict compositions.
%Y A338316 A337665 counts compositions whose distinct parts are pairwise coprime, ranked by A333228.
%Y A338316 A337697 counts pairwise coprime compositions with no 1's.
%Y A338316 Cf. A051424, A056239, A112798, A220377, A289509, A302569, A303282, A318719, A328673, A328867.
%K A338316 nonn
%O A338316 1,2
%A A338316 _Gus Wiseman_, Oct 24 2020