cp's OEIS Frontend

A302568 Odd numbers that are either prime or whose prime indices are pairwise coprime.

%I A302568 #12 Nov 20 2020 17:14:46
%S A302568 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 A302568 71,73,77,79,83,85,89,93,95,97,101,103,107,109,113,119,123,127,131,
%U A302568 137,139,141,143,145,149,151,155,157,161,163,165,167,173,177,179
%N A302568 Odd numbers that are either prime or whose prime indices are pairwise coprime.
%C A302568 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 A302568 Equals A065091 \/ A337984.
%F A302568 Equals A302569 /\ A005408.
%e A302568 The sequence of terms together with their prime indices begins:
%e A302568       3: {2}       43: {14}      89: {24}      141: {2,15}
%e A302568       5: {3}       47: {15}      93: {2,11}    143: {5,6}
%e A302568       7: {4}       51: {2,7}     95: {3,8}     145: {3,10}
%e A302568      11: {5}       53: {16}      97: {25}      149: {35}
%e A302568      13: {6}       55: {3,5}    101: {26}      151: {36}
%e A302568      15: {2,3}     59: {17}     103: {27}      155: {3,11}
%e A302568      17: {7}       61: {18}     107: {28}      157: {37}
%e A302568      19: {8}       67: {19}     109: {29}      161: {4,9}
%e A302568      23: {9}       69: {2,9}    113: {30}      163: {38}
%e A302568      29: {10}      71: {20}     119: {4,7}     165: {2,3,5}
%e A302568      31: {11}      73: {21}     123: {2,13}    167: {39}
%e A302568      33: {2,5}     77: {4,5}    127: {31}      173: {40}
%e A302568      35: {3,4}     79: {22}     131: {32}      177: {2,17}
%e A302568      37: {12}      83: {23}     137: {33}      179: {41}
%e A302568      41: {13}      85: {3,7}    139: {34}      181: {42}
%e A302568 Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of multiset systems.
%e A302568 03: {{1}}
%e A302568 05: {{2}}
%e A302568 07: {{1,1}}
%e A302568 11: {{3}}
%e A302568 13: {{1,2}}
%e A302568 15: {{1},{2}}
%e A302568 17: {{4}}
%e A302568 19: {{1,1,1}}
%e A302568 23: {{2,2}}
%e A302568 29: {{1,3}}
%e A302568 31: {{5}}
%e A302568 33: {{1},{3}}
%e A302568 35: {{2},{1,1}}
%e A302568 37: {{1,1,2}}
%e A302568 41: {{6}}
%e A302568 43: {{1,4}}
%e A302568 47: {{2,3}}
%e A302568 51: {{1},{4}}
%e A302568 53: {{1,1,1,1}}
%t A302568 primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A302568 Select[Range[1,400,2],Or[PrimeQ[#],CoprimeQ@@primeMS[#]]&]
%Y A302568 A005117 is a superset.
%Y A302568 A007359 counts partitions with these Heinz numbers.
%Y A302568 A302569 allows evens, with squarefree version A302798.
%Y A302568 A337694 is the pairwise non-coprime instead of pairwise coprime version.
%Y A302568 A337984 does not include the primes.
%Y A302568 A305713 counts pairwise coprime strict partitions.
%Y A302568 A327516 counts pairwise coprime partitions, ranked by A302696.
%Y A302568 A337462 counts pairwise coprime compositions, ranked by A333227.
%Y A302568 A337561 counts pairwise coprime strict compositions.
%Y A302568 A337667 counts pairwise non-coprime compositions, ranked by A337666.
%Y A302568 A337697 counts pairwise coprime compositions with no 1's.
%Y A302568 Cf. A005408, A051424, A056239, A087087, A112798, A200976, A302797, A303282, A304711, A335235, A338468.
%K A302568 nonn
%O A302568 1,1
%A A302568 _Gus Wiseman_, Apr 10 2018
%E A302568 Extended by _Gus Wiseman_, Oct 29 2020