cp's OEIS Frontend

A339562 Squarefree numbers with no prime index dividing all the other prime indices.

%I A339562 #10 Apr 18 2021 22:41:02
%S A339562 1,15,33,35,51,55,69,77,85,91,93,95,105,119,123,141,143,145,155,161,
%T A339562 165,177,187,195,201,203,205,209,215,217,219,221,231,247,249,253,255,
%U A339562 265,285,287,291,295,299,301,309,323,327,329,335,341,345,355,357,377,381
%N A339562 Squarefree numbers with no prime index dividing all the other prime indices.
%C A339562 First differs from A342193 in lacking 45.
%C A339562 Alternative name: 1 and squarefree numbers with smallest prime index not dividing all the other prime indices.
%C A339562 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 A339562 Also 1 and Heinz numbers of strict integer partitions with smallest part not dividing all the others (counted by A341450). 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.
%e A339562 The sequence of terms together with their prime indices begins:
%e A339562       1: {}         141: {2,15}     219: {2,21}
%e A339562      15: {2,3}      143: {5,6}      221: {6,7}
%e A339562      33: {2,5}      145: {3,10}     231: {2,4,5}
%e A339562      35: {3,4}      155: {3,11}     247: {6,8}
%e A339562      51: {2,7}      161: {4,9}      249: {2,23}
%e A339562      55: {3,5}      165: {2,3,5}    253: {5,9}
%e A339562      69: {2,9}      177: {2,17}     255: {2,3,7}
%e A339562      77: {4,5}      187: {5,7}      265: {3,16}
%e A339562      85: {3,7}      195: {2,3,6}    285: {2,3,8}
%e A339562      91: {4,6}      201: {2,19}     287: {4,13}
%e A339562      93: {2,11}     203: {4,10}     291: {2,25}
%e A339562      95: {3,8}      205: {3,13}     295: {3,17}
%e A339562     105: {2,3,4}    209: {5,8}      299: {6,9}
%e A339562     119: {4,7}      215: {3,14}     301: {4,14}
%e A339562     123: {2,13}     217: {4,11}     309: {2,27}
%t A339562 Select[Range[100],#==1||SquareFreeQ[#]&&With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(p/Min@@p)]&]
%Y A339562 The squarefree complement is A339563.
%Y A339562 These partitions are counted by A341450.
%Y A339562 The not necessarily squarefree version is A342193.
%Y A339562 A000005 counts divisors.
%Y A339562 A000070 counts partitions with a selected part.
%Y A339562 A001221 counts distinct prime factors.
%Y A339562 A005117 lists squarefree numbers.
%Y A339562 A006128 counts partitions with a selected position (strict: A015723).
%Y A339562 A056239 adds up prime indices (row sums of A112798).
%Y A339562 A083710 counts partitions with a dividing part (strict: A097986).
%Y A339562 Cf. A253249, A264401, A257993, A338470, A343337, A343338, A343339, A343379, A343380, A343382.
%K A339562 nonn
%O A339562 1,2
%A A339562 _Gus Wiseman_, Apr 10 2021