cp's OEIS Frontend

A339563 Squarefree numbers > 1 whose smallest prime index divides all the other prime indices.

%I A339563 #12 Apr 18 2021 22:43:54
%S A339563 2,3,5,6,7,10,11,13,14,17,19,21,22,23,26,29,30,31,34,37,38,39,41,42,
%T A339563 43,46,47,53,57,58,59,61,62,65,66,67,70,71,73,74,78,79,82,83,86,87,89,
%U A339563 94,97,101,102,103,106,107,109,110,111,113,114,115,118,122,127
%N A339563 Squarefree numbers > 1 whose smallest prime index divides all the other prime indices.
%C A339563 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 A339563 Also Heinz numbers of strict integer partitions whose smallest part divides all the others (counted by A097986). 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 A339563 The sequence of terms together with their prime indices begins:
%e A339563       2: {1}       29: {10}        59: {17}
%e A339563       3: {2}       30: {1,2,3}     61: {18}
%e A339563       5: {3}       31: {11}        62: {1,11}
%e A339563       6: {1,2}     34: {1,7}       65: {3,6}
%e A339563       7: {4}       37: {12}        66: {1,2,5}
%e A339563      10: {1,3}     38: {1,8}       67: {19}
%e A339563      11: {5}       39: {2,6}       70: {1,3,4}
%e A339563      13: {6}       41: {13}        71: {20}
%e A339563      14: {1,4}     42: {1,2,4}     73: {21}
%e A339563      17: {7}       43: {14}        74: {1,12}
%e A339563      19: {8}       46: {1,9}       78: {1,2,6}
%e A339563      21: {2,4}     47: {15}        79: {22}
%e A339563      22: {1,5}     53: {16}        82: {1,13}
%e A339563      23: {9}       57: {2,8}       83: {23}
%e A339563      26: {1,6}     58: {1,10}      86: {1,14}
%t A339563 Select[Range[2,100],SquareFreeQ[#]&&With[{p=PrimePi/@First/@FactorInteger[#]},And@@IntegerQ/@(p/Min@@p)]&]
%Y A339563 These partitions are counted by A097986 (non-strict: A083710).
%Y A339563 The case with no 1's is counted by A098965 (non-strict: A083711).
%Y A339563 The squarefree complement is A339562, ranked by A341450.
%Y A339563 The complement of the not necessarily squarefree version is A342193.
%Y A339563 A000005 counts divisors.
%Y A339563 A000070 counts partitions with a selected part.
%Y A339563 A001055 counts factorizations.
%Y A339563 A001221 counts distinct prime factors.
%Y A339563 A005117 lists squarefree numbers.
%Y A339563 A006128 counts partitions with a selected position (strict: A015723).
%Y A339563 A056239 adds up prime indices, row sums of A112798.
%Y A339563 A338470 counts partitions with no dividing part.
%Y A339563 Cf. A130714, A253249, A257993, A299702, A339564, A343340, A343378.
%K A339563 nonn
%O A339563 1,1
%A A339563 _Gus Wiseman_, Apr 10 2021