A307516 - OEIS

%I #14 Apr 12 2019 09:33:46

%S 10,14,20,21,22,26,28,30,33,34,38,39,40,42,44,46,50,51,52,55,56,57,58,

%T 60,62,63,65,66,68,69,70,74,76,78,80,82,84,85,86,87,88,90,91,92,93,94,

%U 95,98,99,100,102,104,105,106,110,111,112,114,115,116,117,118

%N Numbers whose maximum prime index and minimum prime index differ by more than 1.

%C 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. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose maximum and minimum parts differ by more than 1. The enumeration of these partitions by sum is given by A000094.

%C Differs from A069900 first at n = 43.

%e The sequence of terms together with their prime indices begins:

%e    10: {1,3}

%e    14: {1,4}

%e    20: {1,1,3}

%e    21: {2,4}

%e    22: {1,5}

%e    26: {1,6}

%e    28: {1,1,4}

%e    30: {1,2,3}

%e    33: {2,5}

%e    34: {1,7}

%e    38: {1,8}

%e    39: {2,6}

%e    40: {1,1,1,3}

%e    42: {1,2,4}

%e    44: {1,1,5}

%e    46: {1,9}

%e    50: {1,3,3}

%e    51: {2,7}

%e    52: {1,1,6}

%e    55: {3,5}

%p with(numtheory):

%p q:= n-> (l-> pi(l[-1])-pi(l[1])>1)(sort([factorset(n)[]])):

%p select(q, [$2..200])[];  # _Alois P. Heinz_, Apr 12 2019

%t Select[Range[100],PrimePi[FactorInteger[#][[-1,1]]]-PrimePi[FactorInteger[#][[1,1]]]>1&]

%Y Positions of numbers > 1 in A243055. Complement of A000961 and A256617.

%Y Cf. A000094, A000245, A001222, A052126, A056239, A061395, A064989, A069900, A105441, A112798, A307517, A325196.

%K nonn

%O 1,1

%A _Gus Wiseman_, Apr 12 2019