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Numbers whose greatest prime factor is a prime with an odd index; n such that A006530(n) is in A031368.
45

%I #16 Feb 11 2021 22:59:56

%S 2,4,5,8,10,11,15,16,17,20,22,23,25,30,31,32,33,34,40,41,44,45,46,47,

%T 50,51,55,59,60,62,64,66,67,68,69,73,75,77,80,82,83,85,88,90,92,93,94,

%U 97,99,100,102,103,109,110,115,118,119,120,121,123,124,125,127,128

%N Numbers whose greatest prime factor is a prime with an odd index; n such that A006530(n) is in A031368.

%C Equally, numbers n for which A061395(n) is odd.

%C A122111 maps each one of these numbers to a unique term of A026424 and vice versa.

%C If the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), these are the Heinz numbers of partitions whose greatest part is odd, counted by A027193. - _Gus Wiseman_, Feb 08 2021

%H Antti Karttunen, <a href="/A244991/b244991.txt">Table of n, a(n) for n = 1..10001</a>

%F For all n, A244989(a(n)) = n.

%e From _Gus Wiseman_, Feb 08 2021: (Start)

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

%e 2: {1} 32: {1,1,1,1,1} 64: {1,1,1,1,1,1}

%e 4: {1,1} 33: {2,5} 66: {1,2,5}

%e 5: {3} 34: {1,7} 67: {19}

%e 8: {1,1,1} 40: {1,1,1,3} 68: {1,1,7}

%e 10: {1,3} 41: {13} 69: {2,9}

%e 11: {5} 44: {1,1,5} 73: {21}

%e 15: {2,3} 45: {2,2,3} 75: {2,3,3}

%e 16: {1,1,1,1} 46: {1,9} 77: {4,5}

%e 17: {7} 47: {15} 80: {1,1,1,1,3}

%e 20: {1,1,3} 50: {1,3,3} 82: {1,13}

%e 22: {1,5} 51: {2,7} 83: {23}

%e 23: {9} 55: {3,5} 85: {3,7}

%e 25: {3,3} 59: {17} 88: {1,1,1,5}

%e 30: {1,2,3} 60: {1,1,2,3} 90: {1,2,2,3}

%e 31: {11} 62: {1,11} 92: {1,1,9}

%e (End)

%t Select[Range[100],OddQ[PrimePi[FactorInteger[#][[-1,1]]]]&] (* _Gus Wiseman_, Feb 08 2021 *)

%o (Scheme, with Antti Karttunen's IntSeq-library)

%o (define A244991 (MATCHING-POS 1 1 (COMPOSE odd? A061395)))

%Y Complement: A244990.

%Y Cf. A006530, A026424, A031368, A122111, A244321, A244322, A244989.

%Y Looking at least instead of greatest prime index gives A026804.

%Y The partitions with these Heinz numbers are counted by A027193.

%Y The case where Omega is odd also is A340386.

%Y A001222 counts prime factors.

%Y A056239 adds up prime indices.

%Y A300063 ranks partitions of odd numbers.

%Y A061395 selects maximum prime index.

%Y A066208 ranks partitions into odd parts.

%Y A112798 lists the prime indices of each positive integer.

%Y A340931 ranks odd-length partitions of odd numbers.

%Y Cf. A000009, A058695, A072233, A160786, A300272, A340101, A340385, A340604.

%K nonn

%O 1,1

%A _Antti Karttunen_, Jul 21 2014