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%I #5 Mar 16 2019 10:12:50
%S 1,0,1,1,1,1,2,3,1,4,4,4,5,6,7,10,9,12,12,16,17,22,22,26,31,35,37,46,
%T 50,55,66,70,82,90,101,114,127,143,159,172,202,215,246,267,301,327,
%U 366,402,447,491,545,600,655,722,795,875,964,1050,1152,1259,1383
%N Number of strict integer partitions of n not containing 1 or any prime indices of the parts.
%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.
%e The a(2) = 1 through a(17) = 12 strict integer partitions (A...H = 10...17):
%e 2 3 4 5 6 7 8 9 A B C D E F G H
%e 42 43 54 64 65 75 76 86 87 97 98
%e 52 63 73 83 84 85 95 96 A6 A7
%e 72 82 542 93 94 A4 A5 C4 B6
%e A2 B2 B3 B4 D3 C5
%e 643 752 C3 E2 D4
%e 842 D2 763 E3
%e 654 943 854
%e 843 A42 863
%e 852 872
%e A52
%e B42
%e An example for n = 60 is (19,14,13,7,5,2), with prime indices:
%e 19: {8}
%e 14: {1,4}
%e 13: {6}
%e 7: {4}
%e 5: {3}
%e 2: {1}
%e None of these prime indices {1,3,4,6,8} belong to the partition, as required.
%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,1]&&Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]],{n,0,30}]
%Y The subset version is A324742, with maximal case is A324763. The non-strict version is A324757. The Heinz number version is A324761. An infinite version is A304360.
%Y Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A290822, A305713, A306844, A324764.
%Y Cf. A324695, A324743, A324748, A324751, A324756, A324758.
%K nonn
%O 0,7
%A _Gus Wiseman_, Mar 16 2019