A370810 - OEIS

%I #7 May 24 2024 03:51:10

%S 1,2,6,9,10,22,25,30,34,42,45,62,63,66,75,82,98,99,102,110,118,121,

%T 134,147,153,166,170,186,210,218,230,246,254,275,279,289,310,314,315,

%U 330,343,354,358,363,369,374,382,390,402,410,422,425,462,482,490,495

%N Numbers n such that only one set can be obtained by choosing a different divisor of each prime index of n.

%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 prime indices of 6591 are {2,6,6,6}, for which the only choice is {1,2,3,6}, so 6591 is in the sequence.

%e The terms together with their prime indices begin:

%e     1: {}

%e     2: {1}

%e     6: {1,2}

%e     9: {2,2}

%e    10: {1,3}

%e    22: {1,5}

%e    25: {3,3}

%e    30: {1,2,3}

%e    34: {1,7}

%e    42: {1,2,4}

%e    45: {2,2,3}

%e    62: {1,11}

%e    63: {2,2,4}

%e    66: {1,2,5}

%e    75: {2,3,3}

%e    82: {1,13}

%e    98: {1,4,4}

%e    99: {2,2,5}

%e   102: {1,2,7}

%e   110: {1,3,5}

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],Length[Union[Sort /@ Select[Tuples[Divisors/@prix[#]],UnsameQ@@#&]]]==1&]

%Y For no choices we have A355740, counted by A370320.

%Y For at least one choice we have A368110, counted by A239312.

%Y Partitions of this type are counted by A370595 and A370815.

%Y For just prime factors we have A370647, counted by A370594.

%Y For more than one choice we have A370811, counted by A370803.

%Y A000005 counts divisors.

%Y A006530 gives greatest prime factor, least A020639.

%Y A027746 lists prime factors, A112798 indices, length A001222.

%Y A355731 counts choices of a divisor of each prime index, firsts A355732.

%Y A355741, A355744, A355745 choose prime factors of prime indices.

%Y A370814 counts factorizations with choosable divisors, complement A370813.

%Y Cf. A133686, A355529, A355739, A355749, A367771, A367904, A370584, A370592, A370808, A370816.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 05 2024