cp's OEIS Frontend

A370810 Numbers n such that only one set can be obtained by choosing a different divisor of each prime index of n.

%I A370810 #7 May 24 2024 03:51:10
%S A370810 1,2,6,9,10,22,25,30,34,42,45,62,63,66,75,82,98,99,102,110,118,121,
%T A370810 134,147,153,166,170,186,210,218,230,246,254,275,279,289,310,314,315,
%U A370810 330,343,354,358,363,369,374,382,390,402,410,422,425,462,482,490,495
%N A370810 Numbers n such that only one set can be obtained by choosing a different divisor of each prime index of n.
%C A370810 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 A370810 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 A370810 The terms together with their prime indices begin:
%e A370810     1: {}
%e A370810     2: {1}
%e A370810     6: {1,2}
%e A370810     9: {2,2}
%e A370810    10: {1,3}
%e A370810    22: {1,5}
%e A370810    25: {3,3}
%e A370810    30: {1,2,3}
%e A370810    34: {1,7}
%e A370810    42: {1,2,4}
%e A370810    45: {2,2,3}
%e A370810    62: {1,11}
%e A370810    63: {2,2,4}
%e A370810    66: {1,2,5}
%e A370810    75: {2,3,3}
%e A370810    82: {1,13}
%e A370810    98: {1,4,4}
%e A370810    99: {2,2,5}
%e A370810   102: {1,2,7}
%e A370810   110: {1,3,5}
%t A370810 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A370810 Select[Range[100],Length[Union[Sort /@ Select[Tuples[Divisors/@prix[#]],UnsameQ@@#&]]]==1&]
%Y A370810 For no choices we have A355740, counted by A370320.
%Y A370810 For at least one choice we have A368110, counted by A239312.
%Y A370810 Partitions of this type are counted by A370595 and A370815.
%Y A370810 For just prime factors we have A370647, counted by A370594.
%Y A370810 For more than one choice we have A370811, counted by A370803.
%Y A370810 A000005 counts divisors.
%Y A370810 A006530 gives greatest prime factor, least A020639.
%Y A370810 A027746 lists prime factors, A112798 indices, length A001222.
%Y A370810 A355731 counts choices of a divisor of each prime index, firsts A355732.
%Y A370810 A355741, A355744, A355745 choose prime factors of prime indices.
%Y A370810 A370814 counts factorizations with choosable divisors, complement A370813.
%Y A370810 Cf. A133686, A355529, A355739, A355749, A367771, A367904, A370584, A370592, A370808, A370816.
%K A370810 nonn
%O A370810 1,2
%A A370810 _Gus Wiseman_, Mar 05 2024