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A328451 Sorted positions of first appearances in A328219, where if n = A000040(i_1) * ... * A000040(i_k), then A328219(n) = LCM(1+i_1,...,1+i_k).

%I A328451 #17 Oct 18 2019 16:45:49
%S A328451 1,2,3,5,6,7,13,14,15,17,19,21,26,29,35,37,38,39,42,47,51,53,58,61,65,
%T A328451 74,78,79,87,89,91,95,101,105,106,107,111,113,119,122,127,133,141,145,
%U A328451 151,158,159,173,174,178,181,182,183,185,195,199,202,203,214,221
%N A328451 Sorted positions of first appearances in A328219, where if n = A000040(i_1) * ... * A000040(i_k), then A328219(n) = LCM(1+i_1,...,1+i_k).
%C A328451 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.
%C A328451 Indices of 1's in the ordinal transform of A328219. - _Antti Karttunen_, Oct 18 2019
%e A328451 The sequence of terms together with their prime indices begins:
%e A328451    1: {}
%e A328451    2: {1}
%e A328451    3: {2}
%e A328451    5: {3}
%e A328451    6: {1,2}
%e A328451    7: {4}
%e A328451   13: {6}
%e A328451   14: {1,4}
%e A328451   15: {2,3}
%e A328451   17: {7}
%e A328451   19: {8}
%e A328451   21: {2,4}
%e A328451   26: {1,6}
%e A328451   29: {10}
%e A328451   35: {3,4}
%e A328451   37: {12}
%e A328451   38: {1,8}
%e A328451   39: {2,6}
%e A328451   42: {1,2,4}
%e A328451   47: {15}
%t A328451 dav=Table[If[n==1,1,LCM@@(PrimePi/@First/@FactorInteger[n]+1)],{n,100}];
%t A328451 Table[Position[dav,i][[1,1]],{i,dav//.{A___,x_,B___,x_,C___}:>{A,x,B,C}}]
%o A328451 (PARI)
%o A328451 up_to = 1024;
%o A328451 ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
%o A328451 A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
%o A328451 A290103(n) = lcm(apply(p->primepi(p),factor(n)[,1]));
%o A328451 A328219(n) = A290103(A003961(n));
%o A328451 vord_trans = ordinal_transform(vector(up_to,n,A328219(n)));
%o A328451 for(n=1,up_to,if(1==vord_trans[n], print1(n,", "))); \\ _Antti Karttunen_, Oct 18 2019
%Y A328451 A subsequence of A005117.
%Y A328451 Sorted positions of first appearances in A328219.
%Y A328451 The GCD of the prime indices of n, all plus 1, is A328169(n).
%Y A328451 The LCM of the prime indices of n, all minus 1, is A328456(n).
%Y A328451 Partitions whose parts plus 1 are relatively prime are A318980.
%Y A328451 Numbers whose prime indices plus 1 are relatively prime are A318981.
%Y A328451 Cf. A001222, A056239, A112798, A289508, A289509, A290103, A328163, A328167, A328168, A328170.
%K A328451 nonn
%O A328451 1,2
%A A328451 _Gus Wiseman_, Oct 17 2019