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