A328451 - OEIS

%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