A328456 -id:A328456 - cp's OEIS frontend

A328219 LCM of the prime indices of n, all plus 1.

1, 2, 3, 2, 4, 6, 5, 2, 3, 4, 6, 6, 7, 10, 12, 2, 8, 6, 9, 4, 15, 6, 10, 6, 4, 14, 3, 10, 11, 12, 12, 2, 6, 8, 20, 6, 13, 18, 21, 4, 14, 30, 15, 6, 12, 10, 16, 6, 5, 4, 24, 14, 17, 6, 12, 10, 9, 22, 18, 12, 19, 12, 15, 2, 28, 6, 20, 8, 30, 20, 21, 6, 22, 26

Offset: 1

Gus Wiseman, Oct 16 2019

nonn

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.

Sorted positions of first appearances are A328451.
LCM of prime indices is A290103.
LCM of prime indices minus 1 is A328456.
GCD of prime indices plus 1 is A328169.
Partitions whose parts plus 1 are relatively prime are A318980.
Numbers whose prime indices plus 1 are relatively prime are A318981,
Cf. A003961, A056239, A112798, A258409, A289508, A289509, A328167, A328168.

Table[If[n==1,1,LCM@@(PrimePi/@First/@FactorInteger[n]+1)],{n,100}]

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A290103(n) = lcm(apply(p->primepi(p),factor(n)[,1]));
A328219(n) = A290103(A003961(n)); \\ Antti Karttunen, Oct 18 2019

a(n) = A290103(A003961(n)).

If n = A000040(i_1) * ... * A000040(i_k), then a(n) = lcm(1+i_1,...,1+i_k).

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).

Original entry on oeis.org

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, 74, 78, 79, 87, 89, 91, 95, 101, 105, 106, 107, 111, 113, 119, 122, 127, 133, 141, 145, 151, 158, 159, 173, 174, 178, 181, 182, 183, 185, 195, 199, 202, 203, 214, 221

Offset: 1

Gus Wiseman, Oct 17 2019

nonn

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.

Indices of 1's in the ordinal transform of A328219. - Antti Karttunen, Oct 18 2019

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   6: {1,2}
   7: {4}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  21: {2,4}
  26: {1,6}
  29: {10}
  35: {3,4}
  37: {12}
  38: {1,8}
  39: {2,6}
  42: {1,2,4}
  47: {15}

A subsequence of A005117.
Sorted positions of first appearances in A328219.
The GCD of the prime indices of n, all plus 1, is A328169(n).
The LCM of the prime indices of n, all minus 1, is A328456(n).
Partitions whose parts plus 1 are relatively prime are A318980.
Numbers whose prime indices plus 1 are relatively prime are A318981.
Cf. A001222, A056239, A112798, A289508, A289509, A290103, A328163, A328167, A328168, A328170.

dav=Table[If[n==1,1,LCM@@(PrimePi/@First/@FactorInteger[n]+1)],{n,100}];
Table[Position[dav,i][[1,1]],{i,dav//.{A___,x_,B___,x_,C___}:>{A,x,B,C}}]

up_to = 1024;
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; };
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A290103(n) = lcm(apply(p->primepi(p),factor(n)[,1]));
A328219(n) = A290103(A003961(n));
vord_trans = ordinal_transform(vector(up_to,n,A328219(n)));
for(n=1,up_to,if(1==vord_trans[n], print1(n,", "))); \\ Antti Karttunen, Oct 18 2019

cp's OEIS Frontend

A328219 LCM of the prime indices of n, all plus 1.

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