A328401 Lexicographically earliest infinite sequence such that a(i) = a(j) => A328400(i) = A328400(j) for all i, j.

1, 2, 2, 3, 2, 2, 2, 4, 3, 2, 2, 5, 2, 2, 2, 6, 2, 5, 2, 5, 2, 2, 2, 7, 3, 2, 4, 5, 2, 2, 2, 8, 2, 2, 2, 3, 2, 2, 2, 7, 2, 2, 2, 5, 5, 2, 2, 9, 3, 5, 2, 5, 2, 7, 2, 7, 2, 2, 2, 5, 2, 2, 5, 10, 2, 2, 2, 5, 2, 2, 2, 11, 2, 2, 5, 5, 2, 2, 2, 9, 6, 2, 2, 5, 2, 2, 2, 7, 2, 5, 2, 5, 2, 2, 2, 12, 2, 5, 5, 3, 2, 2, 2, 7, 2

Offset: 1

Author

Antti Karttunen, Oct 17 2019

Keywords

• nonn

Comments

Restricted growth sequence transform of A328400(n), or equally, of A007947(A181819(n)).

For all i, j:

A101296(i) = A101296(j) => a(i) = a(j),

a(i) = a(j) => A051903(i) = A051903(j) => A008966(i) = A008966(j),

a(i) = a(j) => A051904(i) = A051904(j),

a(i) = a(j) => A052409(i) = A052409(j),

a(i) = a(j) => A072411(i) = A072411(j),

a(i) = a(j) => A071625(i) = A071625(j),

a(i) = a(j) => A267115(i) = A267115(j),

a(i) = a(j) => A267116(i) = A267116(j).

Examples

			Numbers 2 (= 2^1), 3 (= 3^1), 6 = (2^1 * 3^1) and 30 (2^1 * 3^1 * 5^1) all have just one distinct exponent, 1, in the multisets of exponents that occur in their prime factorization, thus they all have the same value a(2) = a(3) = a(6) = a(30) = 2 in this sequence.
Number 4 (2^2), 9 (3^2) and 36 (2^2 * 3^2) all have just one distinct exponent, 2, in the multisets of exponents that occur in their prime factorization, thus they all have the same value a(4) = a(9) = a(36) = 3 in this sequence.
Numbers 12 = 2^2 * 3^1, 18 = 2^1 * 3^2, 60 = 2^2 * 3^1 * 5^1 and 300 = 2^2 * 3^1 * 5^2 all have both 1 and 2 and none other values occurring in the multisets of exponents in their prime factorization, thus they all have the value of a(12) = 5 that was allotted to 12 by the restricted growth sequence transform, as 12 is the smallest number with prime signature (1, 2).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..100000
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A007947, A181819, A328400.

Cf. A005117 (gives indices of terms <= 2), A062503 (after its initial 1, gives indices of 3's).

Cf. A008966, A051903, A051904, A052409, A072411, A071625, A267115, A267116.

Programs

PARI

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007947(n) = factorback(factorint(n)[, 1]);
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
v328401 = rgs_transform(vector(up_to, n, A007947(A181819(n)))); \\ Faster than with A328400(n).
A328401(n) = v328401[n];