A178442 Two numbers k and l we call equivalent if they have the same vector of exponents with positive components in prime power factorization. Let a(1)=1, a(2)=3. Then a(n)>a(n-1) is the smallest number equivalent to n.

1, 3, 5, 9, 11, 14, 17, 27, 49, 51, 53, 63, 67, 69, 74, 81, 83, 98, 101, 116, 118, 119, 127, 135, 169, 177, 343, 356, 359, 366, 367, 3125, 3127, 3131, 3133, 3249, 3251, 3254, 3261, 3272, 3299, 3302, 3307, 3308, 3316, 3317, 3319, 3321, 3481

Offset: 1

Author

Vladimir Shevelev, Dec 22 2010

Keywords

• nonn

Comments

Note that, e.g., 12 and 50 have similar structure in their prime power factorizations, but are not equivalent: their vectors of exponents are (2,1) and (1,2). On the other hand, 6 and 35 are equivalent with the same vector (1,1).

Question. What is the growth of the sequence?

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A172980, A172999

Programs

Mathematica

nxt[{n_,a_}]:=Module[{j=FactorInteger[n+1][[All,2]],k=a+1},While[ j!= FactorInteger[k][[All,2]],k++];{n+1,k}]; Join[{1},NestList[nxt,{2,3},50][[All,2]]] (* Harvey P. Dale, Jul 03 2020 *)

Sage

prime_signature = lambda n: [m for p, m in factor(n)]
@CachedFunction
def A178442(n):
    if n <= 2: return {1:1, 2:3}[n]
    psig_n = prime_signature(n)
    return next(k for k in IntegerRange(A178442(n-1)+1,infinity) if prime_signature(k) == psig_n)
# D. S. McNeil, Dec 22 2010

Extensions

Corrected and extended by D. S. McNeil, Dec 22 2010