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

1,2

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

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 *)

Prog

(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

Crossrefs

Cf. A172980, A172999

Sequence in context: A152259 A219611 A003071 * A319986 A359408 A284310

Adjacent sequences: A178439 A178440 A178441 * A178443 A178444 A178445

Keyword

nonn

Author

Vladimir Shevelev, Dec 22 2010

Extensions

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

Status

approved