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 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. 2
 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 (list; graph; refs; listen; history; text; internal format)
 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 A284310 A109324 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

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Last modified May 12 23:03 EDT 2021. Contains 343829 sequences. (Running on oeis4.)