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A277803 Minimum of A277802(n) and A004709(n).

%I A277803 #19 Aug 05 2024 17:32:45
%S A277803 1,2,3,2,5,6,7,3,10,11,12,13,14,15,17,12,19,20,21,22,23,5,26,28,29,30,
%T A277803 31,33,34,35,6,37,38,39,41,42,43,44,45,46,47,7,20,51,52,53,55,57,58,
%U A277803 59,60,61,62,63,65,66,67,68,69,70,71,73,74,45,76,77,78,79
%N A277803 Minimum of A277802(n) and A004709(n).
%C A277803 With the exception of the first term, each term appears exactly two times.
%C A277803 For how many n < m is A004709(n) < A277802(n)? It seems about m for large m. - _David A. Corneth_, Nov 01 2016
%H A277803 Peter Kagey, <a href="/A277803/b277803.txt">Table of n, a(n) for n = 1..10000</a>
%t A277803 Min /@ Transpose@ {#, Table[k = 1; While[! IntegerQ[(k #)^(1/3)], k++] &@ #[[n]]; k, {n, Length@ #}]} &@ Select[Range@ 80, FreeQ[FactorInteger@ #, {_, k_ /; k > 2}] &] (* _Michael De Vlieger_, Nov 10 2016, after _Jan Mangaldan_ at A004709 *)
%o A277803 \\ List of the first about n terms (probably a bit more).
%o A277803 (PARI) lista(n) = {n = ceil(1.21*n); my(l=List([1]), f); forprime(p=2,n, for(i=1,#l, if(l[i]*p<=n, listput(l, l[i] *p); if(l[i] * p^2<=n, listput(l, l[i]*p^2)))));listsort(l); for(i=2, #l, f=factor(l[i]); f[, 2] = vector(#f[,2], i, 3-(f[i, 2]%3))~; l[i] = min(l[i],factorback(f)));l} \\ _David A. Corneth_, Nov 01 2016
%o A277803 (Python)
%o A277803 from math import prod
%o A277803 from sympy import mobius, factorint, integer_nthroot
%o A277803 def A277803(n):
%o A277803     def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
%o A277803     m, k = n, f(n)
%o A277803     while m != k:
%o A277803         m, k = k, f(k)
%o A277803     return min(m,prod(p**(-e%3) for p, e in factorint(m).items())) # _Chai Wah Wu_, Aug 05 2024
%Y A277803 Cf. A004709, A277802.
%K A277803 nonn
%O A277803 1,2
%A A277803 _Peter Kagey_, Oct 31 2016