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A274459
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Least number of perfect powers that add up to n.
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3
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1, 2, 3, 1, 2, 3, 4, 1, 1, 2, 3, 2, 2, 3, 4, 1, 2, 2, 3, 2, 3, 3, 4, 2, 1, 2, 1, 2, 2, 3, 2, 1, 2, 2, 2, 1, 2, 3, 3, 2, 2, 3, 2, 2, 2, 3, 3, 2, 1, 2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 3, 2, 1, 2, 3, 3, 2, 3, 3, 3, 2, 2, 2, 3, 2, 3, 3, 3, 2, 1, 2, 3, 3, 2
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OFFSET
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1,2
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COMMENTS
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Least number of perfect powers (A001597) needed to add up to n.
This sequence is close to but not exactly equal to A063274.
a(n) is at most 4 since any number can be written as a sum of 4 squares (Lagrange's theorem), but it is possible that for a sufficiently large n, a(n) < 4.
a(n) <= a(i) + a(n-i) for 1 <= i <= n-1. (for computational ease, the maximum value for i can be chosen as floor(n/2)). a(1991) = 4. for 1992 <= k <= 20000, there is no k such that a(k) = 4. - David A. Corneth, Jun 24 2016 [Next such k is 25887, see A113505. - Vaclav Kotesovec, Jun 25 2016]
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LINKS
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EXAMPLE
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a(31) = 2 since 31 can be written as the sum of two (31 = 3^3 + 2^2 = 27 + 4) but no fewer than two perfect powers.
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MATHEMATICA
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nn = 72; t = Select[Range@ nn, # == 1 || GCD @@ FactorInteger[#][[All, 2]] > 1 &]; Table[Min@ Map[Length, Select[IntegerPartitions@ n, AllTrue[#, MemberQ[t, #] &] &]], {n, nn}] (* Michael De Vlieger, Jun 23 2016, after Ant King at A001597 *)
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PROG
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(PARI) lista(n) = {my(v = vector(n)); for(i = 2, sqrtint(n), for(j = 2, logint(n, i), v[i^j] = 1)); v[1]=1; v[2]=2; for(i=3, #v, if(v[i]==0, v[i] = vecmin(vector( i\2, k, v[k] + v[i-k])))); v} \\ David A. Corneth, Jun 24 2016; corrected by Peter Schorn, Jun 09 2022
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CROSSREFS
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Cf. A063275 (indices for which a(n)=3), A113505 (indices for which a(n)=4).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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