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A035166
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Let d(m) = denominator of Sum_{k=1..m} 1/k^2 and consider f(m) = product of primes which appear to odd powers in d(m); sequence lists m such that f(m) is different from f(m-1).
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2
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1, 10, 15, 20, 25, 42, 49, 50, 55, 66, 75, 78, 91, 100, 110, 121, 125, 136, 153, 156, 164, 169, 171, 182, 189, 190, 205, 250, 253, 272, 276, 289, 294, 342, 343, 354, 361, 375, 406, 413, 435, 465, 473, 496, 500, 506, 516, 529, 555, 592, 605, 625
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OFFSET
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1,2
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COMMENTS
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The prime 479 first appears in f(m) at m = 2395, ahead of 71, which first appears in f(2485).
The first occurrence of four distinct primes is at m = 2500, with 5^7, 17^3, 71 and 479.
For 1890 < m < 2006, d(m) is a square (f(m)=1). The lone prime in 1875 .. 1890 is 61 and in 2006 .. 2027 it is 59.
It appears that f(m) can differ from f(m-1) in at most one prime.
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LINKS
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EXAMPLE
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f(10) = 5 is the first time f(m) > 1. The 5 persists until it disappears at m = 15.
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MATHEMATICA
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d[n_] := Denominator[ HarmonicNumber[n, 2]]; f[n_] := Times @@ Select[ FactorInteger[d[n]], OddQ[#[[2]]]&][[All, 1]]; A035166 = Join[{1}, Select[ Range[1000], f[#] != f[#-1]&]] (* Jean-François Alcover, Feb 26 2016 *)
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PROG
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(Macsyma) for k:1 do (subset(factor_number(denom(harmonic(k, 2))), lambda([x], oddp(second(x)))), if old#old:%% then print(k, %%))
(Haskell)
import Data.List (findIndices)
a035166 n = a035166_list !! (n-1)
a035166_list = map (+ 1) $ findIndices (/= 0) $ zipWith (-) (tail gs) gs
where gs = 0 : map a007913 a007407_list
(PARI) d(m) = denominator(sum(k=1, m, 1/k^2));
f(m) = my(f=factor(d(m))); for (k=1, #f~, if (!(f[k, 2] % 2), f[k, 2] = 0)); factorback(f);
isok(m) = if (m==1, 1, f(m) != f(m-1)); \\ Michel Marcus, Sep 06 2023
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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