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A382329 Least positive integer that gives a square of an integer when multiplied by the n-th harmonic number.

%I A382329 #38 May 11 2025 11:45:41
%S A382329 1,6,66,12,8220,20,420,213080,17965080,153720,2320468920,14109480,
%T A382329 412970037480,422245703880,430902992520,6076390320,516336630329520,
%U A382329 161488607280,21362271268818480,866533600973040,97555876321904,186715152624,52866073370045936
%N A382329 Least positive integer that gives a square of an integer when multiplied by the n-th harmonic number.
%H A382329 M. F. Hasler, <a href="/A382329/b382329.txt">Table of n, a(n) for n = 1..150</a>, May 08 2025
%F A382329 a(n) = A002805(n) * A007913(A001008(n)).
%e A382329 6*3/2 = 9, a square of an integer, so a(2) = 6.
%o A382329 (PARI) a(n)={my(s=sum(i=1, n, 1/i)); denominator(s)*core(numerator(s))}; \\ _Yifan Xie_, Mar 22 2025
%o A382329 (PARI) apply( {A382329(n)=denominator(n=harmonic(n))*core(numerator(n))}, [1..20]) \\ _M. F. Hasler_, May 08 2025
%o A382329 (Python)
%o A382329 from math import prod
%o A382329 from sympy import harmonic, factorint
%o A382329 def A382329(n):
%o A382329     a, b = harmonic(n).as_numer_denom()
%o A382329     return b*prod(p for p, e in factorint(a).items() if e&1) # _Chai Wah Wu_, Mar 23 2025
%Y A382329 Cf. A001008, A002805, A007913.
%K A382329 nonn
%O A382329 1,2
%A A382329 _Ali Sada_, Mar 21 2025
%E A382329 More terms from _Yifan Xie_, Mar 22 2025