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A309639 Index of the least harmonic number H_i whose denominator (A002805) is divisible by n.

%I A309639 #30 Jan 11 2020 15:38:53
%S A309639 1,2,3,4,5,3,7,8,9,5,11,4,13,7,5,16,17,9,19,5,9,11,23,9,25,13,27,7,29,
%T A309639 5,31,32,11,17,7,9,37,19,13,8,41,9,43,11,9,23,47,16,49,25,17,13,53,27,
%U A309639 11,8,19,29,59,5,61,31,9,64,13,11,67,17,24,7,71,9,73,37,25
%N A309639 Index of the least harmonic number H_i whose denominator (A002805) is divisible by n.
%C A309639 a(n) is not a divisor of n for n = 21, 24, 42, 69, 84, 105, 115, 120, 138, 168, 171, ..., (A330736).
%C A309639 The sequence for the numerators only has terms for 1, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 33, ..., .
%H A309639 Antti Karttunen, <a href="/A309639/b309639.txt">Table of n, a(n) for n = 1..10000</a>
%H A309639 Antti Karttunen, <a href="/A309639/a309639.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A309639 a(n) = n iff n is a power of a prime (A000961).
%F A309639 a(n) < n iff n is a member of A024619.
%F A309639 a(n) >= A034699(n). - _Robert Israel_, Aug 11 2019
%F A309639 gcd(a(n), n) = A330691(n). - _Antti Karttunen_, Dec 29 2019
%p A309639 H:= 1: B[1]:= 1:
%p A309639 for n from 2 to 200 do H:= H + 1/n; B[n]:= denom(H) od:
%p A309639 f:= proc(n) local F, t0, t;
%p A309639   t0:= max(seq(t[1]^t[2],t=ifactors(n)[2]));
%p A309639   for t from t0 do if B[t] mod n = 0 then return t fi od
%p A309639 end proc:
%p A309639 f(1):= 1:
%p A309639 map(f, [$1..100]); # _Robert Israel_, Aug 11 2019
%t A309639 s = 0; k = 1; t[_] := 0; While[k < 101, s = s + 1/k; lst = Select[ Range@ 100, Mod[Denominator@ s, #] == 0 &]; If[t[#] == 0, t[#] = k] & /@ lst; k++]; t@# & /@ Range@75
%o A309639 (PARI) f(n) = denominator(sum(k=2, n, 1/k)); \\ A002805
%o A309639 a(n) = my(k=1); while(f(k) % n, k++); k; \\ _Michel Marcus_, Aug 11 2019
%o A309639 (PARI) A309639list(up_to) = { my(s=0,v002805=vector(up_to),v309639=vector(up_to)); v002805[1] = 1; for(k=2,up_to,s += 1/k; v002805[k] = denominator(s)); for(n=1,up_to,for(j=1,up_to,if(!(v002805[j]%n),v309639[n] = j; break))); (v309639); }; \\ _Antti Karttunen_, Dec 29 2019
%Y A309639 Cf. A000961, A002805, A024619, A034699, A330691, A330692, A330741, A330753 (ordinal transform), A330734, A330735, A330736.
%K A309639 nonn
%O A309639 1,2
%A A309639 _Robert G. Wilson v_, Aug 11 2019