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

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, 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, 11, 8, 19, 29, 59, 5, 61, 31, 9, 64, 13, 11, 67, 17, 24, 7, 71, 9, 73, 37, 25

Offset

1,2

Comments

a(n) is not a divisor of n for n = 21, 24, 42, 69, 84, 105, 115, 120, 138, 168, 171, ..., (A330736).

The sequence for the numerators only has terms for 1, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 33, ..., .

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Formula

a(n) = n iff n is a power of a prime (A000961).

a(n) < n iff n is a member of A024619.

a(n) >= A034699(n). - Robert Israel, Aug 11 2019

gcd(a(n), n) = A330691(n). - Antti Karttunen, Dec 29 2019

Maple

H:= 1: B[1]:= 1:
for n from 2 to 200 do H:= H + 1/n; B[n]:= denom(H) od:
f:= proc(n) local F, t0, t;
  t0:= max(seq(t[1]^t[2], t=ifactors(n)[2]));
  for t from t0 do if B[t] mod n = 0 then return t fi od
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Aug 11 2019

Mathematica

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

Prog

(PARI) f(n) = denominator(sum(k=2, n, 1/k)); \\ A002805
a(n) = my(k=1); while(f(k) % n, k++); k; \\ Michel Marcus, Aug 11 2019
(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

Crossrefs

Cf. A000961, A002805, A024619, A034699, A330691, A330692, A330741, A330753 (ordinal transform), A330734, A330735, A330736.

Sequence in context: A354933 A346596 A324388 * A327393 A217434 A322035

Adjacent sequences: A309636 A309637 A309638 * A309640 A309641 A309642

Keyword

nonn

Author

Robert G. Wilson v, Aug 11 2019

Status

approved