cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A330754 Number of values of k, 1 <= k <= n, with A330691(k) = A330691(n), where A330691(n) = gcd(n, A309639(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 1, 3, 1, 5, 1, 1, 3, 2, 4, 3, 1, 2, 3, 2, 1, 5, 1, 4, 4, 2, 1, 2, 1, 2, 3, 4, 1, 2, 5, 3, 3, 2, 1, 6, 1, 2, 5, 1, 5, 6, 1, 4, 6, 5, 1, 6, 1, 2, 3, 4, 7, 6, 1, 3, 1, 2, 1, 7, 5, 2, 3, 8, 1, 7, 7, 3, 3, 2, 5, 2, 1, 2, 9, 4, 1, 6, 1, 8, 8
Offset: 1

Views

Author

Antti Karttunen, Dec 30 2019

Keywords

Comments

Ordinal transform of A330691.

Links

Crossrefs

Cf. A309639, A330691, A330753.

Programs

  • Mathematica
    A309639[n_] := For[k = 1, True, k++, If[Divisible[Denominator[ HarmonicNumber[k]], n], Return[k]]];
A330691[n_] := GCD[n, A309639[n]];
Module[{b}, b[_] = 0;
a[n_] := With[{t = A330691[n]}, b[t] = b[t] + 1]];
Array[a, 105] (* Jean-François Alcover, Jan 11 2022 *)
  • PARI
    A330691(n) = gcd(n, A309639(n));
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
v330754 = ordinal_transform(vector(up_to, n, A330691(n)));
A330754(n) = v330754[n];

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

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Aug 11 2019

Keywords

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

Crossrefs

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

Programs

  • 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
  • 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

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

A330692 a(n) = n / gcd(n, A309639(n)), where A309639(n) is the index of the least harmonic number H_i whose denominator (A002805) is divisible by n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 3, 1, 1, 2, 1, 4, 7, 2, 1, 8, 1, 2, 1, 4, 1, 6, 1, 1, 3, 2, 5, 4, 1, 2, 3, 5, 1, 14, 1, 4, 5, 2, 1, 3, 1, 2, 3, 4, 1, 2, 5, 7, 3, 2, 1, 12, 1, 2, 7, 1, 5, 6, 1, 4, 23, 10, 1, 8, 1, 2, 3, 4, 7, 6, 1, 5, 1, 2, 1, 28, 5, 2, 3, 8, 1, 10, 7, 4, 3, 2, 5, 3, 1, 2, 9, 4, 1, 6, 1, 8, 35
Offset: 1

Views

Author

Antti Karttunen, Dec 29 2019

Keywords

Links

Crossrefs

Cf. A000961 (indices of 1's).
Cf. A002805, A309639, A330691, A330742.

Programs

Formula

a(n) = n/A330691(n) = n / gcd(n, A309639(n)).

A330735 a(n) = n mod A309639(n), where A309639(n) is the index of the least harmonic number H_i whose denominator (A002805) is divisible by n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 3
Offset: 1

Views

Author

Antti Karttunen, Jan 10 2020

Keywords

Links

Crossrefs

Cf. A309639, A330691, A330692, A330734, A330736 (indices of nonzero terms).

Programs

Formula

a(n) = n mod A309639(n).

A330734 a(n) = n - A309639(n), where A309639(n) is the index of the least harmonic number H_i whose denominator (A002805) is divisible by n.

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 0, 0, 0, 5, 0, 8, 0, 7, 10, 0, 0, 9, 0, 15, 12, 11, 0, 15, 0, 13, 0, 21, 0, 25, 0, 0, 22, 17, 28, 27, 0, 19, 26, 32, 0, 33, 0, 33, 36, 23, 0, 32, 0, 25, 34, 39, 0, 27, 44, 48, 38, 29, 0, 55, 0, 31, 54, 0, 52, 55, 0, 51, 45, 63, 0, 63, 0, 37, 50, 57, 66, 65, 0, 64, 0, 41, 0, 75, 68, 43, 58, 77, 0, 81
Offset: 1

Views

Author

Antti Karttunen, Jan 10 2020

Keywords

Links

Crossrefs

Cf. A000961 (indices of zeros), A309639, A330691, A330692, A330735.

Programs

Formula

a(n) = n - A309639(n).
Showing 1-5 of 5 results.

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