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 1-9 of 9 results.

A330691 a(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, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 4, 13, 7, 5, 16, 17, 9, 19, 5, 3, 11, 23, 3, 25, 13, 27, 7, 29, 5, 31, 32, 11, 17, 7, 9, 37, 19, 13, 8, 41, 3, 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, 3, 7, 71, 9, 73, 37, 25, 19, 11, 13, 79, 16, 81, 41, 83, 3, 17, 43, 29, 11, 89, 9
Offset: 1

Views

Author

Antti Karttunen, Dec 29 2019

Keywords

Links

Crossrefs

Cf. A000961 (fixed points).
Cf. A002805, A309639, A330692, A330742.

Programs

  • Mathematica
    A309639[n_] := For[k = 1, True, k++, If[Divisible[Denominator[ HarmonicNumber[k]], n], Return[k]]];
a[n_] := GCD[n, A309639[n]];
Array[a, 105]
  • PARI
    A330691(n) = gcd(n, A309639(n));

Formula

a(n) = gcd(n, A309639(n)).
a(n) = n/A330692(n).

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).

A330742 a(n) = n / gcd(A309639(n), A319626(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 6, 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, 15, 2, 1, 6, 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, 30, 7, 4, 3, 2, 5, 6, 1, 2, 9, 4, 1, 6, 1, 8, 105
Offset: 1

Views

Author

Antti Karttunen, Dec 29 2019

Keywords

Links

Crossrefs

Cf. A309639, A319626, A330692, A330741, A330749.

Programs

Formula

a(n) = n / A330741(n) = n / gcd(A309639(n), A319626(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).

A330736 Numbers k such that k is not a multiple of A309639(k), where A309639(k) is the index of the least harmonic number H_i whose denominator (A002805) is divisible by k.

Original entry on oeis.org

21, 24, 42, 69, 84, 105, 115, 120, 138, 168, 171, 201, 207, 210, 225, 230, 276, 301, 329, 342, 345, 402, 407, 414, 420, 450, 451, 460, 473, 483, 505, 515, 602, 603, 605, 639, 658, 684, 690, 759, 804, 805, 814, 828, 840, 855, 869, 891, 897, 900, 902, 903, 913, 920, 946, 966, 987, 1005, 1010, 1030, 1035, 1173, 1197
Offset: 1

Views

Author

Antti Karttunen, Jan 10 2020

Keywords

Links

Crossrefs

Positions of nonzero terms in A330735.
Cf. A002805, A309639.

A330741 a(n) = gcd(A309639(n), A319626(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 2, 13, 7, 5, 16, 17, 9, 19, 5, 3, 11, 23, 3, 25, 13, 27, 7, 29, 5, 31, 32, 11, 17, 7, 9, 37, 19, 13, 8, 41, 3, 43, 11, 3, 23, 47, 8, 49, 25, 17, 13, 53, 27, 11, 8, 19, 29, 59, 5, 61, 31, 9, 64, 13, 11, 67, 17, 3, 7, 71, 9, 73, 37, 25, 19, 11, 13, 79, 16, 81, 41, 83, 3, 17, 43, 29, 11, 89, 3
Offset: 1

Views

Author

Antti Karttunen, Dec 29 2019

Keywords

Links

Crossrefs

Cf. A309639, A319626, A330742, A330749.

Programs

Formula

a(n) = gcd(A309639(n), A319626(n)).
a(n) = n / A330742(n).

A330753 Number of values of k, 1 <= k <= n, with A309639(k) = A309639(n), where A309639 gives the index of the least harmonic number whose denominator is divisible by 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, 5, 1, 2, 3, 2, 1, 6, 1, 4, 7, 2, 1, 2, 1, 2, 3, 4, 1, 2, 5, 3, 3, 2, 1, 6, 1, 2, 8, 1, 5, 6, 1, 4, 1, 5, 1, 9, 1, 2, 3, 4, 7, 6, 1, 3, 1, 2, 1, 10, 5, 2, 3, 8, 1, 11, 7, 3, 3, 2, 5, 2, 1, 2, 9, 4, 1, 6, 1, 8, 12
Offset: 1

Views

Author

Antti Karttunen, Dec 30 2019

Keywords

Comments

Ordinal transform of A309639.
For all n, a(A000961(n)) = 1, but the sequence obtains value 1 also on other n that are not prime powers. In range 1..65537 these extra 1's occur at n = 69, 201, 407, 505, 576, 869, 1791, 5157, 9383, 9691, 10219, 10571, 10991, 12575, 12731, 13343, 13739, 14179, 14483, 14693, 16173, 16723, 23347, 24209, 26233, 26377, 37393, 44407, 46089, 53707, 62063.

Links

Crossrefs

Cf. A000961, A309639.
Cf. also A303759, A330754.

Programs

  • PARI
    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; };
v330753 = ordinal_transform(vector(up_to, n, A309639(n)));
A330753(n) = v330753[n];

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];
Showing 1-9 of 9 results.

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