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.

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A353764 Numbers k for which A353749(sigma(k)) is a multiple of A353749(k), where A353749(k) = phi(k) * A064989(k), and A064989 shifts the prime factorization one step towards lower primes.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 18, 20, 24, 28, 30, 32, 40, 60, 72, 84, 90, 108, 120, 128, 200, 216, 224, 234, 252, 360, 384, 496, 600, 640, 672, 864, 936, 1080, 1120, 1152, 1170, 1488, 1800, 1920, 2016, 2176, 3200, 3360, 3456, 4320, 4464, 4680, 5148, 5600, 5760, 6048, 6528, 6552, 8128, 9600, 10080, 10880, 14976, 16800, 17280
Offset: 1

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Author

Antti Karttunen, May 10 2022

Keywords

Comments

Question: Are there any odd terms after the initial one? See A353789, A353796, A353797.

Links

Crossrefs

Cf. A000010, A000203, A006872, A062401, A064989, A336549, A336550, A353757, A353761, A353766 (characteristic function), A353789, A353796, A353797.
Positions of 1's in A353762. Cf. also A353765.
Subsequence of A353759. Cf. A007691 (a subsequence).

Programs

  • Mathematica
    f[p_, e_] := (p - 1)*p^(e - 1)*If[p == 2, 1, NextPrime[p, -1]^e]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[20000], Divisible[s[DivisorSigma[1, #]], s[#]] &] (* Amiram Eldar, May 10 2022 *)
  • PARI
    A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353749(n) = (eulerphi(n)*A064989(n));
isA353764(n) = { my(s=sigma(n)); !(A353749(s)%A353749(n)); };

A353758 Numbers k for which A353749(k) > A353749(sigma(k)), where A353749(k) = phi(k) * A064989(k), and A064989 shifts the prime factorization one step towards lower primes.

Original entry on oeis.org

3, 5, 7, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 99, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119, 122, 123, 124
Offset: 1

Views

Author

Antti Karttunen, May 10 2022

Keywords

Comments

First odd terms in the intersection of this sequence and A348749 are: 529, 605, 1445, 2825, 6125, 6425, 6875, 7025, ...

Links

Crossrefs

Positions of negative terms in A353757.
Cf. A353749, A353750, A353759 (complement), A353760 (characteristic function), A353765 (conjectured to be a subsequence).
Cf. also A348749, A353684, A353685.

Programs

A353763 a(n) = A353750(n) / gcd(A353749(n), A353750(n)).

Original entry on oeis.org

1, 4, 1, 15, 1, 2, 2, 12, 11, 2, 4, 15, 5, 8, 1, 435, 3, 22, 4, 5, 2, 24, 8, 6, 29, 10, 1, 2, 12, 2, 8, 45, 4, 9, 4, 165, 17, 16, 5, 6, 3, 8, 10, 12, 11, 48, 16, 435, 204, 58, 3, 175, 36, 4, 4, 8, 4, 72, 48, 5, 29, 32, 11, 7119, 5, 24, 208, 45, 8, 8, 48, 66, 31, 34, 29, 20, 16, 10, 16, 145, 2695, 18, 120, 2, 3, 40
Offset: 1

Views

Author

Antti Karttunen, May 10 2022

Keywords

Comments

Denominator of ratio A353749(n) / A353750(n).

Links

Crossrefs

Cf. A000010, A000203, A006872, A062401, A064989, A336549, A336550, A353757, A353761, A353762 (numerators), A353765 (positions of 1's).

Programs

  • PARI
    A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353749(n) = (eulerphi(n)*A064989(n));
A353763(n) = { my(s=sigma(n), u=A353749(s)); (u / gcd(A353749(n), u)); };

Formula

a(n) = A353750(n) / A353761(n) = A353750(n) / gcd(A353749(n), A353750(n)).
Showing 1-3 of 3 results.

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