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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A348941 a(n) = n / gcd(n, A326042(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 10, 11, 6, 13, 7, 15, 16, 17, 18, 19, 20, 21, 22, 23, 4, 25, 13, 27, 14, 29, 15, 31, 32, 33, 34, 35, 36, 37, 19, 39, 40, 41, 21, 43, 4, 45, 23, 47, 24, 49, 25, 17, 13, 53, 27, 11, 28, 57, 58, 59, 30, 61, 62, 63, 64, 65, 33, 67, 68, 23, 35, 71, 24, 73, 37, 75, 38, 77, 39, 79, 80, 81, 82
Offset: 1

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Author

Antti Karttunen, Nov 04 2021

Keywords

Comments

Denominator of ratio A326042(n) / n.
If there are no more 1's in this sequence after the initial one, then there are no odd terms of A336702 (numbers whose abundancy index is a power of 2) larger than one, and neither there are odd terms in A005820 or in A046060. Compare to similar conditions given in A336848, A336849 and A337339.

Links

Crossrefs

Cf. A005820, A046060, A326042, A336702, A336848, A336849, A337339, A348736, A348940, A348942 (numerators).

Programs

  • Mathematica
    f1[2, e_] := 1; f1[p_, e_] := NextPrime[p, -1]^e; s[n_] := Times @@ f1 @@@ FactorInteger[n]; f[p_, e_] := s[((q = NextPrime[p])^(e + 1) - 1)/(q - 1)]; s2[1] = 1; s2[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := n/GCD[n, s2[n]]; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)
  • PARI
    A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A326042(n) = A064989(sigma(A003961(n)));
A348941(n) = (n / gcd(n, A326042(n)));

Formula

a(n) = n / A348940(n) = n / gcd(n, A326042(n)).

A355934 a(n) = sigma(n) / gcd(sigma(n), sigma(A003961(n))), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 3, 2, 7, 3, 1, 2, 3, 13, 9, 6, 14, 7, 1, 1, 31, 9, 39, 5, 21, 4, 9, 4, 1, 31, 7, 10, 14, 15, 3, 16, 9, 4, 27, 1, 7, 19, 5, 14, 9, 21, 1, 11, 6, 39, 3, 8, 62, 3, 31, 3, 49, 9, 5, 9, 1, 5, 45, 30, 7, 31, 12, 26, 127, 7, 3, 17, 63, 8, 3, 36, 39, 37, 19, 62, 35, 4, 7, 20, 93, 11, 63, 14, 28, 27, 11, 5, 9, 45, 117
Offset: 1

Views

Author

Antti Karttunen, Jul 22 2022

Keywords

Comments

Denominator of ratio A003973(n) / A000203(n). See comments in A355933.

Links

Crossrefs

Cf. A000203, A003961, A003973, A355932, A355933 (numerators), A355940, A355941 (positions of 1's).
Cf. also A336849, A349162.

Programs

  • Mathematica
    f[p_, e_] := ((q = NextPrime[p])^(e + 1) - 1)/(q - 1); a[1] = 1; a[n_] := Denominator[Times @@ f @@@ FactorInteger[n] / DivisorSigma[1, n]]; Array[a, 100] (* Amiram Eldar, Jul 22 2022 *)
  • PARI
    A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
A355934(n) = { my(u=sigma(n)); (u/gcd(A003973(n), u)); };

Formula

a(n) = A000203(n) / A355932(n) = A000203(n) / gcd(A000203(n), A003973(n)).

A361468 a(n) = A249670(A003961(n)).

Original entry on oeis.org

1, 12, 30, 117, 56, 40, 132, 1080, 775, 672, 182, 390, 306, 176, 1680, 9801, 380, 9300, 552, 6552, 3960, 2184, 870, 144, 2793, 408, 19500, 1716, 992, 2240, 1406, 88452, 5460, 4560, 7392, 90675, 1722, 736, 9180, 60480, 1892, 5280, 2256, 126, 43400, 1160, 2862, 32670, 16093, 3724, 456, 442, 3540, 26000
Offset: 1

Views

Author

Antti Karttunen, Mar 20 2023

Keywords

Links

Crossrefs

Cf. A000290, A003961, A249670, A336850, A361467.

Programs

  • PARI
    A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A249670(n) = { my(ab = sigma(n)/n); numerator(ab)*denominator(ab); };
A361468(n) = A249670(A003961(n));

Formula

a(n) = A249670(A003961(n)) = A336849(n) * A341525(n).
a(n) = A361467(n) / A000290(A336850(n)).

A378996 Denominator of sigma(2*n)/(2*n).

Original entry on oeis.org

2, 4, 1, 8, 5, 3, 7, 16, 6, 10, 11, 2, 13, 1, 5, 32, 17, 36, 19, 4, 7, 11, 23, 12, 50, 26, 9, 7, 29, 5, 31, 64, 11, 34, 35, 24, 37, 19, 13, 40, 41, 3, 43, 22, 5, 23, 47, 8, 98, 100, 17, 52, 53, 27, 55, 14, 19, 58, 59, 1, 61, 31, 21, 128, 65, 11, 67, 68, 23, 5, 71, 144, 73, 74, 25, 38, 77, 39, 79, 80, 54, 82, 83, 7
Offset: 1

Views

Author

Antti Karttunen, Dec 13 2024

Keywords

Links

Crossrefs

Even bisection of A017666.
Topmost row of array A341606.
Cf. A378994, A378995 (numerators).
Cf. also A336849.

Programs

  • Mathematica
    a[n_] := Denominator[DivisorSigma[-1, 2*n]]; Array[a, 100] (* Amiram Eldar, Dec 14 2024 *)
  • PARI
    A378996(n) = denominator(sigma(2*n)/(2*n));

Formula

a(n) = A017666(2*n).
a(n) = 2*n / A378994(n).
Previous Showing 11-14 of 14 results.

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