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 1-4 of 4 results.

A064807 Numbers which are divisible by their digital root (A010888).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 19, 20, 21, 24, 27, 28, 30, 36, 37, 38, 39, 40, 42, 45, 46, 48, 50, 54, 55, 56, 57, 60, 63, 64, 66, 70, 72, 73, 74, 75, 76, 78, 80, 81, 82, 84, 90, 91, 92, 93, 95, 96, 99, 100, 102, 108, 109, 110, 111, 112, 114, 117, 118
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 21 2001

Keywords

Comments

All numbers 9m, m > 0, belong to this sequence.
All numbers 6m, m > 0, belong to this sequence. - Christian Schulz, Oct 30 2013
All numbers 280m, m > 0, belong to this sequence. Only 6, 9, 280, and their multiples have this property. - Charles R Greathouse IV, Dec 26 2013
Conjecture: All k-multiply perfect numbers belong to this sequence. - Ivan N. Ianakiev, May 10 2016
The asymptotic density of this sequence is 1321/2520 = 0.524206... (see A074947 and A074949 for the values in other base representations). - Amiram Eldar, Nov 24 2022
The even perfect numbers are a subsequence. It is an open question whether the odd perfect numbers are a subsequence; this would involve ruling out 148 residue classes mod 2520 as OPNs. - Charles R Greathouse IV, Jan 03 2023

Examples

			48: 4 + 8 = 12 -> 1 + 2 = 3. 48 = 3 * 16 therefore 48 = a(28).

Links

Crossrefs

Cf. A010888, A074947, A074949.

Programs

  • Haskell
    a064807 n = a064807_list !! (n-1)
a064807_list = filter (\x -> x `mod` a010888 x == 0) [1..]
-- Reinhard Zumkeller, Jan 03 2014
  • Maple
    A064807 := proc(n) option remember: local k: if(n=1)then return 1:fi: for k from procname(n-1)+1 do if(k mod (((k-1) mod 9) + 1) = 0)then return k: fi: od: end: seq(A064807(n),n=1..100); # Nathaniel Johnston, May 05 2011
  • Mathematica
    Select[Range[125], Divisible[#, Mod[# - 1, 9] + 1] &] (* Alonso del Arte, Nov 01 2013 *)
  • PARI
    is(n)=n%((n-1)%9+1)==0 \\ Charles R Greathouse IV, Dec 26 2013

Formula

a(n) = a(n-1321) + 2520. - Charles R Greathouse IV, Dec 26 2013
2520n/1321 - 10 < a(n) <= 2520n/1321. (In fact, if you exclude n = 10 mod 1321, you can replace 10 with 9.) - Charles R Greathouse IV, Jan 03 2023
a(n) = a(n-1) + a(n-1321) - a(n-1322). - Charles R Greathouse IV, Apr 20 2023

Extensions

Offset changed from 0 to 1 by Harry J. Smith, Sep 26 2009

A074947 Numerators of Sum_{k=1..n} 1/lcm(n,k).

Original entry on oeis.org

1, 1, 5, 5, 37, 47, 69, 263, 1321, 1429, 9901, 17249, 113741, 161191, 32867, 158363, 3157279, 5777183, 18358381, 20649997, 9258477, 10610101, 24266365, 2411391361, 2299685867, 1072410923, 22804031069, 27841579901, 395718022103
Offset: 1

Views

Author

Benoit Cloitre, Oct 05 2002

Keywords

Comments

a(n)/A074949(n) is the asymptotic density of numbers that are divisible by their digital root in base n+1 (e.g., A064807 for base 10). - Amiram Eldar, Nov 24 2022

Examples

			1, 1, 5/6, 5/6, 37/60, 47/60, 69/140, 263/420, 1321/2520, 1429/2520, ...

Links

Crossrefs

Cf. A064807, A074949 (denominators).

Programs

  • Mathematica
    a[n_] := Numerator @ Sum[1/LCM[k, n], {k, 1, n}]; Array[a, 30] (* Amiram Eldar, Jan 31 2021 *)
  • PARI
    a(n)=numerator(sum(i=1,n,1/lcm(n,i)))

A333695 Numerators of coefficients in expansion of Sum_{k>=1} phi(k) * log(1/(1 - x^k)).

Original entry on oeis.org

1, 3, 7, 11, 21, 7, 43, 43, 61, 63, 111, 77, 157, 129, 49, 171, 273, 61, 343, 231, 43, 333, 507, 301, 521, 471, 547, 473, 813, 147, 931, 683, 259, 819, 129, 671, 1333, 1029, 1099, 903, 1641, 43, 1807, 111, 427, 1521, 2163, 399, 2101, 1563, 637, 1727, 2757, 547, 2331
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 02 2020

Keywords

Examples

			1, 3/2, 7/3, 11/4, 21/5, 7/2, 43/7, 43/8, 61/9, 63/10, 111/11, 77/12, 157/13, 129/14, 49/5, ...

Crossrefs

Cf. A000010, A000203, A001157, A018804, A057660, A071708, A072155, A074947, A074949, A333696 (denominators), A346602.

Programs

  • Mathematica
    nmax = 55; CoefficientList[Series[Sum[EulerPhi[k] Log[1/(1 - x^k)], {k, 1, nmax}], {x, 0, nmax}], x] // Numerator // Rest
Table[Sum[EulerPhi[n/d]/d, {d, Divisors[n]}], {n, 55}] // Numerator
Table[Sum[1/GCD[n, k], {k, n}], {n, 55}] // Numerator
Table[DivisorSigma[2, n^2]/(n DivisorSigma[1, n^2]), {n, 55}] // Numerator
  • PARI
    a(n) = numerator(sumdiv(n, d, eulerphi(n/d) / d)); \\ Michel Marcus, Apr 03 2020

Formula

a(n) = numerator of Sum_{d|n} phi(n/d) / d.
a(n) = numerator of Sum_{k=1..n} 1 / gcd(n,k).
a(n) = numerator of sigma_2(n^2) / (n * sigma_1(n^2)).
a(p) = p^2 - p + 1 where p is prime.
From Amiram Eldar, Nov 21 2022: (Start)
a(n) = numerator(A057660(n)/n).
Sum_{k=1..n} a(k)/A333696(k) ~ c * n^2, where c = zeta(3)/(2*zeta(2)) = 0.365381... (A346602). (End)

A333696 Denominators of coefficients in expansion of Sum_{k>=1} phi(k) * log(1/(1 - x^k)).

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 7, 8, 9, 10, 11, 12, 13, 14, 5, 16, 17, 6, 19, 20, 3, 22, 23, 24, 25, 26, 27, 28, 29, 10, 31, 32, 11, 34, 5, 36, 37, 38, 39, 40, 41, 2, 43, 4, 15, 46, 47, 16, 49, 50, 17, 52, 53, 18, 55, 56, 57, 58, 59, 20, 61, 62, 63, 64, 65, 22, 67, 68, 23, 10
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 02 2020

Keywords

Examples

			1, 3/2, 7/3, 11/4, 21/5, 7/2, 43/7, 43/8, 61/9, 63/10, 111/11, 77/12, 157/13, 129/14, 49/5, ...

Crossrefs

Cf. A000010, A000203, A001157, A018804, A057660, A070999, A071000 (fixed points), A071708, A072155, A074947, A074949, A333695 (numerators).

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Sum[EulerPhi[k] Log[1/(1 - x^k)], {k, 1, nmax}], {x, 0, nmax}], x] // Denominator // Rest
Table[Sum[EulerPhi[n/d]/d, {d, Divisors[n]}], {n, 70}] // Denominator
Table[Sum[1/GCD[n, k], {k, n}], {n, 70}] // Denominator
Table[DivisorSigma[2, n^2]/(n DivisorSigma[1, n^2]), {n, 70}] // Denominator
  • PARI
    a(n) = denominator(sumdiv(n, d, eulerphi(n/d) / d)); \\ Michel Marcus, Apr 03 2020

Formula

a(n) = denominator of Sum_{d|n} phi(n/d) / d.
a(n) = denominator of Sum_{k=1..n} 1 / gcd(n,k).
a(n) = denominator of sigma_2(n^2) / (n * sigma_1(n^2)).
Showing 1-4 of 4 results.

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