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 1-6 of 6 results.

A083542 a(n) = phi(n+1)*phi(n), product of totients of two consecutive integers.

Original entry on oeis.org

1, 2, 4, 8, 8, 12, 24, 24, 24, 40, 40, 48, 72, 48, 64, 128, 96, 108, 144, 96, 120, 220, 176, 160, 240, 216, 216, 336, 224, 240, 480, 320, 320, 384, 288, 432, 648, 432, 384, 640, 480, 504, 840, 480, 528, 1012, 736, 672, 840, 640, 768, 1248, 936, 720, 960, 864, 1008
Offset: 1

Views

Author

Labos Elemer, May 21 2003

Keywords

Links

Crossrefs

Cf. A000010, A002378, A058515, A065474, A066813, A330319 (partial sums).
Cf. A083481, A083539, A092517.

Programs

  • Haskell
    a083542 n = a000010 n * a000010 (n + 1)
a083542_list = zipWith (*) (tail a000010_list) a000010_list
-- Reinhard Zumkeller, Apr 22 2012
  • Maple
    a:= n-> (p-> p(n)*p(n+1))(numtheory[phi]):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 21 2022
  • Mathematica
    Times @@ EulerPhi@ # & /@ Partition[Range@ 58, 2, 1] (* Michael De Vlieger, Mar 25 2017 *)
Times@@@Partition[EulerPhi[Range[60]],2,1] (* Harvey P. Dale, Oct 29 2019 *)
  • PARI
    a(n) = eulerphi(n) * eulerphi(n+1); \\ Amiram Eldar, Jul 10 2024

Formula

a(n) = A000010(A002378(n)). - Amiram Eldar, Jul 10 2024
Sum_{k=1..n} a(k) = c * n^3 / 3 + O((n*log(n))^2), where c = Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474). - Amiram Eldar, Dec 09 2024
a(n) = A058515(n)*A066813(n). - Amiram Eldar, May 07 2025

A083539 a(n) = sigma(n) * sigma(n+1): product of sigma-values for consecutive integers.

Original entry on oeis.org

3, 12, 28, 42, 72, 96, 120, 195, 234, 216, 336, 392, 336, 576, 744, 558, 702, 780, 840, 1344, 1152, 864, 1440, 1860, 1302, 1680, 2240, 1680, 2160, 2304, 2016, 3024, 2592, 2592, 4368, 3458, 2280, 3360, 5040, 3780, 4032, 4224, 3696, 6552, 5616, 3456, 5952
Offset: 1

Views

Author

Labos Elemer, May 21 2003

Keywords

Links

Crossrefs

Cf. A000203, A002378, A060781, A060780, A083538, A330322 (partial sums).
Cf. A083481, A083542, A092517.

Programs

  • Mathematica
    f[x_] := DivisorSigma[1, x]; t=Table[f[w+1]*f[w], {w, 1, 128}]
Times@@@Partition[DivisorSigma[1,Range[50]],2,1] (* Harvey P. Dale, May 21 2014 *)
  • PARI
    a(n)=sigma(n)*sigma(n+1) \\ Charles R Greathouse IV, Feb 14 2013

Formula

a(n) = A000203(A002378(n)). - Amiram Eldar, Jul 10 2024

A083482 Square root of smallest square of the type n(n+1)*k.

Original entry on oeis.org

2, 6, 6, 10, 30, 42, 28, 12, 30, 110, 66, 78, 182, 210, 60, 68, 102, 114, 190, 210, 462, 506, 276, 60, 130, 234, 126, 406, 870, 930, 248, 264, 1122, 1190, 210, 222, 1406, 1482, 780, 820, 1722, 1806, 946, 330, 690, 2162, 564, 84, 70, 510, 1326, 1378, 954, 990
Offset: 1

Views

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 03 2003

Keywords

Comments

Squares pertaining to A083481.
a(n) == (p*q*r... ) where p,q,r are prime factors of n(n+1).

Links

Crossrefs

Cf. A083481.

Programs

  • Mathematica
    Table[Times @@ ((a = Transpose[FactorInteger[n (n + 1)]])[[1]]^Quotient[a[[2]] + 1, 2]), {n, 54}] (* Ivan Neretin, May 20 2015 *)
  • PARI
    a(n)=sqrt(n*(n+1)*core(n*(n+1)))
  • Python
    from math import prod
from sympy import factorint
def A083482(n): return n*(n+1)//prod(p**(q>>1) for p, q in factorint(n*(n+1)).items()) # Chai Wah Wu, Mar 20 2023

Formula

a(n) = sqrt(A002378(n)*A083481(n)) = sqrt(A002378(n)*A007913(A002378(n))). a(n) = A019554(A002378(n)). - David Wasserman, Nov 16 2004

Extensions

More terms from Benoit Cloitre, May 04 2003
More terms from David Wasserman, Nov 16 2004

A361670 Squarefree part of the n-th triangular number.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 7, 1, 5, 55, 66, 78, 91, 105, 30, 34, 17, 19, 190, 210, 231, 253, 69, 3, 13, 39, 42, 406, 435, 465, 31, 33, 561, 595, 70, 74, 703, 741, 195, 205, 861, 903, 946, 110, 115, 1081, 282, 6, 1, 51, 1326, 1378, 159, 165, 385, 399, 1653, 1711, 1770, 1830, 1891, 217, 14, 130, 2145, 2211, 2278
Offset: 1

Views

Author

R. J. Mathar, Mar 20 2023

Keywords

Comments

a(n) / A083481(n) is either 2 or 1/2 depending on A136480(n) being even or odd, which is indicated by A039963(n).
a(n) = 1 for n>0 in A001108. - Michel Marcus, Mar 22 2023

Crossrefs

Cf. A000217, A007913, A083481 (of oblong), A361671 (of tetrahedral).
Cf. A001108, A039963.

Programs

  • Maple
    a:= n-> mul(i[1]^irem(i[2], 2), i=ifactors(n*(n+1)/2)[2]):
seq(a(n), n=1..70);  # Alois P. Heinz, Mar 20 2023
  • PARI
    a(n) = core(n*(n+1)/2); \\ Michel Marcus, Mar 22 2023
  • Python
    from sympy.ntheory.factor_ import core
def A361670(n): return core(n*(n+1)>>1) # Chai Wah Wu, Mar 20 2023

Formula

a(n) = A007913(A000217(n)).

A361671 Squarefree part of the n-th tetrahedral number.

Original entry on oeis.org

1, 1, 10, 5, 35, 14, 21, 30, 165, 55, 286, 91, 455, 35, 170, 51, 969, 285, 1330, 385, 1771, 506, 23, 26, 13, 91, 406, 1015, 4495, 310, 341, 374, 6545, 1785, 7770, 2109, 9139, 2470, 2665, 2870, 12341, 3311, 14190, 3795, 16215, 1081, 94, 1, 17, 221, 23426, 689, 2915, 770, 7315, 7714, 32509, 8555
Offset: 1

Views

Author

R. J. Mathar, Mar 20 2023

Keywords

Crossrefs

Cf. A007913, A000292, A361670 (of triangular), A083481 (of oblong).
Cf. A003556 (squarefree part is 1).

Programs

  • PARI
    a(n) = core(n*(n+1)*(n+2)/6); \\ Michel Marcus, Mar 22 2023
  • Python
    from sympy.ntheory.factor_ import core
def A361671(n): return core(n*(n*(n + 3) + 2)//6) # Chai Wah Wu, Mar 20 2023

Formula

a(n) = A007913(A000292(n)).

A306415 Numbers k such that A179682(k) <> A033996(k).

Original entry on oeis.org

0, 8, 24, 48, 49, 80, 120, 168, 224, 242, 288, 360, 440, 528, 624, 675, 728, 840, 960, 1088, 1224, 1368, 1444, 1520, 1680, 1681, 1848, 2024, 2208, 2400, 2600, 2645, 2808, 3024, 3248, 3480, 3720, 3968, 4224, 4374, 4488, 4760, 5040, 5328, 5624, 5928, 6240, 6560, 6727, 6888, 7224, 7568, 7920, 8280, 8648, 9024, 9408, 9800, 10200, 10608
Offset: 1

Views

Author

Robert Israel, Feb 15 2019

Keywords

Comments

0 and numbers k such that for some j with k < j < 4*k*(k+1), k*(k+1)*j*(j+1) is a square.
If k > 0 is a member, then so is A179682(k).
Includes A033996.
Conjecture: every member of the sequence is a member of A033996 or is A179682(k) for some k in the sequence.
A number k in this list indicates that A083481(k) is the same as some A083481(k') at an earlier place k'A083481(8) = A083481(1). 24 appears because A083481(24) = A083481(2). 242 appears because A083481(242) = A083481(24) = A083481(2). - R. J. Mathar, Mar 16 2023

Examples

			24 is a term because A179682(24) = 242: 24 < 242 < 4*24*25 and 24*25*242*243 = 5940^2.

Crossrefs

Cf. A033996, A179682.

Programs

  • Maple
    A179682:= proc(n) local F, t, p, k0, d, k, a, j;
  p:= max(map(t -> `if`(t[2]::odd, t[1], NULL), [op(ifactors(n)[2]), op(ifactors(n+1)[2])]));
  if n mod p = 0 then k0:= n+p-1; d:= 1;
    else  k0:= n+1; d:= p-1;
  fi;
  t:= n*(n+1)/4;
  for a from k0 by p do
    for k in [a, a+d] do
       if issqr(k*(k+1)*t) then return k fi
  od od
end proc:
f(0):= 1:
select(t -> A179682(t) <> 4*t*(t+1), [$0..11000]);
Showing 1-6 of 6 results.

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