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.

A062875 Records in A046112 (or A006339).

Original entry on oeis.org

1, 5, 25, 125, 3125, 15625, 390625, 1953125, 48828125, 6103515625, 30517578125, 3814697265625, 95367431640625, 476837158203125, 11920928955078125, 1490116119384765625, 186264514923095703125, 931322574615478515625, 116415321826934814453125
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

A111333 gives where records occur in A046112; A102781 gives where records occur in A006339.

Links

Crossrefs

Cf. A006339, A046112, A062876, A102781, A111333.

Programs

  • Python
    from sympy import prime
def A062875(n): return 5**(prime(n)-1>>1) # Chai Wah Wu, Aug 02 2024

Formula

a(n) = A046112(A111333(n)) = A006339(A102781(n)).
a(n) = 5^(A111333(n)-1) = 5^A102781(n).

Extensions

Edited and extended by Ray Chandler, Jan 05 2012

A046109 Number of lattice points (x,y) on the circumference of a circle of radius n with center at (0,0).

Original entry on oeis.org

1, 4, 4, 4, 4, 12, 4, 4, 4, 4, 12, 4, 4, 12, 4, 12, 4, 12, 4, 4, 12, 4, 4, 4, 4, 20, 12, 4, 4, 12, 12, 4, 4, 4, 12, 12, 4, 12, 4, 12, 12, 12, 4, 4, 4, 12, 4, 4, 4, 4, 20, 12, 12, 12, 4, 12, 4, 4, 12, 4, 12, 12, 4, 4, 4, 36, 4, 4, 12, 4, 12, 4, 4, 12, 12, 20, 4, 4, 12, 4, 12, 4, 12, 4, 4, 36
Offset: 0

Views

Author

Eric W. Weisstein

Keywords

Comments

Also number of Gaussian integers x + yi having absolute value n. - Alonso del Arte, Feb 11 2012
The indices of terms not equaling 4 or 12 correspond to A009177, n>0. - Bill McEachen, Aug 14 2025

Examples

			a(5) = 12 because the circumference of the circle with radius 5 will pass through the twelve points (5, 0), (4, 3), (3, 4), (0, 5), (-3, 4), (-4, 3), (-5, 0), (-4, -3), (-3, -4), (0, -5), (3, -4) and (4, -3). Alternatively, we can say the twelve Gaussian integers 5, 4 + 3i, ... , 4 - 3i all have absolute value of 5.

Links

Crossrefs

Cf. A004018, A046080, A046110, A046111, A046112, A063014, A256452.
Cf. A000328, A051132, A009177.

Programs

  • Haskell
    a046109 n = length [(x,y) | x <- [-n..n], y <- [-n..n], x^2 + y^2 == n^2]
-- Reinhard Zumkeller, Jan 23 2012
  • Maple
    N:= 1000: # to get a(0) to a(N)
A:= Array(0..N):
A[0]:= 1:
for x from 1 to N do
  A[x]:= A[x]+4;
  for y from 1 to min(x-1,floor(sqrt(N^2-x^2))) do
     z:= x^2+y^2;
     if issqr(z) then
       t:= sqrt(z);
       A[t]:= A[t]+8;
     fi
  od
od:
seq(A[i],i=0..N); # Robert Israel, May 08 2015
  • Mathematica
    Table[Length[Select[Flatten[Table[r + I i, {r, -n, n}, {i, -n, n}]], Abs[#] == n &]], {n, 0, 49}] (* Alonso del Arte, Feb 11 2012 *)
  • PARI
    a(n)=if(n==0, return(1)); my(f=factor(n)); 4*prod(i=1,#f~, if(f[i,1]%4==1, 2*f[i,2]+1, 1)) \\ Charles R Greathouse IV, Feb 01 2017
  • PARI
    a(n)=if(n==0, return(1)); t=0; for(x=1, n-1, y=n^2-x^2; if(issquare(y), t++)); return(4*t+4) \\ Arkadiusz Wesolowski, Nov 14 2017
  • Python
    from sympy import factorint
def a(n):
    r = 1
    for p, e in factorint(n).items():
        if p%4 == 1: r *= 2*e + 1
    return 4*r if n > 0 else 0
# Orson R. L. Peters, Jan 31 2017

Formula

a(n) = A000328(n) - A051132(n).
a(n) = 8*A046080(n) + 4 for n > 0.
a(n) = A004018(n^2).
From Jean-Christophe Hervé, Dec 01 2013: (Start)
a(A084647(k)) = 28.
a(A084648(k)) = 36.
a(A084649(k)) = 44. (End)
a(n) = 4 * Product_{i=1..k} (2*e_i + 1) for n > 0, given that p_i^e_i is the i-th factor of n with p_i = 1 mod 4. - Orson R. L. Peters, Jan 31 2017
a(n) = [x^(n^2)] theta_3(x)^2, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 20 2018
From Hugo Pfoertner, Sep 21 2023: (Start)
a(n) = 8*A063014(n) - 4 for n > 0.
a(n) = 4*A256452(n) for n > 0. (End)

A006339 Least hypotenuse of n distinct Pythagorean triangles.

Original entry on oeis.org

1, 5, 25, 125, 65, 3125, 15625, 325, 390625, 1953125, 1625, 48828125, 4225, 1105, 6103515625, 30517578125, 40625, 21125, 3814697265625, 203125, 95367431640625, 476837158203125, 5525, 11920928955078125, 274625, 5078125, 1490116119384765625, 528125, 25390625, 186264514923095703125
Offset: 0

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Except for offset, same as A046112.

Programs

  • Mathematica
    oneModFourPrimes[1] = 5;
oneModFourPrimes[n_] := oneModFourPrimes[n] = NestWhile[NextPrime, NextPrime[oneModFourPrimes[n - 1]], Mod[#, 4] != 1 & ];
factorizations[1, limit_] = {{}};
factorizations[n_, limit_] := factorizations[n, limit] = Join @@ Table[Prepend[#, d]& /@ factorizations[n/d, d], {d, Select[Rest[Divisors[n]], # <= limit & ]}];
leastHypotenuse[n_] := Min[(Times @@ (Array[oneModFourPrimes, Length[#]]^((# - 1)/2)) & ) /@ factorizations[2*n + 1, 2*n + 1]];
Array[leastHypotenuse, 30, 0]
(* Albert H. Mao, Jan 06 2012 *)

A111333 Number of odd numbers <= n-th prime.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 10, 12, 15, 16, 19, 21, 22, 24, 27, 30, 31, 34, 36, 37, 40, 42, 45, 49, 51, 52, 54, 55, 57, 64, 66, 69, 70, 75, 76, 79, 82, 84, 87, 90, 91, 96, 97, 99, 100, 106, 112, 114, 115, 117, 120, 121, 126, 129, 132, 135, 136, 139, 141, 142, 147, 154, 156, 157
Offset: 1

Views

Author

Giovanni Teofilatto, Nov 05 2005

Keywords

Comments

This is A006254 (numbers n such that 2n-1 is prime) with a leading 1. - Lambert Klasen (lambert.klasen(AT)gmx.net), Nov 06 2005
Same as smallest k such that prime(n) divides C(2k,k). - Jonathan Sondow, Jan 20 2016
Positions of records in A046112. - Hugo Pfoertner, Jul 11 2019

Links

Crossrefs

Cf. A006254, A046112, A049990, A062875.

Programs

Formula

a(n) = ceiling((prime_n)/2). - Robert G. Wilson v, Nov 07 2005

Extensions

More terms from Robert G. Wilson v, Nov 07 2005

A097756 Table read by rows of A054994 ordered by A046080.

Original entry on oeis.org

1, 5, 25, 125, 65, 625, 3125, 15625, 325, 78125, 390625, 1953125, 1625, 9765625, 48828125, 4225, 244140625, 1105, 8125, 1220703125, 6103515625, 30517578125, 40625, 152587890625, 21125, 762939453125, 3814697265625, 203125
Offset: 1

Views

Author

Ray Chandler, Aug 26 2004

Keywords

Comments

Row n of table is A054994(k) such that A046080(A054994(k)) = n; number of terms in row n is A001055(2n+1).
Column 1 of table gives A006339 (or A046112).

Examples

			Table begins:
0: 1,
1: 5,
2: 25,
3: 125,
4: 65,625,
5: 3125,
6: 15625,
7: 325,78125,
8: 390625,
9: 1953125,
10: 1625,9765625,
11: 48828125,
12: 4225,244140625,
13: 1105,8125,1220703125,
14: 6103515625,
15: 30517578125,
16: 40625,152587890625,
17: 21125,762939453125,
18: 3814697265625,
19: 203125,19073486328125,
20: 95367431640625,
...

Links

Crossrefs

Cf. A097751, A097752, A097753, A097754, A097755, A054994.

A062876 Numbers of lattice points corresponding to incrementally largest circle radii in A062875.

Original entry on oeis.org

4, 12, 20, 28, 44, 52, 68, 76, 92, 116, 124, 148, 164, 172, 188, 212, 236, 244, 268, 284, 292, 316, 332, 356, 388, 404, 412, 428, 436, 452, 508, 524, 548, 556, 596, 604, 628, 652, 668, 692, 716, 724, 764, 772, 788, 796, 844, 892, 908, 916, 932, 956, 964
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

For n = 1 and n >= 3, a(n) is the smallest nonsquarefree number divisible by prime(n). - David James Sycamore, Jun 15 2024

Links

Crossrefs

Cf. A046112, A062875.
Cf. A017113, A111333.
Cf. A000040, A103929

Programs

Formula

a(n) = A017113(A111333(n)-1) = 8*A111333(n) - 4.
For n >= 2 a(n) = 4*A000040(n) (a term in A013929). - David James Sycamore, Jun 15 2024

Extensions

Edited and extended by Ray Chandler, Jan 05 2012
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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