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.

Previous Showing 11-14 of 14 results.

A057791 Sum[k^(n-k)], where sum is over positive integers, k, where k <= n and gcd(k,n) = 1.

Original entry on oeis.org

1, 1, 3, 4, 22, 6, 209, 376, 1835, 2540, 49863, 94944, 1151914, 2190666, 12079274, 95722288, 1150653920, 3217888350, 47454745803, 130819911320, 846278385786, 8064305838350, 126356632390297, 288019285668096, 6861189820377586, 32739025597541220, 377062456413683219
Offset: 1

Views

Author

Leroy Quet, Nov 04 2000

Keywords

Examples

			a(8) = 1^7 + 3^5 + 5^3 + 7^1 = 376, since 1, 3, 5 and 7 are the positive integers relatively prime to 8 and <= 8.

Links

Crossrefs

Cf. A057789, A066840, A124440.

Programs

Extensions

a(26) onwards from John Tyler Rascoe, May 14 2025

A340714 a(n) is the sum of (n-2*j) for j < n/2 coprime to n.

Original entry on oeis.org

0, 0, 1, 2, 4, 4, 9, 8, 13, 12, 25, 12, 36, 24, 32, 32, 64, 28, 81, 40, 66, 60, 121, 48, 124, 84, 121, 84, 196, 56, 225, 128, 170, 144, 216, 108, 324, 180, 240, 160, 400, 120, 441, 220, 272, 264, 529, 192, 513, 252, 416, 312, 676, 244, 560, 336, 522, 420, 841, 240, 900, 480, 570, 512, 792, 320
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Jan 17 2021

Keywords

Comments

Sum of differences j-i for 0 < i < j coprime to n with i+j = n.
If p is an odd prime, a(p^k) = (p-1)*(p^(2*k-1)-1)/4.
Primes in this sequence are a(4) = 2 and a(3^k) = (3^(2*k-1)-1)/2 where 2*k-1 is in A028491.

Examples

			For n = 10, a(10) = (10-2*1) + (10-2*3) = 12.

Links

Crossrefs

Cf. A023896, A028491, A066840.

Programs

  • Maple
    f:= proc(n) local j; add(n-2*j, j= select(t -> igcd(t,n)=1, [$1..(n-1)/2])) end proc:
map(f, [$1..100]);
  • Mathematica
    Table[Sum[(n - 2 i) Floor[1/GCD[n - i, n]], {i, Floor[(n-1)/2]}], {n, 80}] (* Wesley Ivan Hurt, Jan 18 2021 *)

Formula

a(n) = A023896(n) - 2*A066840(n) for n >= 3.
a(n) = Sum_{k=1..floor((n-1)/2)} floor(1/gcd(n,n-k)) * (n-2*k). - Wesley Ivan Hurt, Jan 18 2021

A341032 Numbers k such that A124440(k) is a square.

Original entry on oeis.org

1, 2, 10, 17, 469, 646, 1542, 1601, 24939, 25090, 43690, 50925, 77577, 84002, 131087, 156817, 174755, 182106, 220974, 293930, 371307, 389130, 394290, 401573, 440819, 492886, 584326, 609301, 839590, 935685, 1727207, 1775622, 1939666, 1948705, 2235041, 2267650
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Feb 03 2021

Keywords

Comments

Terms in common with A341031, i.e. numbers such that both A066840(k) and A124440(k) are squares, include 1, 2, 10, 17, and 25090.

Examples

			a(4) = 17 is a term because A124440(17) = 100 = 10^2.

Links

Crossrefs

Cf. A066840, A122440, A341031.

Programs

  • Maple
    N:= 40000: # for terms <= N
G:= add(numtheory:-mobius(n)*n*x^(2*n)/((1-x^n)*(1-x^(2*n))^2), n=1..N/2):
S:= series(G, x, N+1):
A66840:= [seq(coeff(S, x, j), j=1..N)]:
f:= proc(n) n*numtheory:-phi(n)/2 - A66840[n] end proc:
f(1):= 1: f(2):= 1:
select(t -> issqr(f(t)), [$1..N]);

Extensions

More terms from Daniel Suteu, Feb 03 2021

A341015 Numbers k such that A124446(k) = 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 18, 25, 27, 54, 81, 125, 162, 243, 486, 625, 729, 1458, 2187, 3125, 4374, 6561, 13122, 15625, 19683, 39366, 59049, 78125, 118098, 177147, 354294, 390625, 531441, 1062882, 1594323, 1953125, 3188646, 4782969, 9565938, 9765625, 14348907, 28697814
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Feb 02 2021

Keywords

Comments

Numbers k such that A066840(k) and A124440(k) are coprime.
Contains all numbers of the forms 3^j, 2*3^j and 5^j.
Conjecture: the only term not of one of those forms is 4.

Examples

			18 is a term because A066840(18) = 13 and A124440(18) = 41 are coprime.

Crossrefs

Cf. A066840, A124440, A124446.

Programs

  • Maple
    N:= 2*10^4: # for terms <= N
G:= add(numtheory:-mobius(n)*n*x^(2*n)/((1-x^n)*(1-x^(2*n))^2),n=1..N/2):
S:= series(G,x,N+1):
A66840:= [seq(coeff(S,x,j),j=1..N)]:
filter:= n -> igcd(A66840[n], n*numtheory:-phi(n)/2)=1:
filter(1):= true:
select(filter, [$1..N]);

Formula

A124446(a(n)) = 1.

Extensions

More terms from Jinyuan Wang, Feb 07 2021
Previous Showing 11-14 of 14 results.

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