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 1-7 of 7 results.

A124446 a(n) = gcd(A066840(n), A124440(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 4, 1, 4, 5, 6, 3, 3, 2, 16, 4, 1, 9, 20, 6, 5, 11, 24, 1, 12, 1, 42, 7, 8, 15, 64, 10, 16, 6, 54, 9, 9, 6, 80, 10, 6, 21, 110, 2, 11, 23, 96, 3, 4, 8, 156, 13, 1, 10, 168, 18, 28, 29, 120, 15, 15, 6, 256, 24, 10, 33, 272, 22, 24, 35, 216, 18, 36, 2, 342, 30, 24, 39
Offset: 1

Views

Author

Leroy Quet, Nov 01 2006

Keywords

Examples

			Those positive integers which are coprime to 8 and are <= 8/2, are 1 and 3. Those integers which are coprime to 8 and are between 8/2 and 8, are 5 and 7.
So a(8) = gcd(1+3, 5+7) = gcd(4, 12) = 4.

Links

Crossrefs

Cf. A066840, A124440, A124447.

Programs

  • Maple
    N:= 100: # for a(1)..a(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)]:
[1,1,seq(igcd(A66840[n], n*numtheory:-phi(n)/2),n=3..N)]; # Robert Israel, Feb 02 2021
  • Mathematica
    f1[n_] := Plus @@ Select[Range[Floor[n/2]], GCD[ #,n] == 1 &]; f2[n_] := Plus @@ Select[Range[Ceiling[n/2], n], GCD[ #, n] == 1 &];Table[GCD[f1[n], f2[n]], {n, 80}] (* Ray Chandler, Nov 12 2006 *)

Extensions

Extended by Ray Chandler, Nov 12 2006

A124447 a(n) = lcm(A066840(n), A124440(n)).

Original entry on oeis.org

0, 1, 2, 3, 21, 5, 30, 12, 140, 16, 120, 18, 399, 99, 322, 48, 900, 533, 630, 60, 480, 425, 1122, 72, 11781, 360, 11102, 126, 4515, 352, 2760, 192, 2000, 832, 5406, 162, 9405, 2349, 6726, 240, 12810, 2046, 7392, 330, 27202, 4235, 9660, 288, 66306, 11656
Offset: 1

Views

Author

Leroy Quet, Nov 01 2006

Keywords

Examples

			Those positive integers which are coprime to 8 and are <= 8/2, are 1 and 3. Those integers which are coprime to 8 and are between 8/2 and 8, are 5 and 7.
So a(8) = lcm(1+3,5+7) = lcm(4,12) = 12.

Crossrefs

Cf. A066840, A124440, A124446.

Programs

  • Mathematica
    f1[n_] := Plus @@ Select[Range[Floor[n/2]], GCD[ #,n] == 1 &]; f2[n_] := Plus @@ Select[Range[Ceiling[n/2], n], GCD[ #, n] == 1 &];Table[LCM[f1[n], f2[n]], {n, 51}] (* Ray Chandler, Nov 12 2006 *)

Extensions

Extended by Ray Chandler, Nov 12 2006

A341016 Numbers k such that A124440(k) is a multiple of A066840(k).

Original entry on oeis.org

2, 3, 4, 6, 8, 10, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232, 236, 240, 244
Offset: 1

Views

Author

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

Keywords

Comments

Numbers k such that k*A000010(k)/2 is a multiple of A066840(k).
Includes all multiples of 4.
Are 2, 3, 6 and 10 the only terms that are not multiples of 4?

Links

Crossrefs

Cf. A000010, A066840, A124440.

Programs

  • Maple
    N:= 1000: # 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 -> n*numtheory:-phi(n)/2 mod A66840[n] = 0:
select(filter, [$2..N]);

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

A066840 Sum of positive integers k where k <= n/2 and gcd(k,n) = 1.

Original entry on oeis.org

0, 1, 1, 1, 3, 1, 6, 4, 7, 4, 15, 6, 21, 9, 14, 16, 36, 13, 45, 20, 30, 25, 66, 24, 63, 36, 61, 42, 105, 32, 120, 64, 80, 64, 102, 54, 171, 81, 114, 80, 210, 66, 231, 110, 134, 121, 276, 96, 258, 124, 200, 156, 351, 121, 270, 168, 252, 196, 435, 120, 465, 225, 282
Offset: 1

Views

Author

Leroy Quet, Jan 20 2002

Keywords

Comments

a(n) = n iff n = 16, 20, 24. - Bernard Schott, Mar 17 2021

Examples

			a(8) = 4 = 1 + 3 because 1 and 3 are the positive integers <= 8 / 2 = 4 and relatively prime to 8.
a(36) = 54. First, factor 36 = 2^2 * 3^2. look at distinct prime factors, 2 and 3. Add all positive integers up to floor(36/2) = 18, gives binomial(18 + 1, 2) = 171. Subtract all multiples of 2, i.e., subtract 2 * binomial(1+floor(18/2), 2) = 90, gives 171 - 90 = 81. Subtract all multiples of 3, i.e., subtract 3 * binomial(1+floor(18/3), 2) = 63, gives 81 - 63 = 18. Multiples of 2 * 3 = 6 were subtracted twice so add them, i.e., add 6 * binomial(1+floor(18/6), 2) = 36. Gives 18 + 36 = 54.
a(29#) = 826398242058977280 where p# is as in A002110. - _David A. Corneth_, Apr 14 2015

Links

Crossrefs

Cf. A124440, A124446, A124447, A282600, A282601.

Programs

  • Maple
    seq(convert(select(k->igcd(k,n)=1, [$1..floor(n/2)]),`+`),n=1..100); # Robert Israel, Apr 12 2015
  • Mathematica
    f[n_] := Plus @@ Select[Range[Floor[n/2]], GCD[ #, n] == 1 &];Table[f[n], {n, 65}] (* Ray Chandler, Dec 06 2006 *)
  • PARI
    for (n=1, 1000, k=1; s=0; while(k<=n/2, if (gcd(k, n) == 1, s+=k); k++); write("b066840.txt", n, " ", s) ) \\ Harry J. Smith, Apr 01 2010
  • PARI
    a(n) = sum(k=1, n\2, if (gcd(n,k)==1, k)); \\ Michel Marcus, Nov 12 2014
  • PARI
    a(n) = {my(h=n\2, d, b, r=0); f=factor(n)[,1]; for(i=0,2^#f - 1, b=binary(i); d=#f-#b; p=prod(j=1,#b,f[j+#f-#b]^b[j]); r += (-1)^vecsum(b) * p * binomial(1+h\p,2));r} \\ David A. Corneth, Apr 14 2015

Formula

For odd prime p, a(p) = (p^2-1)/8. - Thomas Ordowski, Nov 12 2014
Conjecture: a(n) = n*phi(n)/8 + O(n). - Thomas Ordowski, Nov 12 2014
G.f.: Sum_{n>=1} mu(n)*n*x^(2*n)/((1-x^n)*(1-x^(2*n))^2), where mu(n)=A008683(n). - Mamuka Jibladze, Apr 05 2015
For prime p: a(2p) = floor(p/2)^2. a(3p) = (p-1)(3p-1)/4, p>3. a(4p) = (p-1)*p, p>2. a(5p) = (p-1)(5p-1)/2, p=3 or p>5, ... - M. F. Hasler, Apr 09 2015
For odd prime p and e>0, a(p^e) = (p^(2e-1)+1)*(p-1)/8; a(2^e) = 4^(e-2) for e>1 (and a(2)=1). - Mamuka Jibladze, Apr 10 2015
If n == 0 (mod 4) then a(n) = n*phi(n)/8. - Robert Israel, Apr 13 2015
If n is prime then a(n) = binomial(1 + floor(n/2), 2). - David A. Corneth, Apr 14 2015
a(n) = (1/2) * Sum_{k=1..n} k * [gcd(k,2*n-k) = 2], where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jun 07 2021

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

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
Showing 1-7 of 7 results.

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