A066840 -id:A066840 - cp's OEIS frontend

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

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

Leroy Quet, Nov 01 2006

nonn

			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.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A066840, A124440, A124447.

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

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 *)

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

Leroy Quet, Nov 01 2006

nonn

			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.

Cf. A066840, A124440, A124446.

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 *)

Extended by Ray Chandler, Nov 12 2006

A341031 Numbers k such that A066840(k) is a square.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 14, 16, 17, 22, 26, 32, 34, 38, 46, 54, 58, 62, 64, 74, 82, 86, 94, 106, 118, 122, 128, 134, 142, 146, 158, 166, 171, 178, 194, 202, 206, 214, 218, 226, 254, 255, 256, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 451, 454, 458, 466

Offset: 1

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

nonn

			a(6) = 8 is a term because A066840(8) = 4 = 2^2.

Robert Israel, Table of n, a(n) for n = 1..2500

Includes A000079 and A100484.

Cf. A066840.

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)]:
select(t -> issqr(A66840[t]), [$1..N]);

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

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

nonn

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?

Robert Israel, Table of n, a(n) for n = 1..5000

Cf. A000010, A066840, A124440.

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]);

A124440 a(n) = Sum_{n/2<=k<=n, gcd(k,n)=1} k.

Original entry on oeis.org

1, 1, 2, 3, 7, 5, 15, 12, 20, 16, 40, 18, 57, 33, 46, 48, 100, 41, 126, 60, 96, 85, 187, 72, 187, 120, 182, 126, 301, 88, 345, 192, 250, 208, 318, 162, 495, 261, 354, 240, 610, 186, 672, 330, 406, 385, 805, 288, 771, 376, 616, 468, 1027, 365, 830, 504, 774, 616, 1276

Offset: 1

Leroy Quet, Nov 01 2006

nonn

			The integers which are >= 10/2 and are <= 10 and which are coprime to 10 are 7 and 9. So a(10) = 7 + 9 = 16.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A066840, A124446, A124447.

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)]:
f:= proc(n) n*numtheory:-phi(n)/2 - A66840[n] end proc:
f(1):= 1: f(2):= 1:
map(f, [$1..N]); # Robert Israel, Feb 02 2021

a[n_] := Plus @@ Select[Range[Ceiling[n/2], n], GCD[ #, n] == 1 &];Table[a[n], {n, 60}] (* Ray Chandler, Nov 12 2006 *)

a(n) = sum(k=ceil(n/2), n, if (gcd(n, k)==1, k)); \\ Michel Marcus, Feb 03 2021

For n > 2, a(n) = phi(n)*n/2 - A066840(n).

Extended by Ray Chandler, Nov 12 2006

A282600 a(n) = Sum_(k=1..phi(n)) floor(d_k/2) where d_k are the totatives of n.

Original entry on oeis.org

0, 0, 1, 1, 4, 2, 9, 6, 12, 8, 25, 10, 36, 18, 28, 28, 64, 24, 81, 36, 60, 50, 121, 44, 120, 72, 117, 78, 196, 56, 225, 120, 160, 128, 204, 102, 324, 162, 228, 152, 400, 120, 441, 210, 264, 242, 529, 184, 504, 240, 400, 300, 676, 234, 540, 324, 504, 392, 841, 232

Offset: 1

Michel Marcus, Feb 19 2017

The totatives of n are the numbers k <= n with gcd(k,n) = 1.

Robert Israel, Table of n, a(n) for n = 1..10000
David Zmiaikou, Origamis and permutation groups, Thesis, 2011. See p. 65.

Cf. A000010, A023896, A038566, A066840, A282601.

a(n) = sum(k=1, n, (k\2)*(gcd(n, k)==1));

If n is odd, a(n) = A023896(n)/2 - A000010(n)/4;

If n is even, a(n) = A023896(n)/2 - A000010(n)/2.

A295574 a(n) = Sum_{1 <= j <= n/2, gcd(j,n)=1} j^2.

Original entry on oeis.org

0, 1, 1, 1, 5, 1, 14, 10, 21, 10, 55, 26, 91, 35, 70, 84, 204, 75, 285, 140, 210, 165, 506, 196, 525, 286, 549, 406, 1015, 340, 1240, 680, 880, 680, 1190, 654, 2109, 969, 1482, 1080, 2870, 966, 3311, 1650, 2010, 1771, 4324, 1544, 4214, 2050

Offset: 1

N. J. A. Sloane, Dec 08 2017

nonn

n does not divide a(n) iff n = (2^k)*(q^m) with k > 0, m >= 0 and q odd prime such that q == 3 (mod 4) or n = (2^k)*(3^L)*Product_{q} q^(v_q) with k >= 0, L > 0, v_q >= 0 and all q odd primes such that q == 5 (mod 6). - René Gy, Oct 21 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
John D. Baum, A Number-Theoretic Sum, Mathematics Magazine 55.2 (1982): 111-113.
René Gy, The sum of product pairs of integers prime to n, Math StackExchange.

In the Baum (1982) paper, S_1, S_2, S_3, S_4 are A023896, A053818, A053819, A053820, and S'_1, S'_2, S'_3, S'_4 are A066840, A295574, A295575, A295576.

Cf. A023022.

R:=proc(n,k) local x,t1,S;
t1:={}; S:=0;
for x from 1 to floor(n/2) do if gcd(x,n)=1 then t1:={op(t1),x^k}; S:=S+x^k; fi; od;
S; end;
s:=k->[seq(R(n,k),n=1..50)];
s(2);

f[n_] := Plus @@ (Select[ Range[n/2], GCD[#, n] == 1 &]^2); Array[f, 50] (* Robert G. Wilson v, Dec 10 2017 *)

a(n) = sum(j=1, n\2, (gcd(j, n)==1)*j^2); \\ Michel Marcus, Dec 10 2017

A295575 a(n) = Sum_{1 <= j <= n/2, gcd(j,n)=1} j^3.

Original entry on oeis.org

0, 1, 1, 1, 9, 1, 36, 28, 73, 28, 225, 126, 441, 153, 416, 496, 1296, 469, 2025, 1100, 1710, 1225, 4356, 1800, 4959, 2556, 5581, 4410, 11025, 3872, 14400, 8128, 11090, 8128, 15822, 8910, 29241, 13041, 21996, 16400, 44100, 15426, 53361, 27830, 33716, 29161, 76176, 27936, 77652, 37828

Offset: 1

N. J. A. Sloane, Dec 08 2017

nonn

If p is an odd prime, a(p) = (n^2-1)^2/64. - Robert Israel, Dec 10 2017

Seiichi Manyama, Table of n, a(n) for n = 1..10000
John D. Baum, A Number-Theoretic Sum, Mathematics Magazine 55.2 (1982): 111-113.

In the Baum (1982) paper, S_1, S_2, S_3, S_4 are A023896, A053818, A053819, A053820, and S'_1, S'_2, S'_3, S'_4 are A066840, A295574, A295575, A295576.

f:= n -> add(t^3, t = select(t->igcd(t,n)=1, [$1..n/2])) :
map(f, [$1..100]); # Robert Israel, Dec 10 2017

f[n_] := Plus @@ (Select[Range[n/2], GCD[#, n] == 1 &]^3); Array[f, 50] (* Robert G. Wilson v, Dec 10 2017 *)

a(n) = sum(j=1, n\2, (gcd(j, n)==1)*j^3); \\ Michel Marcus, Dec 10 2017

A295576 a(n) = Sum_{1 <= j <= n/2, gcd(j,n)=1} j^4.

Original entry on oeis.org

0, 1, 1, 1, 17, 1, 98, 82, 273, 82, 979, 626, 2275, 707, 2674, 3108, 8772, 3027, 15333, 9044, 14994, 9669, 39974, 17668, 50085, 24310, 60597, 50470, 127687, 45604, 178312, 103496, 149908, 103496, 225302, 129750, 432345, 187017, 349830, 266088, 722666

Offset: 1

N. J. A. Sloane, Dec 08 2017

nonn

If p is an odd prime, a(p) = p*(p^2-1)*(3*p^2-7)/480. - Robert Israel, Dec 10 2017

Seiichi Manyama, Table of n, a(n) for n = 1..10000
John D. Baum, A Number-Theoretic Sum, Mathematics Magazine 55.2 (1982): 111-113.

In the Baum (1982) paper, S_1, S_2, S_3, S_4 are A023896, A053818, A053819, A053820, and S'_1, S'_2, S'_3, S'_4 are A066840, A295574, A295575, A295576.

f:= n -> add(t^4, t = select(t->igcd(t,n)=1, [$1..n/2])):
map(f, [$1..100]); # Robert Israel, Dec 10 2017

f[n_] := Plus @@ (Select[Range[n/2], GCD[#, n] == 1 &]^4); Array[f, 41] (* Robert G. Wilson v, Dec 10 2017 *)

a(n) = sum(j=1, n\2, (gcd(j, n)==1)*j^4); \\ Michel Marcus, Dec 10 2017

A282601 a(n) = Sum_(k=1..phi(n)/2) floor(d_k/2) where d_k are the totatives of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 2, 1, 3, 1, 6, 2, 9, 3, 6, 6, 16, 5, 20, 8, 14, 10, 30, 10, 29, 15, 28, 18, 49, 14, 56, 28, 38, 28, 48, 24, 81, 36, 54, 36, 100, 30, 110, 50, 64, 55, 132, 44, 124, 57, 96, 72, 169, 56, 130, 78, 122, 91, 210, 56, 225, 105, 136, 120, 186, 80, 272, 128, 182, 102

Offset: 1

Michel Marcus, Feb 19 2017

nonn

The totatives of n are the numbers k <= n with gcd(k,n) = 1.

David Zmiaikou, Origamis and permutation groups, Thesis, 2011. See p. 65.

Cf. A000010, A023896, A038566, A066840, A282600.

a(n) = {vn = vector(n, k, k); vt = select(x->(gcd(x,n) == 1), vn); sum(k=1, #vt\2, vt[k]\2);}

cp's OEIS Frontend

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

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

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A341031 Numbers k such that A066840(k) is a square.

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

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A124440 a(n) = Sum_{n/2<=k<=n, gcd(k,n)=1} k.

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A282600 a(n) = Sum_(k=1..phi(n)) floor(d_k/2) where d_k are the totatives of n.

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A295574 a(n) = Sum_{1 <= j <= n/2, gcd(j,n)=1} j^2.

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A295575 a(n) = Sum_{1 <= j <= n/2, gcd(j,n)=1} j^3.

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