A049690 -id:A049690 - cp's OEIS frontend

A190815 A bisection of A049690.

0, 3, 9, 17, 31, 45, 63, 87, 111, 139, 173, 205, 243, 287, 329, 373, 435, 487, 535, 607, 663, 727, 809, 877, 955, 1037, 1117, 1205, 1293, 1385, 1475, 1595, 1695, 1783, 1913, 2005, 2123, 2267, 2379, 2487, 2629, 2763, 2893, 3041, 3177, 3313, 3473, 3625, 3761

Offset: 1

N. J. A. Sloane, May 20 2011

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A000010, A049690, A099958.

[n eq 1 select 0 else (&+[EulerPhi(2*j): j in [1..2*(n-1)]]) : n in [1..60] ]; // G. C. Greubel, Dec 03 2023

A190815 := proc(n) option remember: if(n=1)then return 0:fi: return procname(n-1)+numtheory[phi](4*n-6)+numtheory[phi](4*n-4): end: seq(A190815(n),n=1..49); # Nathaniel Johnston, May 24 2011

a[n_] := Sum[EulerPhi[2k], {k, 1, 2n-2}];
Table[a[n], {n, 1, 49}] (* Jean-François Alcover, Apr 16 2023 *)

[sum(euler_phi(2*j) for j in range(1,2*n-1)) for n in range(1,61)] # G. C. Greubel, Dec 03 2023

More terms from Nathaniel Johnston, May 24 2011

A061571 Duplicate of A049690.

Original entry on oeis.org

0, 1, 3, 5, 9, 13, 17, 23, 31, 37, 45, 55, 63, 75, 87, 95, 111, 127, 139, 157, 173, 185, 205

Offset: 0

dead

A068773 Alternating sum phi(1) - phi(2) + phi(3) - phi(4) + ... + ((-1)^(n+1))*phi(n).

Original entry on oeis.org

1, 0, 2, 0, 4, 2, 8, 4, 10, 6, 16, 12, 24, 18, 26, 18, 34, 28, 46, 38, 50, 40, 62, 54, 74, 62, 80, 68, 96, 88, 118, 102, 122, 106, 130, 118, 154, 136, 160, 144, 184, 172, 214, 194, 218, 196, 242, 226, 268, 248, 280, 256, 308, 290, 330, 306, 342, 314, 372, 356

Offset: 1

Klaus Brockhaus, Feb 28 2002

			a(3) = phi(1) - phi(2) + phi(3) = 1 - 1 + 2 = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A000010, A067929.

Cf. A002088, A049690.

with(numtheory): seq(add((-1)^(k+1)*phi(k),k=1..n), n=1..80); # Ridouane Oudra, Mar 22 2024

Accumulate[Array[(-1)^(# + 1) * EulerPhi[#] &, 100]] (* Amiram Eldar, Oct 14 2022 *)

a068773(m)=local(s,n); s=0; for(n=1,m, if(n%2==0,s=s-eulerphi(n),s=s+eulerphi(n)); print1(s,","))

# uses code from A002088 and A049690
def A068773(n): return A002088(n)-(A049690(n>>1)<<1) # Chai Wah Wu, Aug 04 2024

a(n) = Sum_{k=1..n} (-1)^(k+1)*phi(k).
a(n) = n^2/Pi^2 + O(n * log(n)^(2/3) * log(log(n))^(4/3)) (Tóth, 2017). - Amiram Eldar, Oct 14 2022
a(n) = 3*A002088(n) - 2*A049690(n). - Ridouane Oudra, Mar 22 2024
a(n) = A002088(n) - 2*A049690(floor(n/2)). - Chai Wah Wu, Aug 04 2024

A372619 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = 1/(phi(k)) * Sum_{j=1..n} phi(k*j).

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 2, 5, 6, 1, 3, 5, 9, 10, 1, 2, 5, 7, 13, 12, 1, 3, 4, 9, 11, 17, 18, 1, 2, 6, 6, 13, 14, 23, 22, 1, 3, 4, 10, 11, 17, 20, 31, 28, 1, 2, 5, 6, 14, 13, 23, 24, 37, 32, 1, 3, 5, 9, 10, 20, 19, 31, 33, 45, 42, 1, 2, 5, 7, 13, 12, 26, 23, 37, 37, 55, 46

Offset: 1

Seiichi Manyama, May 07 2024

			Square array T(n,k) begins:
   1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
   2,  3,  2,  3,  2,  3,  2,  3,  2,  3, ...
   4,  5,  5,  5,  4,  6,  4,  5,  5,  5, ...
   6,  9,  7,  9,  6, 10,  6,  9,  7,  9, ...
  10, 13, 11, 13, 11, 14, 10, 13, 11, 14, ...
  12, 17, 14, 17, 13, 20, 12, 17, 14, 18, ...
  18, 23, 20, 23, 19, 26, 19, 23, 20, 24, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=1..10 give: A002088, A049690, A372621, A049690, A372622, A372637, A372638, A049690, A372621, A372639.
Main diagonal gives A070639.
Cf. A000010, A372606.
Cf. A078615, A322360.

T[n_, k_] := Sum[EulerPhi[k*j], {j, 1, n}] / EulerPhi[k]; Table[T[k, n-k+1], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2024 *)

T(n, k) = sum(j=1, n, eulerphi(k*j))/eulerphi(k);

T(n,k) ~ (3/Pi^2) * c(k) * n^2, where c(k) = A078615(k)/A322360(k) is the multiplicative function defined by c(p^e) = p^2/(p^2-1). - Amiram Eldar, May 09 2024

A099957 a(n) = Sum_{k=0..n-1} phi(2k+1).

Original entry on oeis.org

1, 3, 7, 13, 19, 29, 41, 49, 65, 83, 95, 117, 137, 155, 183, 213, 233, 257, 293, 317, 357, 399, 423, 469, 511, 543, 595, 635, 671, 729, 789, 825, 873, 939, 983, 1053, 1125, 1165, 1225, 1303, 1357, 1439, 1503, 1559, 1647, 1719, 1779, 1851, 1947

Offset: 1

Hugo Pfoertner, Nov 13 2004

nonn

The n-th term is the number of notes of the (2n-1)-limit tonality diamond. This is a term from music theory and means the scale consisting of the rational numbers r, 1 <= r < 2, such that the odd part of both the numerator and the denominator of r, when reduced to lowest terms, is less than or equal to the fixed odd number 2n-1. - Gene Ward Smith, Mar 27 2006
(1/4)*Number of distinct angular positions under which an observer positioned at the center of a square of a square lattice can see the (2n) X (2n) points symmetrically surrounding his position.
(1/8)*number of distinct angular positions under which an observer positioned at a lattice point of a square lattice can see the (2n+1)X(2n+1) points symmetrically surrounding his position gives A002088.
(1/2)*number of distinct angular positions under which an observer positioned at the center of an edge of a square lattice can see the (2n)X(2n-1) points symmetrically surrounding his position gives A099958.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Lv Chuan, On the Mean Value of an Arithmetical Function, in Zhang Wenpeng (ed.), Research on Smarandache Problems in Number Theory (collected papers), 2004, pp. 89-92.
Wikipedia, Tonality diamond.

Bisection of A274401.
Partial sums of A037225.
Cf. A000010, A002088, A099958, A049687, A049690.

Accumulate[EulerPhi[2*Range[0,50]+1]] (* Harvey P. Dale, Aug 20 2021 *)

apply( {A099957(n)=sum(k=1,n, eulerphi(2*k-1))}, [1..55]) \\ M. F. Hasler, Apr 03 2023

a(n+1) - a(n) = phi(2n+1) (A037225).
a(n) = (8/Pi^2)*n^2 + O(n^(3/2+eps)) (Lemma 1 in Lv Chuan, 2004). - Amiram Eldar, Aug 02 2022, corrected by M. F. Hasler, Mar 26 2023
a(n) = A002088(2*n-1) - A049690(n-1). - Chai Wah Wu, Aug 04 2024

A372606 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} phi(k*j).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 2, 4, 5, 6, 4, 6, 10, 9, 10, 2, 8, 10, 14, 13, 12, 6, 6, 16, 18, 22, 17, 18, 4, 12, 12, 24, 26, 28, 23, 22, 6, 12, 24, 20, 44, 34, 40, 31, 28, 4, 12, 20, 36, 28, 52, 46, 48, 37, 32, 10, 12, 30, 36, 60, 40, 76, 62, 66, 45, 42, 4, 20, 20, 42, 52, 72, 52, 92, 74, 74, 55, 46

Offset: 1

Seiichi Manyama, May 07 2024

			Square array T(n,k) begins:
   1,  1,  2,  2,  4,  2,   6, ...
   2,  3,  4,  6,  8,  6,  12, ...
   4,  5, 10, 10, 16, 12,  24, ...
   6,  9, 14, 18, 24, 20,  36, ...
  10, 13, 22, 26, 44, 28,  60, ...
  12, 17, 28, 34, 52, 40,  72, ...
  18, 23, 40, 46, 76, 52, 114, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=1..2 give: A002088, A049690.
Main diagonal gives A372608.
Cf. A000010, A372619.
Cf. A007947, A048250, A332880, A332881.

T[n_, k_] := Sum[EulerPhi[k*j], {j, 1, n}]; Table[T[k, n-k+1], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, May 10 2024 *)

PARI
```
T(n, k) = sum(j=1, n, eulerphi(k*j));
```

T(n,k) ~ (3/Pi^2) * c(k) * n^2, where c(k) = k * A007947(k)/A048250(k) = k * A332881(k) / A332880(k) is the multiplicative function defined by c(p^e) = p^(e+1)/(p+1). - Amiram Eldar, May 10 2024

A099958 (1/2)*number of distinct angular positions under which an observer positioned at the center of an edge of a square lattice can see the (2n)X(2n-1) points symmetrically surrounding his position.

Original entry on oeis.org

1, 5, 13, 23, 37, 55, 75, 95, 127, 157, 185, 227, 263, 305, 357, 403, 455, 511, 571, 631, 703, 769, 833, 923, 997, 1069, 1169, 1245, 1329, 1443, 1535, 1631, 1743, 1849, 1957, 2075, 2195, 2307, 2439, 2565, 2683, 2845, 2957, 3097, 3265, 3385

Offset: 1

Hugo Pfoertner, Nov 13 2004

nonn

See A099957 for further information. Cf. A049687, A049690, A190815.

This is a bisection of A049690, that is, a(n) = Sum[k=1..2n+1, phi(2k)]. - Ralf Stephan, Nov 13 2004.

A106481 An Euler phi transform of 1/(1 - x^2).

Original entry on oeis.org

0, 1, 1, 3, 3, 7, 5, 13, 9, 19, 13, 29, 17, 41, 23, 49, 31, 65, 37, 83, 45, 95, 55, 117, 63, 137, 75, 155, 87, 183, 95, 213, 111, 233, 127, 257, 139, 293, 157, 317, 173, 357, 185, 399, 205, 423, 227, 469, 243, 511, 263, 543, 287, 595, 305, 635, 329, 671, 357, 729

Offset: 0

Paul Barry, May 03 2005

Cf. A049690, A099957.

a(n) = sum(k=0, n, if (k%2, eulerphi(n-k+1))); \\ Michel Marcus, Jun 12 2024

# uses programs from A002088 and A049690
def A106481(n): return A002088(n+1)-A049690(n+1>>1) if n&1 else A049690(n>>1) # Chai Wah Wu, Aug 04 2024

a(n) = Sum_{k=0..n} phi(n-k+1)*(k mod 2).
Euler transform of period 7 sequence [3,-2,-1,-1,-2,3,0,...].
a(2n) = A049690(n).
a(2n+1) = A099957(n).

A274401 Number of balanced linear 4-partitions of the n X n grid.

Original entry on oeis.org

0, 1, 8, 3, 16, 7, 32, 13, 48, 19, 80, 29, 96, 41, 144, 49, 176, 65, 224, 83, 256, 95, 336, 117, 368, 137, 464, 155, 512, 183, 576, 213, 640, 233, 768, 257, 816, 293, 960, 317, 1024, 357, 1120, 399, 1200, 423, 1376, 469, 1440

Offset: 1

Max Alekseyev, Jun 20 2016

D. M. Acketa, N. Divljak, J. Zunic. On balanced linear 4-partitions of the (n,n)-grid. Novi Sad Journal of Mathematics 24:1 (1994), 309-319.

Cf. A002088, A049690, A099957.

{ a(n) = if(n%2, 8*sum(m=1,(n-1)\2,eulerphi(m)), sum(m=0,(n-1)\2,eulerphi(2*m+1)) ); }

# uses programs from A002088 and A049690
def A274401(n): return (A002088(n>>1)<<3) if n&1 else A002088(n-1)-A049690((n>>1)-1) # Chai Wah Wu, Aug 04 2024

For odd n, a(n) = 8 * A002088((n-1)/2).

For even n, a(n) = A099957(n/2).

A083239 First order recursion: a(0) = 1; a(n) = phi(n) - a(n-1) = A000010(n) - a(n-1).

Original entry on oeis.org

1, 0, 1, 1, 1, 3, -1, 7, -3, 9, -5, 15, -11, 23, -17, 25, -17, 33, -27, 45, -37, 49, -39, 61, -53, 73, -61, 79, -67, 95, -87, 117, -101, 121, -105, 129, -117, 153, -135, 159, -143, 183, -171, 213, -193, 217, -195, 241, -225, 267, -247, 279, -255, 307, -289, 329, -305, 341, -313, 371, -355, 415, -385, 421, -389, 437, -417

Offset: 0

Labos Elemer, Apr 23 2003

Provides interesting decomposition: phi(n) = u+w, where u and w consecutive terms of this sequence. Depends also on initial value.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000010, A068773, A083236, A083237, A083238.

A083239 := proc(n)
    option remember ;
    if n = 0 then
        1 ;
    else
        numtheory[phi](n)-procname(n-1) ;
    end if;
end proc:
seq(A083239(n),n=0..100) ; # R. J. Mathar, Jun 20 2021

a[n_] := a[n] = EulerPhi[n] -a[n-1]; a[0] = 1; Table[a[n], {n, 0, 100}]

# uses programs from A002088 and A049690
def A083239(n): return A002088(n)-(A049690(n>>1)<<1)-1 if n&1 else 1+(A049690(n>>1)<<1)-A002088(n) # Chai Wah Wu, Aug 04 2024

a(n) + a(n-1) = A000010(n).

a(n) = (-1)^n * (1 - A068773(n)) for n >= 1. - Amiram Eldar, Mar 05 2024

a(0)=1 prepended by R. J. Mathar, Jun 20 2021

cp's OEIS Frontend

A190815 A bisection of A049690.

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A061571 Duplicate of A049690.

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A068773 Alternating sum phi(1) - phi(2) + phi(3) - phi(4) + ... + ((-1)^(n+1))*phi(n).

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A372619 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = 1/(phi(k)) * Sum_{j=1..n} phi(k*j).

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A099957 a(n) = Sum_{k=0..n-1} phi(2k+1).

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A372606 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} phi(k*j).

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A099958 (1/2)*number of distinct angular positions under which an observer positioned at the center of an edge of a square lattice can see the (2n)X(2n-1) points symmetrically surrounding his position.

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A106481 An Euler phi transform of 1/(1 - x^2).

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A274401 Number of balanced linear 4-partitions of the n X n grid.

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