A002088 -id:A002088 - cp's OEIS frontend

A340793 Sequence whose partial sums give A000203.

1, 2, 1, 3, -1, 6, -4, 7, -2, 5, -6, 16, -14, 10, 0, 7, -13, 21, -19, 22, -10, 4, -12, 36, -29, 11, -2, 16, -26, 42, -40, 31, -15, 6, -6, 43, -53, 22, -4, 34, -48, 54, -52, 40, -6, -6, -24, 76, -67, 36, -21, 26, -44, 66, -48, 48, -40, 10, -30, 108, -106, 34, 8

Offset: 1

Omar E. Pol, Jan 21 2021

Essentially a duplicate of A053222.
Convolved with the nonzero terms of A000217 gives A175254, the volume of the stepped pyramid described in A245092.
Convolved with the nonzero terms of A046092 gives A244050, the volume of the stepped pyramid described in A244050.
Convolved with A000027 gives A024916.
Convolved with A000041 gives A138879.
Convolved with A000070 gives the nonzero terms of A066186.
Convolved with the nonzero terms of A002088 gives A086733.
Convolved with A014153 gives A182738.
Convolved with A024916 gives A000385.
Convolved with A036469 gives the nonzero terms of A277029.
Convolved with A091360 gives A276432.
Convolved with A143128 gives the nonzero terms of A000441.
For the correspondence between divisors and partitions see A336811.

1 together with A053222.
Cf. A000203 (partial sums).
Cf. A000027, A000041, A000070, A000217, A000385, A000441, A002088, A002961, A014153, A024916, A036469, A046092, A053223, A053224, A053225, A053226, A053227, A066186, A086733, A091360, A138879, A143128, A175254, A182738, A237593, A244050, A245092, A276432, A277029, A336811.

a:= n-> (s-> s(n)-s(n-1))(numtheory[sigma]):
seq(a(n), n=1..77);  # Alois P. Heinz, Jan 21 2021

Join[{1}, Differences @ Table[DivisorSigma[1, n], {n, 1, 100}]] (* Amiram Eldar, Jan 21 2021 *)

a(n) = if (n==1, 1, sigma(n)-sigma(n-1)); \\ Michel Marcus, Jan 22 2021

a(n) = A053222(n-1) for n>1. - Michel Marcus, Jan 22 2021

A100449 Number of ordered pairs (i,j) with |i| + |j| <= n and gcd(i,j) <= 1.

Original entry on oeis.org

1, 5, 9, 17, 25, 41, 49, 73, 89, 113, 129, 169, 185, 233, 257, 289, 321, 385, 409, 481, 513, 561, 601, 689, 721, 801, 849, 921, 969, 1081, 1113, 1233, 1297, 1377, 1441, 1537, 1585, 1729, 1801, 1897, 1961, 2121, 2169, 2337, 2417, 2513, 2601, 2785, 2849, 3017

Offset: 0

N. J. A. Sloane, Nov 21 2004

nonn

Note that gcd(0,m) = m for any m.
I would also like to get the sequences of the numbers of distinct sums i+j (also distinct products i*j) over all ordered pairs (i,j) with |i| + |j| <= n; also over all ordered pairs (i,j) with |i| + |j| <= n and gcd(i,j) <= 1.
From Robert Price, May 10 2013: (Start)
List of sequences that address these extensions:
Distinct sums i+j with or without the GCD qualifier results in a(n)=2n+1 (A005408).
Distinct products i*j without the GCD qualifier is given by A225523.
Distinct products i*j with    the GCD qualifier is given by A225526.
With the restriction i,j >= 0 ...
Distinct sums or products equal to n is trivial and always equals one (A000012).
Distinct sums <=n with or without the GCD qualifier results in a(n)=n (A001477).
Distinct products <=n without the GCD qualifier is given by A225527.
Distinct products <=n with    the GCD qualifier is given by A225529.
Ordered pairs with the sum = n without the GCD qualifier is a(n)=n+1.
Ordered pairs with the sum = n with    the GCD qualifier is A225530.
Ordered pairs with the sum <=n without the GCD qualifier is A000217(n+1).
Ordered pairs with the sum <=n with    the GCD qualifier is A225531.
(End)
This sequence (A100449) without the GCD qualifier results in A001844. - Robert Price, Jun 04 2013

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A018805, A100448, A100450, A027430, etc.

f:=proc(n) local i,j,k,t1,t2,t3; t1:=0; for i from -n to n do for j from -n to n do if abs(i) + abs(j) <= n then t2:=gcd(i,j); if t2 <= 1 then t1:=t1+1; fi; fi; od: od: t1; end;
# second Maple program:
b:= proc(n) b(n):= numtheory[phi](n)+`if`(n=0, 0, b(n-1)) end:
a:= n-> 1+4*b(n):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 01 2013

f[n_] := Length[ Union[ Flatten[ Table[ If[ Abs[i] + Abs[j] <= n && GCD[i, j] <= 1, {i, j}, {0, 0}], {i, -n, n}, {j, -n, n}], 1]]]; Table[ f[n], {n, 0, 49}] (* Robert G. Wilson v, Dec 14 2004 *)

a(n) = 1+4*sum(k=1, n, eulerphi(k) ); \\ Joerg Arndt, May 10 2013

from functools import lru_cache
@lru_cache(maxsize=None)
def A100449(n):
    if n == 0:
        return 1
    c, j = 0, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*((A100449(k1)-3)//2)
        j, k1 = j2, n//j2
    return 2*(n*(n-1)-c+j)+1 # Chai Wah Wu, Mar 29 2021

a(n) = 1 + 4*Sum(phi(k), k=1..n) = 1 + 4*A002088(n). - Vladeta Jovovic, Nov 25 2004

More terms from Vladeta Jovovic, Nov 25 2004

A135342 Number of distinct means of nonempty subsets of {1,...,n}.

Original entry on oeis.org

1, 3, 5, 9, 15, 25, 37, 55, 77, 105, 137, 179, 225, 283, 347, 419, 499, 595, 697, 817, 945, 1085, 1235, 1407, 1587, 1787, 1999, 2229, 2471, 2741, 3019, 3327, 3651, 3995, 4355, 4739, 5135, 5567, 6017, 6491, 6981, 7511, 8053, 8637, 9241, 9869, 10519, 11215, 11927, 12681

Offset: 1

Jacob A. Siehler, Feb 16 2008

			a(4) = 9: the possible means for a set drawn from {1, 2, 3, 4} are {1, 3/2, 2, 7/3, 5/2, 8/3, 3, 7/2, 4}.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
R. J. Mathar, Derivation of formula for 2nd differences

First differences are A002088, second differences A000010.

a:= proc(n) option remember; `if`(n<4, [0, 1, 3, 5][n+1],
      2*a(n-1)-a(n-2)+numtheory[phi](n-1))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 13 2019

a[n_] := Sum[EulerPhi[k] (n - k), {k, 1, n - 1}] + Min[n, 2]

M135342=List([1,3,5]);
A135342(n)=while(n>#M135342, listput(M135342, [-1,2]*Col(M135342[-2..-1])+eulerphi(#M135342))); M135342[n];
apply(A135342, [1..55]) \\ M. F. Hasler, Jan 24 2023

from sympy import totient
def A135342(n, A=[1,3,5]):
    while n>len(A): A.append(2*A[-1]-A[-2]+totient(len(A)))
    return A[n-1] # M. F. Hasler, Jan 24 2023

a(n) = Sum_{k=1..n-1} [(n-k) * phi(k)] + min(n,2) = A103116(n-1)+ min(n,2); a(1)=1; a(2)=3; a(3)=5.
a(n) = 2*a(n-1) - a(n-2) + phi(n-1) for n>3.
a(n)-a(n-1) = A002088(n-1), n>=3. (Note the previous formula just says that the 2nd differences are A000010, and this is a trivial consequence.) - R. J. Mathar, Jan 27 2023

A011755 a(n) = Sum_{k=1..n} k*phi(k).

Original entry on oeis.org

1, 3, 9, 17, 37, 49, 91, 123, 177, 217, 327, 375, 531, 615, 735, 863, 1135, 1243, 1585, 1745, 1997, 2217, 2723, 2915, 3415, 3727, 4213, 4549, 5361, 5601, 6531, 7043, 7703, 8247, 9087, 9519, 10851, 11535, 12471, 13111, 14751, 15255, 17061, 17941, 19021, 20033

Offset: 1

N. J. A. Sloane

nonn

a(n) = Sum_{(x,y): 1<=x<=y<=n, 1=gcd(x,y)} y. Sum_{(x,y): 1<=x<=y<=n, 1=gcd(x,y)} x = (a(n)+1)/2. - Vladeta Jovovic, Jan 02 2003

Equals row sums of triangle A110663. Example: a(4) = 17 = (6 + 5 + 4 + 2), where row 4 of triangle A110663 = (6, 5, 4, 2). - Gary W. Adamson, Jul 26 2008

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..2001 from Indranil Ghosh)

Partial sums of A002618.

Cf. A000010, A002088, A110663, A319087.

Accumulate[Table[k*EulerPhi[k], {k, 1, 50}]] (* Vaclav Kotesovec, Sep 10 2018 *)

a(n) = sum(k=1, n, k*eulerphi(k)); \\ Michel Marcus, Feb 13 2017

from sympy import totient
def A011755(n): return sum(k*totient(k) for k in range(1,n+1)) # Chai Wah Wu, Feb 21 2023

Asymptotically: a(n) ~ C*n^3 with C=0.20264.... - Benoit Cloitre, Jan 14 2002
Asymptotically: a(n) ~ (2/Pi^2)*n^3. - Vladeta Jovovic, Jan 02 2003
a(n) = Sum_{k=1..n} phi(k^2). - Vaclav Kotesovec, May 08 2024

A015631 Number of ordered triples of integers from [ 1..n ] with no global factor.

Original entry on oeis.org

1, 3, 8, 15, 29, 42, 69, 95, 134, 172, 237, 287, 377, 452, 552, 652, 804, 915, 1104, 1252, 1450, 1635, 1910, 2106, 2416, 2674, 3007, 3301, 3735, 4027, 4522, 4914, 5404, 5844, 6432, 6870, 7572, 8121, 8805, 9389, 10249, 10831, 11776, 12506

Offset: 1

Olivier Gérard

nonn

Number of integer-sided triangles with at least two sides <= n and sides relatively prime. - Henry Bottomley, Sep 29 2006

			a(4) = 15 because the 15 triples in question are in lexicographic order: [1,1,1], [1,1,2], [1,1,3], [1,1,4], [1,2,2], [1,2,3], [1,2,4], [1,3,3], [1,3,4], [1,4,4], [2,2,3], [2,3,3], [2,3,4], [3,3,4] and [3,4,4]. - _Wolfdieter Lang_, Apr 04 2013
The a(4) = 15 triangles with at least two sides <= 4 and sides relatively prime (see _Henry Bottomley_'s comment above) are: [1,1,1], [1,2,2], [2,2,3], [1,3,3], [2,3,3], [2,3,4], [3,3,4], [3,3,5], [1,4,4], [2,4,5], [3,4,4], [3,4,5], [3,4,6], [4,4,5], [4,4,7]. - _Alois P. Heinz_, Feb 14 2020

R. J. Mathar, Table of n, a(n) for n = 1..10000

Column k=3 of A177976.

Cf. A000292, A002088, A015616, A015634, A015650, A134543.

[n eq 1 select 1 else Self(n-1)+ &+[MoebiusMu(n div d) *d*(d+1)/2:d in Divisors(n)]:n in [1..50]]; // Marius A. Burtea, Feb 14 2020

with(numtheory):
b:= proc(n) option remember;
       add(mobius(n/d)*d*(d+1)/2, d=divisors(n))
    end:
a:= proc(n) option remember;
      b(n) + `if`(n=1, 0, a(n-1))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Feb 09 2011

a[1] = 1; a[n_] := a[n] = Sum[MoebiusMu[n/d]*d*(d+1)/2, {d, Divisors[n]}] + a[n-1]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jan 20 2014, after Maple *)
Accumulate[Table[Sum[MoebiusMu[n/d]*d*(d + 1)/2, {d, Divisors[n]}], {n, 1, 50}]] (* Vaclav Kotesovec, Jan 31 2019 *)

a(n) = sum(k=1, n, sumdiv(k, d, moebius(k/d)*binomial(d+1, 2))); \\ Seiichi Manyama, Jun 12 2021

a(n) = binomial(n+2, 3)-sum(k=2, n, a(n\k)); \\ Seiichi Manyama, Jun 12 2021

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*x^k/(1-x^k)^3)/(1-x)) \\ Seiichi Manyama, Jun 12 2021

from functools import lru_cache
@lru_cache(maxsize=None)
def A015631(n):
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A015631(k1)
        j, k1 = j2, n//j2
    return n*(n-1)*(n+4)//6-c+j # Chai Wah Wu, Mar 30 2021

a(n) = (A071778(n)+3*A018805(n)+2)/6. - Vladeta Jovovic, Dec 01 2004
Partial sums of the Moebius transform of the triangular numbers (A007438). - Steve Butler, Apr 18 2006
a(n) = 2*A123324(n) - A046657(n) for n>1. - Henry Bottomley, Sep 29 2006
Row sums of triangle A134543. - Gary W. Adamson, Oct 31 2007
a(n) ~ n^3 / (6*Zeta(3)). - Vaclav Kotesovec, Jan 31 2019
G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - x^k)^3. - Ilya Gutkovskiy, Feb 14 2020
a(n) = n*(n+1)*(n+2)/6 - Sum_{j=2..n} a(floor(n/j)) = A000292(n) - Sum_{j=2..n} a(floor(n/j)). - Chai Wah Wu, Mar 30 2021

A057434 a(n) = Sum_{k=1..n} phi(k)^2.

Original entry on oeis.org

1, 2, 6, 10, 26, 30, 66, 82, 118, 134, 234, 250, 394, 430, 494, 558, 814, 850, 1174, 1238, 1382, 1482, 1966, 2030, 2430, 2574, 2898, 3042, 3826, 3890, 4790, 5046, 5446, 5702, 6278, 6422, 7718, 8042, 8618, 8874, 10474, 10618, 12382, 12782

Offset: 1

N. J. A. Sloane, Sep 08 2000

nonn

Partial sums of A127473. - R. J. Mathar, Sep 29 2008

Ivan Neretin, Table of n, a(n) for n = 1..10000
U. Balakrishnan & Y.-F. S. Pétermann, The Dirichlet series of zeta(s)*zeta(s+1)^alpha*f(s+1): On an error term associated with its coefficients, Acta Arith. 75 (1996), 39--69.

Cf. A000010, A002088, A061502, A072379, A074789.

FoldList[Plus, 1, EulerPhi[Range[2, 50]]^2] (* Ivan Neretin, May 30 2015 *)

a(n) = sum(k=1, n, eulerphi(k)^2); \\ Michel Marcus, Dec 20 2015

We can derive an asymptotic formula from a general formula given in the reference, namely: a(n) = C*n^3 + O(log(x)^(4/3)log(log(x))^(8/3)) where C = (1/3)/zeta(2)^2*Product_{p prime}(1+1/(p-1)/(p+1)^2) = 0.142749835225698(...). - Benoit Cloitre, Dec 22 2015

a(n) ~ c * n^3 / 3, where c = A065464 = Product_{primes p} (1 - 2/p^2 + 1/p^3) = 0.4282495056770944402187657075818235461212985133559361440319... - Vaclav Kotesovec, Dec 18 2019

A276187 Number of subsets of {1,..,n} of cardinality >= 2 such that the elements of each counted subset are pairwise coprime.

Original entry on oeis.org

0, 1, 4, 7, 18, 21, 48, 63, 94, 105, 220, 235, 482, 529, 600, 711, 1438, 1501, 3020, 3211, 3594, 3849, 7720, 7975, 11142, 11877, 14628, 15459, 30946, 31201, 62432, 69855, 76126, 80221, 89820, 91611, 183258, 192601, 208600, 214231, 428502, 431573, 863188, 900563

Offset: 1

Robert C. Lyons, Aug 23 2016

nonn

n is prime if and only if a(n) = 2*a(n-1)+n-1. - Robert Israel, Aug 24 2016

			From _Gus Wiseman_, May 08 2021: (Start)
The a(2) = 1 through a(6) = 21 sets:
  {1,2}   {1,2}    {1,2}     {1,2}      {1,2}
          {1,3}    {1,3}     {1,3}      {1,3}
          {2,3}    {1,4}     {1,4}      {1,4}
         {1,2,3}   {2,3}     {1,5}      {1,5}
                   {3,4}     {2,3}      {1,6}
                  {1,2,3}    {2,5}      {2,3}
                  {1,3,4}    {3,4}      {2,5}
                             {3,5}      {3,4}
                             {4,5}      {3,5}
                            {1,2,3}     {4,5}
                            {1,2,5}     {5,6}
                            {1,3,4}    {1,2,3}
                            {1,3,5}    {1,2,5}
                            {1,4,5}    {1,3,4}
                            {2,3,5}    {1,3,5}
                            {3,4,5}    {1,4,5}
                           {1,2,3,5}   {1,5,6}
                           {1,3,4,5}   {2,3,5}
                                       {3,4,5}
                                      {1,2,3,5}
                                      {1,3,4,5}
(End)

Robert Israel, Table of n, a(n) for n = 1..340

Cf. A000010, A002088, A018805.
The case of pairs is A015614.
The indivisible instead of coprime version is A051026(n) - n.
Allowing empty sets and singletons gives A084422.
The relatively prime instead of pairwise coprime version is A085945(n) - 1.
Allowing all singletons gives A187106.
Allowing only the singleton {1} gives A320426.
Row sums of A320436, each minus one.
The maximal case is counted by A343659.
The version for sets of divisors is A343655(n) - 1.
A000005 counts divisors.
A186972 counts pairwise coprime k-sets containing n.
A186974 counts pairwise coprime k-sets.
A326675 ranks pairwise coprime non-singleton sets.
Cf. A007360, A063647, A186971, A302696, A305713, A320423, A320430, A326077, A327516, A337485.

f:= proc(S) option remember;
    local s, Sp;
    if S = {} then return 1 fi;
    s:= S[-1];
    Sp:= S[1..-2];
    procname(Sp) + procname(select(t -> igcd(t,s)=1, Sp))
end proc:
seq(f({$1..n}) - n - 1, n=1..50); # Robert Israel, Aug 24 2016

f[S_] := f[S] = Module[{s, Sp}, If[S == {}, Return[1]]; s = S[[-1]]; Sp = S[[1;;-2]]; f[Sp] + f[Select[Sp, GCD[#, s] == 1&]]];
Table[f[Range[n]] - n - 1, {n, 1, 50}] (* Jean-François Alcover, Sep 15 2022, after Robert Israel *)

f(n,k=1)=if(n==1, return(2)); if(gcd(k,n)==1, f(n-1,n*k)) + f(n-1,k)
a(n)=f(n)-n-1 \\ Charles R Greathouse IV, Aug 24 2016

from sage.combinat.subsets_pairwise import PairwiseCompatibleSubsets
def is_coprime(x, y): return gcd(x, y) == 1
max_n = 40
seq = []
for n in range(1, max_n+1):
    P = PairwiseCompatibleSubsets(range(1,n+1), is_coprime)
    a_n = len([1 for s in P.list() if len(s) > 1])
    seq.append(a_n)
print(seq)

a(n) = A320426(n) - 1. - Gus Wiseman, May 08 2021

Name and example edited by Robert Israel, Aug 24 2016

A063985 Partial sums of cototient sequence A051953.

Original entry on oeis.org

0, 1, 2, 4, 5, 9, 10, 14, 17, 23, 24, 32, 33, 41, 48, 56, 57, 69, 70, 82, 91, 103, 104, 120, 125, 139, 148, 164, 165, 187, 188, 204, 217, 235, 246, 270, 271, 291, 306, 330, 331, 361, 362, 386, 407, 431, 432, 464, 471, 501, 520, 548, 549, 585, 600, 632, 653, 683

Offset: 1

Labos Elemer, Sep 06 2001

nonn

Number of elements in the set {(x,y): 1 <= x <= y <= n, 1 = gcd(x,y)}; a(n) = A000217(n) - A002088(n) = A100613(n) - A185670(n). - Reinhard Zumkeller, Jan 21 2013

8*a(n) is the number of dots not in direct reach via a straight line from the center of a 2*n+1 X 2*n+1 array of dots. - Kiran Ananthpur Bacche, May 25 2022

Harry J. Smith, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Cf. A000010, A000217, A002088, A048290, A051953.

a063985 n = length [()| x <- [1..n], y <- [x..n], gcd x y > 1]
-- Reinhard Zumkeller, Jan 21 2013

// Save the file as A063985.java to compile and run
import java.util.stream.IntStream;
import java.util.*;
public class A063985 {
  public static int getInvisiblePoints(int n) {
    Set slopes = new HashSet();
    IntStream.rangeClosed(1, n).forEach(i ->
      {IntStream.rangeClosed(1, n).forEach(j ->
        slopes.add(Float.valueOf((float)i/(float)j))); });
    return (n * n - slopes.size() + n - 1) / 2;
  }
  public static void main(String args[]) throws Exception {
    IntStream.rangeClosed(1, 30).forEach(i ->
      System.out.println(getInvisiblePoints(i)));
  }
} // Kiran Ananthpur Bacche, May 25 2022

f[n_] := n(n + 1)/2 - Sum[ EulerPhi@i, {i, n}]; Array[f, 58] (* Robert G. Wilson v *)
Accumulate[Table[n-EulerPhi[n],{n,1,60}]] (* Harvey P. Dale, Aug 19 2015 *)

{ a=0; for (n=1, 1000, write("b063985.txt", n, " ", a+=n - eulerphi(n)) ) } \\ Harry J. Smith, Sep 04 2009

from sympy.ntheory import totient
def a(n): return sum(x - totient(x) for x in range(1,n + 1))
[a(n) for n in range(1, 51)] # Indranil Ghosh, Mar 18 2017

from functools import lru_cache
@lru_cache(maxsize=None)
def A063985(n): # based on second formula in A018805
    if n == 0:
        return 0
    c, j = 0, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*(k1*(k1+1)-2*A063985(k1)-1)
        j, k1 = j2, n//j2
    return (2*n+c-j)//2 # Chai Wah Wu, Mar 24 2021

a(n) = Sum_{x=1..n} (x - phi(x)) = Sum(x) - Sum(phi(x)) = A000217(n) - A002088(n), phi(n) = A000010(n), cototient(n) = A051953(n).
a(n) = n^2 - A091369(n). - Enrique Pérez Herrero, Feb 25 2012
G.f.: x/(1 - x)^3 - (1/(1 - x))*Sum_{k>=1} mu(k)*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Mar 18 2017
a(n) = (1/2 - 3/Pi^2)*n^2 + O(n*log(n)). - Amiram Eldar, Jul 26 2022

Corrected by Robert G. Wilson v, Dec 13 2006

A053570 Sum of totient functions over arguments running through reduced residue system of n.

Original entry on oeis.org

1, 1, 2, 3, 6, 5, 12, 13, 18, 15, 32, 21, 46, 35, 42, 49, 80, 49, 102, 71, 88, 85, 150, 89, 156, 125, 164, 137, 242, 113, 278, 213, 230, 217, 272, 191, 396, 275, 320, 261, 490, 237, 542, 369, 386, 401, 650, 355, 640, 431, 560, 507, 830, 449, 704, 551, 696, 643

Offset: 1

Labos Elemer, Jan 17 2000

nonn

Phi summation results over numbers not exceeding n are given in A002088 while summation over the divisor set of n would give n. This is a further way of Phi summation.

Equals row sums of triangle A143620. - Gary W. Adamson, Aug 27 2008

			Given n = 36, its reduced residue system is {1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35}; the Euler phi of these terms are {1, 4, 6, 10, 12, 16, 18, 22, 20, 28, 30, 24}. Summation over this last set gives 191. So a(36) = 191.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000010, A002088.

Cf. A143620. - Gary W. Adamson, Aug 27 2008

A038566_row := proc(n)
    a := {} ;
    for m from 1 to n do
        if igcd(n,m) =1 then
            a := a union {m} ;
        end if;
    end do:
    a ;
end proc:
A053570 := proc(n)
    add(numtheory[phi](r),r=A038566_row(n)) ;
end proc:
seq(A053570(n),n=1..30) ; # R. J. Mathar, Jan 09 2017

Join[{1}, Table[Sum[EulerPhi[i] * KroneckerDelta[GCD[i, n], 1], {i, n - 1}], {n, 2, 60}]] (* Alonso del Arte, Nov 02 2014 *)

a(n) = Sum_{k>=1} A000010(A038566(n,k)). - R. J. Mathar, Jan 09 2017

A177754 Partial sums of A047994.

Original entry on oeis.org

1, 2, 4, 7, 11, 13, 19, 26, 34, 38, 48, 54, 66, 72, 80, 95, 111, 119, 137, 149, 161, 171, 193, 207, 231, 243, 269, 287, 315, 323, 353, 384, 404, 420, 444, 468, 504, 522, 546, 574, 614, 626, 668, 698, 730, 752, 798, 828, 876, 900, 932, 968, 1020, 1046, 1086

Offset: 1

Jonathan Vos Post, May 12 2010

Partial sums of unitary totient (or unitary phi) function uphi(n). This is to A047994 as A002088 is to A000010. The subsequence of primes in the partial sum begins: 2, 7, 11, 13, 19, 137, 149, 193, 269, 353, 1523, 1543, 1609, 1657.

			a(7) = 1 + 1 + 2 + 3 + 4 + 2 + 6 = 19.

Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Mathematische Zeitschrift, Vol. 74, No. 1 (1960), pp. 66-80.
R. Sitaramachandrarao and D. Suryanarayana, On Sum_{n<=x} sigma*(n) and Sum_{n<=x} phi*(n), Proceedings of the American Mathematical Society, Vol. 41, No. 1 (1973), pp. 61-66.

Cf. A000010, A002088, A003271, A047994, A065463.

uphi[1] = 1; uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); s = 0; Accumulate[Array[uphi, 60]] (* Amiram Eldar, Dec 18 2018*)

a(n) = Sum_{i=1..n} A047994(i).

a(n) ~ alpha * n^2/2 + O(n*log^2(n)) where alpha = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442... (A065463). - Amiram Eldar, Dec 18 2018

cp's OEIS Frontend

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