A037213 -id:A037213 - cp's OEIS frontend

A340774 Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s-1) * zeta(2*s-1).

1, 2, 3, 6, 5, 6, 7, 12, 12, 10, 11, 18, 13, 14, 15, 28, 17, 24, 19, 30, 21, 22, 23, 36, 30, 26, 36, 42, 29, 30, 31, 56, 33, 34, 35, 72, 37, 38, 39, 60, 41, 42, 43, 66, 60, 46, 47, 84, 56, 60, 51, 78, 53, 72, 55, 84, 57, 58, 59, 90, 61, 62, 84, 120, 65, 66, 67

Offset: 1

Werner Schulte, Jan 20 2021

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A069290, A055615, A037213.
Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k =
0..10: A046951 (k=0), this sequence (k=1), A351600 (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).

a:= n-> mul((i[1]^(i[2]+1)-i[1]^iquo(i[2]+1, 2))/(i[1]-1), i=ifactors(n)[2]):
seq(a(n), n=1..77);  # Alois P. Heinz, Jan 20 2021

f[p_, e_] := (p^(e + 1) - p^Floor[(e + 1)/2])/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 20 2021 *)

A340774(n) = { my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); ((p^(e+1)-(p^((e+1)\2))) / (p-1))); }; \\ Antti Karttunen, Aug 19 2021

Multiplicative with a(p^e) = (p^(e+1)-p^floor((e+1)/2))/(p-1).
Dirichlet convolution of A000010 and A069290.
Dirichlet convolution with A055615 equals A037213.
G.f.: Sum_{k>=1} k * x^(k^2) / (1 - x^(k^2))^2. - Ilya Gutkovskiy, Aug 19 2021
Sum_{k=1..n} a(k) ~ zeta(3)*n^2/2. - Vaclav Kotesovec, Aug 19 2021
a(n) = n * Sum_{d^2|n} 1/d. - Wesley Ivan Hurt, Feb 14 2022

A227291 Characteristic function of squarefree numbers squared (A062503).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0

Offset: 1

Ralf Stephan, Jul 05 2013

			a(3) = 0 because 3 is not the square of a squarefree number.
a(4) = 1 because sqrt(4) = 2, a squarefree number.

Antti Karttunen, Table of n, a(n) for n = 1..100000 (first 10000 terms from Reinhard Zumkeller)
Index entries for characteristic functions
Index entries for sequences computed from exponents in factorization of n

Cf. A000196, A007427, A008836, A008683, A008966, A010052, A027750, A037213, A225546, A225569, A225817, A307430, A322327, A355448.

Absolute values of A271102.

a227291 n = fromEnum $ (sum $ zipWith (*) mds (reverse mds)) == 1
   where mds = a225817_row n
-- Reinhard Zumkeller, Jul 30 2013, Jul 07 2013

A227291 := proc(n)
    local pe;
    if n = 0 then
        1;
    else
        for pe in ifactors(n)[2] do
            if op(2,pe) <> 2 then
                return 0 ;
            end if;
        end do:
    end if;
    1 ;
end proc:
seq(A227291(n),n=1..100) ; # R. J. Mathar, Feb 07 2023

Table[Abs[Sum[MoebiusMu[n/d], {d,Select[Divisors[n], SquareFreeQ[#] &]}]], {n, 1, 200}] (* Geoffrey Critzer, Mar 18 2015 *)
Module[{nn=120,len,sfr},len=Ceiling[Sqrt[nn]];While[!SquareFreeQ[len],len++];sfr=(Select[Range[len],SquareFreeQ])^2; Table[If[MemberQ[ sfr,n],1,0],{n,nn}]] (* Harvey P. Dale, Nov 27 2024 *)

a(n)=if(n<1, 0, direuler(p=2, n, 1+X^2)[n])

A227291(n) = factorback(apply(e->(2==e), factor(n)[,2])); \\ Antti Karttunen, Jul 14 2022

A227291(n) = (issquare(n) && issquarefree(sqrtint(n))); \\ Antti Karttunen, Jul 14 2022

(define (A227291 n) (if (= 1 n) n (* (if (= 2 (A067029 n)) 1 0) (A227291 (A028234 n))))) ;; Antti Karttunen, Jul 28 2017

Dirichlet g.f.: zeta(2s)/zeta(4s) = prod[prime p: 1+p^(-2s) ], see A008966.
a(n) = A008966(A037213(n)), when assumed A008966(0) = 0. - Reinhard Zumkeller, Jul 07 2013
a(n) = A063524(sum(A225817(n,k)*A225817(n,A000005(n)+1-k): k=1..A000005(n))). - Reinhard Zumkeller, Aug 01 2013
Multiplicative with a(p^e) = 1 if e=2, a(p^e) = 0 if e=1 or e>2. - Antti Karttunen, Jul 28 2017
Sum_{k=1..n} a(k) ~ 6*sqrt(n) / Pi^2. - Vaclav Kotesovec, Feb 02 2019
a(n) = A225569(A225546(n)-1). - Peter Munn, Oct 31 2019
From Antti Karttunen, Jul 18 2022: (Start)
a(n) = A010052(n) * A008966(A000196(n)).
a(n) = Sum_{d|n} A008836(n/d) * A307430(d).
a(n) = Sum_{d|n} A007427(n/d) * A322327(d).
(End)

A094048 Let p(n) be the n-th prime congruent to 1 mod 4. Then a(n) = the least m for which m^2+1=p(n)*k^2 has a solution.

Original entry on oeis.org

2, 18, 4, 70, 6, 32, 182, 29718, 1068, 500, 5604, 10, 8890182, 776, 1744, 113582, 4832118, 1118, 1111225770, 1764132, 14, 1710, 23156, 71011068, 16, 82, 8920484118, 1063532, 2482, 126862368, 352618

Offset: 1

Matthijs Coster, Apr 29 2004

nonn

Subsequence of A191860. [Reinhard Zumkeller, Jun 18 2011]

Cf. A002144, A094049 (associated k), A130226, A137351, A179073.

Cf. A000196, A037213, A000290.

a094048 n = head [m | m <- map (a037213 . subtract 1 . (* a002144 n))
                               (tail a000290_list), m > 0]
-- Reinhard Zumkeller, Jun 13 2015

f[n_] := Block[{y = 1}, While[ !IntegerQ[ Sqrt[n*y^2 - 1]], y++ ]; Sqrt[n*y^2 - 1]]; lst = {}; Do[p = Prime@ n; If[ Mod[p, 4] == 1, AppendTo[lst, f@p]; Print[{n, Prime@n, f@p}]], {n, 66}]; lst

Edited by Don Reble, Apr 30 2004

A045698 Number of ways n can be written as the sum of two squares of primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 0

Felice Russo

a(A214879(n)) = 0; a(A045636(n)) > 0; a(A214723(n)) = 1; a(A214511(n)) = n and a(m) < n for m < A214511(n). - Reinhard Zumkeller, Jul 29 2012

The smallest value of n such that a(n) = 2 is 338. (This helps distinguish it from the characteristic function of A045636.) - Wesley Ivan Hurt, Jun 13 2013

			For example, a(29) = 1 because 29 = 2^2 + 5^2. a(3) = 0 because there is no way to write 3 as sum of two squares of primes.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to sums of squares

Cf. A001248, A010051, A037213.

a045698 n = length $ filter (\x -> x > 0 && a010051' x == 1) $
map (a037213 . (n -)) $
takeWhile (<= div n 2) a001248_list
-- Reinhard Zumkeller, Jul 29 2012

a(n)=my(s=0,q);forprime(p=2,sqrtint(n\2),if(issquare(n-p^2,&q)&&isprime(q),s++));s \\ Charles R Greathouse IV, Jun 04 2014

More terms from Erich Friedman

A060866 Sum of (d+d') over all unordered pairs (d,d') with d*d' = n.

Original entry on oeis.org

2, 3, 4, 9, 6, 12, 8, 15, 16, 18, 12, 28, 14, 24, 24, 35, 18, 39, 20, 42, 32, 36, 24, 60, 36, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 97, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 64, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 135, 84, 144, 68, 126, 96

Offset: 1

Jason Earls, May 04 2001

Paraphrasing the Jovovic formula: if n is not a square then a(n) = sigma(n), the sum of divisors of n, otherwise a(n) = sigma(n) + sqrt(n). - Omar E. Pol, Jun 23 2009

Row sums of A161901. - Omar E. Pol, Jan 06 2014

			a(4)=9 because pairs of factors are 1*4 and 2*2 and 1+4+2+2=9. a(6)=12 because pairs of factors are 1*6 and 2*3 and 1+6+2+3=12.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000203, A037213, A060872, A066839, A070038.

A060866 := proc(n)
        numtheory[sigma](n) ;
        if issqr(n) then
                %+sqrt(n) ;
        else
                % ;
        end if;
end proc: # R. J. Mathar, Oct 24 2011

Table[Sum[(i^2 + n) (1 - Ceiling[n/i] + Floor[n/i])/i, {i, Floor[Sqrt[n]]}], {n, 100}] (* Wesley Ivan Hurt, Jul 14 2014 *)
Array[If[IntegerQ@ #2, #3 + #2, #3] & @@ {#, Sqrt@ #, DivisorSigma[1, #]} &, 69] (* Michael De Vlieger, Nov 23 2017 *)

A037213(n) = if(issquare(n,&n),n,0);
A060866(n) = (sigma(n)+A037213(n)); \\ Antti Karttunen, Nov 23 2017, after Jan 25 2003 formula of Vladeta Jovovic

a(n) = A066839(n)+A070038(n) = A000203(n)+A037213(n). G.f.: Sum_{n>0} n*x^n*(x^(n*(n-1))-x^(n^2)+1)/(1-x^n). - Vladeta Jovovic, Jan 25 2003

a(n) = sum_{i=1..floor(sqrt(n))} (n+i^2)*(1-ceiling(n/i)+floor(n/i))/i. - Wesley Ivan Hurt, Jul 14 2014

More terms from Erich Friedman, Jun 03 2001

A156678 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B) = 1, A < BA020884(n)).

Original entry on oeis.org

4, 12, 24, 15, 40, 60, 35, 84, 112, 63, 144, 180, 21, 99, 220, 264, 143, 312, 364, 45, 195, 420, 480, 255, 56, 544, 612, 77, 323, 684, 80, 760, 399, 840, 924, 117, 483, 1012, 1104, 55, 575, 1200, 140, 1300, 165, 675, 1404, 1512, 783, 176, 1624, 1740, 91, 221, 899

Offset: 1

Ant King, Feb 15 2009

The ordered sequence of A values is A020884(n) and the ordered sequence of B values is A020883(n) (allowing repetitions) and A024354(n) (excluding repetitions)

			As the first four primitive Pythagorean triples (ordered by increasing A) are (3,4,5), (5,12,13), (7,24,25) and (8,15,17), then a(1)=4, a(2)=12, a(3)=24 and a(4)=15.

Beiler, Albert H.: Recreations In The Theory Of Numbers, Chapter XIV, The Eternal Triangle, Dover Publications Inc., New York, 1964, pp. 104-134.
Sierpinski, W.; Pythagorean Triangles, Dover Publications, Inc., Mineola, New York, 2003.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Ron Knott, Right-angled Triangles and Pythagoras' Theorem

A020884, A020883, A024354, A156679, A042965, A156680, A156681, A042965.

Cf. A037213.

a156678 n = a156678_list !! (n-1)
a156678_list = f 1 1 where
   f u v | v > uu `div` 2        = f (u + 1) (u + 2)
         | gcd u v > 1 || w == 0 = f u (v + 2)
         | otherwise             = v : f u (v + 2)
         where uu = u ^ 2; w = a037213 (uu + v ^ 2)
-- Reinhard Zumkeller, Nov 09 2012

PrimitivePythagoreanTriplets[n_]:=Module[{t={{3,4,5}},i=4,j=5},While[i

a(n) = A020884(n) + A156680(n).

A156679 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B) = 1, A < BA020884(n)).

Original entry on oeis.org

5, 13, 25, 17, 41, 61, 37, 85, 113, 65, 145, 181, 29, 101, 221, 265, 145, 313, 365, 53, 197, 421, 481, 257, 65, 545, 613, 85, 325, 685, 89, 761, 401, 841, 925, 125, 485, 1013, 1105, 73, 577, 1201, 149, 1301, 173, 677, 1405, 1513, 785, 185, 1625, 1741, 109, 229

Offset: 1

Ant King, Feb 15 2009

The ordered sequence of A values is A020884(n) and the ordered sequence of C values is A020882(n) (allowing repetitions) and A008846(n) (excluding repetitions).

			As the first four primitive Pythagorean triples (ordered by increasing A) are (3,4,5), (5,12,13), (7,24,25) and (8,15,17), then a(1)=5, a(2)=13, a(3)=25 and a(4)=17.

Beiler, Albert H.: Recreations In The Theory Of Numbers, Chapter XIV, The Eternal Triangle, Dover Publications Inc., New York, 1964, pp. 104-134.
Sierpinski, W.; Pythagorean Triangles, Dover Publications, Inc., Mineola, New York, 2003.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Ron Knott, Right-angled Triangles and Pythagoras' Theorem

Cf. A020884, A020882, A008846, A156678, A156682.

Cf. A037213.

a156679 n = a156679_list !! (n-1)
a156679_list = f 1 1 where
   f u v | v > uu `div` 2        = f (u + 1) (u + 2)
         | gcd u v > 1 || w == 0 = f u (v + 2)
         | otherwise             = w : f u (v + 2)
         where uu = u ^ 2; w = a037213 (uu + v ^ 2)
-- Reinhard Zumkeller, Nov 09 2012

PrimitivePythagoreanTriplets[n_]:=Module[{t={{3,4,5}},i=4,j=5},While[iHarvey P. Dale, May 10 2020 *)

A208133 Total number of subgroups of rank <= 2 of a certain class of abelian groups of order n defined as direct products of Z/(mZ) X Z/(kZ) with m|k.

Original entry on oeis.org

1, 2, 2, 8, 2, 4, 2, 12, 9, 4, 2, 16, 2, 4, 4, 31, 2, 18, 2, 16, 4, 4, 2, 24, 11, 4, 14, 16, 2, 8, 2, 42, 4, 4, 4, 72, 2, 4, 4, 24, 2, 8, 2, 16, 18, 4, 2, 62, 13, 22, 4, 16, 2, 28, 4, 24, 4, 4, 2, 32, 2, 4, 18, 90, 4, 8, 2, 16, 4, 8, 2, 108, 2, 4, 22, 16

Offset: 1

R. J. Mathar, Mar 29 2012

Level function l_tau^2(n) of Bhowmik and Wu.

Records occur at 1, 2, 4, 8, 12, 16, 32, 36, 64, 72, 108, 128, 144, 288, 432, 576, 1152, 1296, 2304, 3600, 5184, 7200, 9216, 10368, 14112, 14400, 20736, 28224, 28800, 32400, 57600, ... and they are: 1, 2, 8, 12, 16, 31, 42, 72, 90, 108, 112, 116, 279, 378, 434, 810, 1044, 1302, 2025, 3069, 3780, 4158, 4644, 4872, 4914, 8910, 9450, 10530, 11484, 14322, 22275, ... - Antti Karttunen, Mar 21 2018

A. Laurincikas, The universality of Dirichlet series attached to finite Abelian groups, in "Number Theory", Proc. Turku Sympos. on Number Theory, May 31-June 4, 1999, p 179.

Antti Karttunen, Table of n, a(n) for n = 1..65537
G. Bhowmik, Jie Wu, Zeta function of subgroups of abelian groups and average orders, J. reine angew. Math. 530 (2001) 1-15.
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Cf. A010052, A037213, A300828.

L300828 := [ 1, 0, 0, 1, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0
] ;
L010052 := [ 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ];
L037213 := [ 1, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ;
Lx := DIRICHLET(L300828,L037213) ;
Lx := DIRICHLET(Lx,L010052) ;
Lx := DIRICHLET(Lx,L010052) ;
Lx := MOBIUSi(Lx) ;
Lx := MOBIUSi(Lx) ;
# Name of initial list L1 changed to L300828 to refer to sequence A300828 by Antti Karttunen, Mar 21 2018

A037213(n) = if(issquare(n),sqrtint(n),0);
A300828(n) = { if(1==n, return(1)); my(val=1, v=factor(n), d=matsize(v)[1]); for(i=1,d, if(v[i,2] < 2 || v[i,2] > 3, return(0)); if (v[i,2] == 3, val *= -2)); return(val); };
a208133s1(n) = sumdiv(n,d,A037213(n/d)*A300828(d));
a208133s2(n) = sumdiv(n,d,issquare(n/d)*a208133s1(d));
a208133s3(n) = sumdiv(n,d,issquare(n/d)*a208133s2(d));
a208133s4(n) = sumdiv(n,d,a208133s3(d));
A208133(n) = sumdiv(n,d,a208133s4(d)); \\ Antti Karttunen, Mar 21 2018, after R. J. Mathar's Maple code

for(n=1, 100, print1(direuler(p=2, n, (1 + X + 2*X^2)/(1 - X)^3/(1 + X)^2/(1 - p*X^2))[n], ", ")) \\ Vaclav Kotesovec, Jun 18 2020

Dirichlet g.f.: zeta(s)^2*zeta(2s)^2*zeta(2s-1)*Product_{primes p} (1 + 1/p^(2s) - 2/p^(3s)).
Sum_{k=1..n} a(k) ~ c * Pi^4 * log(n)^2 * n / 144, where c = A330594 = Product_{primes p} (1 + 1/p^2 - 2/p^3) = 1.10696011195321767665117913000743959294954883365812241904313404497877733241... - Vaclav Kotesovec, Dec 18 2019
More precise asymptotics: Let f(s) = Product_{primes p} (1 + 1/p^(2*s) - 2/p^(3*s)), then Sum_{k=1..n} a(k) ~ n*Pi^2 * (Pi^2 * f(1) * log(n)^2 + 2*Pi^2 * log(n) * (f(1) * (-1 + 8*gamma - 48*log(A) + 4*log(2*Pi)) + f'(1)) + Pi^2 * (2*f(1)*(1 + 25*gamma^2 + 576*log(A)^2 + log(A) * (48 - 96*log(2*Pi)) - 8*gamma * (1 + 36*log(A) - 3*log(2*Pi)) - 4*log(2*Pi) + 4*log(2*Pi)^2 - 6*sg1) + 2*(-1 + 8*gamma - 48*log(A) + 4*log(2*Pi))*f'(1) + f''(1)) + 48*f(1)*zeta''(2)) / 144, where f(1) = A330594, f'(1) = A330594 * (-A335705) = f(1) * Sum_{primes p} = -2*(p-3) * log(p) / (p^3 + p - 2) = -0.087825458097278818094375273108270679512035928574..., f''(1) = A330594 * (A335705^2 + A335706) = f'(1)^2/f(1) + f(1) * Sum_{primes p} = 2*p*(2*p^3 - 9*p^2 - 1) * log(p)^2) / (p^3 + p - 2)^2 =  0.26722508718782634450711076996710402451611235402675360769..., zeta''(2) = A201994, A is the Glaisher-Kinkelin constant A074962, gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Jun 18 2020

A318462 a(n) = Sum_{d|n, d < n/d} (d XOR n/d) + {square root of n when n is a square}.

Original entry on oeis.org

1, 3, 2, 7, 4, 8, 6, 15, 11, 18, 10, 24, 12, 20, 20, 31, 16, 35, 18, 30, 24, 32, 22, 52, 29, 42, 36, 44, 28, 56, 30, 63, 40, 54, 36, 87, 36, 56, 52, 90, 40, 80, 42, 80, 68, 68, 46, 116, 55, 93, 68, 86, 52, 112, 68, 112, 72, 90, 58, 144, 60, 92, 98, 127, 72, 136, 66, 122, 88, 128, 70, 171, 72, 114, 110, 136, 88, 152, 78

Offset: 1

Antti Karttunen, Aug 28 2018

Cf. A000203, A003987, A037213, A318457, A318460, A318461, A318463.

A318462(n) = { my(xors=0); fordiv(n,d, if(d<(n/d), xors += bitxor(d,n/d), if(d==(n/d), xors += d))); (xors); };

a(n) = A318460(n) + A037213(n) = A000203(n) - 2*A318463(n).

A348608 a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(d + n/d) * d.

Original entry on oeis.org

1, -1, 1, 1, 1, -3, 1, 1, 4, -3, 1, -2, 1, -3, 4, 5, 1, -6, 1, -3, 4, -3, 1, 2, 6, -3, 4, -3, 1, -11, 1, 5, 4, -3, 6, 0, 1, -3, 4, 0, 1, -12, 1, -3, 9, -3, 1, 8, 8, -8, 4, -3, 1, -12, 6, -2, 4, -3, 1, -5, 1, -3, 11, 13, 6, -12, 1, -3, 4, -15, 1, 0, 1, -3, 9, -3, 8, -12, 1, 8

Offset: 1

Ilya Gutkovskiy, Oct 25 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000593, A006005, A046897, A066839, A109506, A305152, A333782, A037213, A348953.

Table[DivisorSum[n, (-1)^(# + n/#) # &, # <= Sqrt[n] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[k x^(k^2)/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, if (d<=sqrt(n), (-1)^(d + n/d)*d)); \\ Michel Marcus, Oct 25 2021

G.f.: Sum_{k>=1} k * x^(k^2) / (1 + x^k).
a(n) = 1 if n = 1 or n is an odd prime (A006005) or n = 4 or n = 8. - Bernard Schott, Dec 18 2021
a(n) = A037213(n) - A348953(n). - Ridouane Oudra, Aug 21 2025

cp's OEIS Frontend

A340774 Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s-1) * zeta(2*s-1).

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A227291 Characteristic function of squarefree numbers squared (A062503).

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A094048 Let p(n) be the n-th prime congruent to 1 mod 4. Then a(n) = the least m for which m^2+1=p(n)*k^2 has a solution.

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A045698 Number of ways n can be written as the sum of two squares of primes.

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A060866 Sum of (d+d') over all unordered pairs (d,d') with d*d' = n.

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A156678 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B) = 1, A < BA020884(n)).

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A156679 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B) = 1, A < BA020884(n)).

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