A037213 -id:A037213 - cp's OEIS frontend

A348953 a(n) = -Sum_{d|n, d < sqrt(n)} (-1)^(d + n/d) * d.

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

Offset: 1

Ilya Gutkovskiy, Nov 04 2021

sign

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

Cf. A000593, A046897, A070039, A109506, A333810, A348608, A348951, A348952, A348954, A348955, A348956, A037213.

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

A348953(n) = -sumdiv(n,d,if((d*d)Antti Karttunen, Nov 05 2021

G.f.: Sum_{k>=1} k * x^(k*(k + 1)) / (1 + x^k).

a(n) = A037213(n) - A348608(n). - Ridouane Oudra, Aug 21 2025

A062320 Nonsquarefree numbers squared. A013929(n)^2.

Original entry on oeis.org

16, 64, 81, 144, 256, 324, 400, 576, 625, 729, 784, 1024, 1296, 1600, 1936, 2025, 2304, 2401, 2500, 2704, 2916, 3136, 3600, 3969, 4096, 4624, 5184, 5625, 5776, 6400, 6561, 7056, 7744, 8100, 8464, 9216, 9604, 9801, 10000, 10816, 11664, 12544, 13456

Offset: 1

Jason Earls, Jul 05 2001

A008966(A037213(a(n))) = 0. - Reinhard Zumkeller, Sep 03 2015

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A013929, A008966, A037213, A320965.

Squares in A046099.

a062320 = (^ 2) . a013929 -- Reinhard Zumkeller, Sep 03 2015

for(n=1,55, if(issquarefree(n), n+1,print(n^2)))

n=-1; for (m=1, 10^9, if (!issquarefree(m), write("b062320.txt", n++, " ", m^2); if (n==1000, break))) \\ Harry J. Smith, Aug 04 2009

is(n)=issquare(n,&n) && !issquarefree(n) \\ Charles R Greathouse IV, Sep 18 2015

from math import isqrt
from sympy import mobius
def A062320(n):
    def f(x): return n+1+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f)**2 # Chai Wah Wu, Aug 31 2024

Sum_{n>=1} 1/a(n) = Pi^2/6 - 15/Pi^2. - Amiram Eldar, Jul 16 2020

More terms from Larry Reeves (larryr(AT)acm.org), Jul 11 2001

Offset corrected by Andrew Howroyd, Sep 18 2024

A173517 a(n) = k if n is the k-th nonsquare, zero otherwise.

Original entry on oeis.org

0, 0, 1, 2, 0, 3, 4, 5, 6, 0, 7, 8, 9, 10, 11, 12, 0, 13, 14, 15, 16, 17, 18, 19, 20, 0, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 0, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 0, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 0, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67

Offset: 0

Reinhard Zumkeller, Feb 20 2010

a(A000037(n)) = n; a(A000290(n)) = 0.

a(n)*A037213(n) = 0 for all n.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A000037, A000196, A000290, A010052, A028391, A037213.

a173517 n = (1 - a010052 n) * a028391 n
-- Reinhard Zumkeller, Dec 15 2013

a(n)=if(issquare(n), 0, n-sqrtint(n)) \\ Charles R Greathouse IV, Sep 02 2015

a(n) = (1 - A010052(n))*A028391(n).

Definition revised by Reinhard Zumkeller, Dec 15 2013, at the suggestion of Antti Karttunen, who further edited the name

A076948 Smallest k such that nk-1 is a square, or 0 if no such number exists.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 5, 0, 0, 0, 0, 5, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 10, 0, 0, 0, 0, 5, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 10, 13, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 1, 0

Offset: 1

Amarnath Murthy, Oct 20 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A008784.

Cf. A037213.

a076948 1 = 1
a076948 n = if null qs then 0 else head qs
            where qs = filter ((> 0) . a037213 . subtract 1 . (* n)) [1..n]
-- Reinhard Zumkeller, Oct 25 2015

a[n_] := Module[{r, j, k}, r = Solve[j>0 && k>0 && n k - 1 == j^2, {j, k}, Integers]; If[r === {}, Return[0], Return[k /. (r /. C[1] -> 0) // Min]]]; a[1] = 1;
Array[a, 100] (* Jean-François Alcover, Apr 27 2020 *)

a(n) = if (!issquare(Mod(-1, n)), 0, my(k=1); while (!issquare(n*k-1), k++); k); \\ Michel Marcus, Apr 27 2020

a(n) != 0 if and only if n is a term of A008784. - Joerg Arndt, Apr 27 2020

a(n) = 1 if and only if n is a term of A002522. - Bernard Schott, Apr 27 2020

Edited and extended by Robert G. Wilson v, Oct 21 2002

A318460 a(n) = Sum_{d|n, d < n/d} (d XOR n/d), where XOR is bitwise-xor (A003987).

Original entry on oeis.org

0, 3, 2, 5, 4, 8, 6, 15, 8, 18, 10, 24, 12, 20, 20, 27, 16, 35, 18, 30, 24, 32, 22, 52, 24, 42, 36, 44, 28, 56, 30, 63, 40, 54, 36, 81, 36, 56, 52, 90, 40, 80, 42, 80, 68, 68, 46, 116, 48, 93, 68, 86, 52, 112, 68, 112, 72, 90, 58, 144, 60, 92, 98, 119, 72, 136, 66, 122, 88, 128, 70, 171, 72, 114, 110, 136, 88, 152, 78

Offset: 1

Antti Karttunen, Aug 28 2018

Cf. A003987, A037213, A318461, A318462, A318463.

A318460(n) = { my(xors=0); fordiv(n,d, if(d<(n/d), xors += bitxor(d,n/d))); (xors); };

a(n) = (1/2) * Sum_{d|n} (d XOR n/d).

a(n) = A318462(n) - A037213(n).

A347176 G.f.: Sum_{k>=1} (-1)^(k+1) * k * x^(k^2) / (1 - x^(k^2)).

Original entry on oeis.org

1, 1, 1, -1, 1, 1, 1, -1, 4, 1, 1, -1, 1, 1, 1, -5, 1, 4, 1, -1, 1, 1, 1, -1, 6, 1, 4, -1, 1, 1, 1, -5, 1, 1, 1, -4, 1, 1, 1, -1, 1, 1, 1, -1, 4, 1, 1, -5, 8, 6, 1, -1, 1, 4, 1, -1, 1, 1, 1, -1, 1, 1, 4, -13, 1, 1, 1, -1, 1, 1, 1, -4, 1, 1, 6, -1, 1, 1, 1, -5, 13, 1, 1, -1, 1, 1, 1, -1, 1, 4

Offset: 1

Ilya Gutkovskiy, Aug 21 2021

Excess of sum of square roots of odd square divisors of n over sum of square roots of even square divisors of n.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002129, A002162, A037213, A069290, A344299, A344300.

nmax = 90; CoefficientList[Series[Sum[(-1)^(k + 1) k x^(k^2)/(1 - x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(# + 1) #^(1/2) &, IntegerQ[#^(1/2)] &], {n, 1, 90}]
f[p_, e_] := (p^(Floor[e/2] + 1) - 1)/(p - 1); f[2, e_] := 3 - 2^(Floor[e/2] + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)

a(n) = sumdiv(n, d, if (issquare(d), (-1)^((d%2)+1)*sqrtint(d))); \\ Michel Marcus, Aug 22 2021

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1]==2, 3 - 2^(floor(f[i,2]/2) + 1), (f[i,1]^(floor(f[i,2]/2) + 1) - 1)/(f[i,1] - 1)));} \\ Amiram Eldar, Nov 15 2022

Multiplicative with a(2^e) = 3 - 2^(floor(e/2) + 1), and a(p^e) = (p^(floor(e/2) + 1) - 1)/(p - 1) for p > 2. - Amiram Eldar, Nov 15 2022

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(2) (A002162). - Amiram Eldar, Mar 01 2023

A320471 a(n) = floor(sqrt(n)) mod ceiling(sqrt(n)).

Original entry on oeis.org

0, 1, 1, 0, 2, 2, 2, 2, 0, 3, 3, 3, 3, 3, 3, 0, 4, 4, 4, 4, 4, 4, 4, 4, 0, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 0, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 0, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 0, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 0, 9, 9, 9, 9, 9, 9

Offset: 1

Kritsada Moomuang, Oct 13 2018

nonn

Sequence consists of zeros interleaved with the positive integers, each positive integer k appearing 2k times.

Cf. A000196, A003059, A037213, A049240, A173517.

[Binomial(Ceiling(Sqrt(n)), Floor(Sqrt(n))) - 1: n in [1..100]]; // Vincenzo Librandi, Dec 02 2018

a:= proc(n) modp(floor(sqrt(n)),ceil(sqrt(n))) end: seq(a(n),n=1..100); # Muniru A Asiru, Oct 17 2018

Array[Mod[Floor@ #, Ceiling@ #] &@ Sqrt@ # &, 99] (* or *)
Array[IntegerPart@ # - If[IntegerQ@ #, #, 0] &@ Sqrt@ # &, 99] (* or *)
Flatten@ Array[{0}~Join~ConstantArray[#, 2 #] &, 9] (* Michael De Vlieger, Oct 15 2018 *)

a(n) = sqrtint(n) % (1+sqrtint(n-1)); \\ Michel Marcus, Nov 04 2018

a(n) = sqrtint(n-1) * !issquare(n) \\ David A. Corneth, Nov 04 2018

from math import isqrt
def A320471(n): return 0 if (m:=isqrt(n))**2==n else m # Chai Wah Wu, Jul 29 2022

a(n) = A000196(n) - A037213(n).
a(n) = A000196(n)*A049240(n).
a(n) = A000196(n) mod A003059(n).
a(n) = (n - A173517(n)) - A037213(n)^2.
a(n) = binomial(ceiling(sqrt(n)),floor(sqrt(n))) - 1.
From David A. Corneth, Nov 04 2018: (Start)
a(k^2) = 0.
a(m) = floor(sqrt(m)) for nonsquare m. (End)

Corrected by Michel Marcus, Jun 14 2022

A056926 a(n) = sqrt(n) if n is a square, otherwise 1.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1

Offset: 0

Henry Bottomley, Jul 12 2000

Like A037213, but has 1's instead of 0's for nonsquare n > 0. - Antti Karttunen, Jul 22 2018

			a(24) = 1 because 24 is not a square, a(25) = 5 because 25 = 5^2.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A000196, A037213.

sq1[n_]:=Module[{sn=Sqrt[n]},If[IntegerQ[sn],sn,1]]; Array[sq1,110] (* Harvey P. Dale, Jul 25 2011 *)

A056926(n) = if(issquare(n,&n),n,1); \\ Antti Karttunen, Jul 22 2018

a(n) = A007955(n)/A056925(n) = 1 + (sqrt(n)-1)*A010052(n), for all n >= 1.

a(n) = n^((d(n) mod 2)/2) for n>=1. - Wesley Ivan Hurt, Jun 07 2023

Term a(0) = 0 prepended by Antti Karttunen, Jul 22 2018

A108655 Primes that are sums of the squares of two semiprimes.

Original entry on oeis.org

97, 181, 241, 277, 421, 457, 541, 641, 661, 709, 757, 821, 1109, 1117, 1237, 1301, 1381, 1597, 1621, 1669, 1709, 1901, 2069, 2341, 2381, 2417, 2437, 2557, 2617, 2677, 2741, 2797, 3041, 3061, 3221, 3557, 3637, 3701, 3733, 3989, 4241, 4261, 4421, 4517

Offset: 1

Reinhard Zumkeller, Jul 07 2005

nonn

Subsequence of A002144.

a(n) = A074985(i) + A074985(j) for appropriate i and j.

			A000040(733) = 5557 = 81 + 5476 = (3*3)^2 + (2*37)^2 =
A001358(3)^2 + A001358(25)^2 = A074985(3) + A074985(25),
therefore 5557 is a term.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000404, A001358, A000290.

a108655 n = a108655_list !! (n-1)
a108655_list = filter f a000040_list where
   f p = any (> 0) $ map (a064911 . a037213 . (p -)) $
                         takeWhile (< p) a074985_list
-- Reinhard Zumkeller, Aug 09 2012

A156689 Inradii of primitive Pythagorean triples a^2+b^2=c^2, 0A020884).

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 6, 9, 10, 11, 11, 12, 13, 10, 13, 14, 15, 15, 12, 16, 17, 14, 17, 18, 15, 19, 19, 20, 21, 18, 21, 22, 23, 15, 23, 24, 21, 25, 22, 25, 26, 27, 27, 24, 28, 29, 21, 26, 29, 30

Offset: 1

Ant King, Feb 18 2009

The inradius is given by r=1/2 (a+b-c)=ab/(a+b+c)=area/semiperimeter, and the inradii ordered by increasing r are in A020888.

			The eighth primitive Pythagorean triple ordered by increasing a is (13,84,85). As this has inradius 1/2 (13+84-85)=6, we have a(8)=6.

Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.
D. G. Rogers, Putting Pythagoras in the frame, Mathematics Today, The Institute of Mathematics and its Applications, Vol. 44, No. 3, June 2008, pp. 123-125.

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

Cf. A020884, A020888, A156678, A156679.

Cf. A037213.

a156689 n = a156689_list !! (n-1)
a156689_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             = (u + v - w) `div` 2 : 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

A156689(n)=1/2 (A020884(n)+A156678(n)-A156679(n))

cp's OEIS Frontend

A348953 a(n) = -Sum_{d|n, d < sqrt(n)} (-1)^(d + n/d) * d.

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A062320 Nonsquarefree numbers squared. A013929(n)^2.

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A173517 a(n) = k if n is the k-th nonsquare, zero otherwise.

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A076948 Smallest k such that nk-1 is a square, or 0 if no such number exists.

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A318460 a(n) = Sum_{d|n, d < n/d} (d XOR n/d), where XOR is bitwise-xor (A003987).

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A347176 G.f.: Sum_{k>=1} (-1)^(k+1) * k * x^(k^2) / (1 - x^(k^2)).

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Mathematica

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A320471 a(n) = floor(sqrt(n)) mod ceiling(sqrt(n)).

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