Peter Schorn - cp's OEIS frontend

A380816 Number of pairs (x, y) with 0 < x < y < n such that x^y = y^x modulo n.

0, 0, 0, 0, 1, 1, 5, 2, 2, 3, 5, 6, 8, 8, 6, 18, 11, 7, 20, 16, 15, 17, 28, 28, 15, 23, 32, 27, 24, 22, 35, 88, 20, 31, 19, 34, 32, 43, 35, 72, 33, 40, 37, 52, 45, 51, 57, 134, 36, 37, 38, 73, 65, 73, 61, 118, 72, 52, 59, 94, 61, 74, 111, 428, 67, 65, 69

Offset: 1

Peter Schorn, Feb 04 2025

			For n < 5 there are no (x, y) with 0 < x < y < n such that x^y = y^x modulo n. Therefore a(n) = 0.
For n = 5 and 6 there is only 2^4 = 4^2 modulo n which makes a(5) = a(6) = 1.
For n = 7, there is 2^4=4^2, 2^5=5^2, 2^6=6^2, 4^5=5^4 and 4^6=6^4 modulo 7 which makes a(7) = 5.

Paolo Xausa, Table of n, a(n) for n = 1..1000

A380816[n_] := Sum[Boole[PowerMod[x, y, n] == PowerMod[y, x, n]], {x, 2, n-2}, {y, x+1, n-1}];
Array[A380816, 100] (* Paolo Xausa, Mar 17 2025 *)

a(n)={my(c=0);for(x=1,n-1,for(y=x+1,n-1,if(Mod(x,n)^y==Mod(y,n)^x,c++)));c}

def A380816(n): return sum(1 for x in range(1,n-1) for y in range(x+1,n) if pow(x,y,n)==pow(y,x,n)) # Chai Wah Wu, Feb 12 2025

A363340 a(n) is the smallest positive integer such that a(n) * n is the sum of two squares.

Original entry on oeis.org

1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 11, 3, 1, 7, 3, 1, 1, 1, 19, 1, 21, 11, 23, 3, 1, 1, 3, 7, 1, 3, 31, 1, 33, 1, 7, 1, 1, 19, 3, 1, 1, 21, 43, 11, 1, 23, 47, 3, 1, 1, 3, 1, 1, 3, 11, 7, 57, 1, 59, 3, 1, 31, 7, 1, 1, 33, 67, 1, 69, 7, 71, 1, 1, 1, 3, 19, 77, 3, 79

Offset: 1

Peter Schorn, May 28 2023

Using Fermat's two-squares theorem it is easy to see that a(n) is the product of all prime factors of n that are congruent to 3 modulo 4 and have an odd exponent.
This implies that a(n) is also the smallest positive integer such that n / a(n) is the sum of two squares.
Equivalently, a(n) is the product of all primes of the form 4k+3 that divide the squarefree part of n. If we use the squarefree kernel instead, we get A170819. - Peter Munn, Aug 06 2023

			a(1) = a(2) = 1 since 1 and 2 are sums of two squares.
a(3) = 3 since 3 and 6 are not sums of two squares but 3*3 is.
a(6) = 3 since 6 and 12 are not sums of two squares but 3*6 = 3^2 + 3^2.

Cf. A001481 (positions of 1's), A167181 (range of values).
Fixed points: A167181.
Cf. A000265, A002145, A007913, A059897, A097706, A170819.

a(n) = my(r=1); foreach(mattranspose(factor(n)), f, if(f[1]%4==3&&f[2]%2==1, r*=f[1])); r

Multiplicative with a(p^e) = p if p^e == 3 (mod 4), otherwise 1. - Peter Munn, Jul 03 2023
From Peter Munn, Aug 06 2023: (Start)
a(n) = A007913(A097706(n)) = A097706(A007913(n)).
a(n) == A000265(n) (mod 4).
a(A059897(n, k)) = A059897(a(n), a(k)).
(End)

A354761 Least number of squares and cubes that add up to n.

Original entry on oeis.org

1, 2, 3, 1, 2, 3, 4, 1, 1, 2, 3, 2, 2, 3, 4, 1, 2, 2, 3, 2, 3, 3, 4, 2, 1, 2, 1, 2, 2, 3, 2, 2, 2, 2, 2, 1, 2, 3, 3, 2, 2, 3, 2, 2, 2, 3, 3, 3, 1, 2, 3, 2, 2, 2, 3, 3, 2, 2, 3, 3, 2, 3, 2, 1, 2, 3, 3, 2, 3, 3, 3, 2, 2, 2, 3, 2, 3, 3, 3, 2, 1, 2, 3, 3, 2, 3

Offset: 1

Peter Schorn, Jun 06 2022

nonn

a(n) <= 4 since any number can be written as a sum of 4 squares (Lagrange's theorem).

Sequence first differs from A063274, A225926 and A274459 at n = 32 since 32 is a powerful number, a prime power and a perfect power but neither a square nor a cube.

			a(1) = 1, a(4) = 1 (4 = 2^2), a(7) = 4 (7 = 2^2 + 1^2 + 1^2 + 1^2), a(8) = 1 (8 = 2^3), a(12) = 2 (12 = 2^3 + 2^2), a(17) = 2 (17 = 4^2 + 1^2), a(32) = 2 (32 = 4^2 + 4^2).

Cf. A002760, A063274, A225926, A274459, A002376, A002828, A000290, A000578.

lista(n) = {my(v = vector(n)); for(j = 2, 3, for(i = 2, sqrtnint(n, j), v[i^j] = 1)); v[1]=1; v[2]=2; for(i=3, #v, if(v[i]==0, v[i] = vecmin(vector(i\2, k, v[k] + v[i-k])))); v}

A343923 If n = Product (p_j^k_j) then a(n) = Sum (abs(p_j-k_j)) (a(1) = 0 by convention).

Original entry on oeis.org

0, 1, 2, 0, 4, 3, 6, 1, 1, 5, 10, 2, 12, 7, 6, 2, 16, 2, 18, 4, 8, 11, 22, 3, 3, 13, 0, 6, 28, 7, 30, 3, 12, 17, 10, 1, 36, 19, 14, 5, 40, 9, 42, 10, 5, 23, 46, 4, 5, 4, 18, 12, 52, 1, 14, 7, 20, 29, 58, 6, 60, 31, 7, 4, 16, 13, 66, 16, 24, 11, 70, 2, 72

Offset: 1

Peter Schorn, May 04 2021

nonn

			a(24) = a(2^3 * 3) = abs(2 - 3) + abs(3 - 1) = 3. a(27) = a(3^3) = 0.

Cf. A008474.

a[n_] := Plus @@ (Abs[#[[1]] - #[[2]]] & /@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, May 04 2021 *)

a(n)=local(t); if(n<1, 0, t=factor(n); vecsum(abs(t[,1]-t[,2])))

Additive with a(p^e) = abs(p-e).

A342176 Tower of primes modulo n: a(n) = (2^3^5^7^ ... ^prime(n)) mod n.

Original entry on oeis.org

0, 0, 2, 0, 3, 2, 1, 0, 8, 8, 8, 8, 8, 8, 8, 0, 8, 8, 18, 8, 8, 8, 2, 8, 8, 8, 26, 8, 26, 8, 8, 0, 8, 8, 8, 8, 6, 18, 8, 8, 8, 8, 32, 8, 8, 2, 7, 32, 29, 8, 8, 8, 18, 26, 8, 8, 56, 26, 42, 8, 8, 8, 8, 0, 8, 8, 58, 8, 2, 8, 18, 8, 1, 6, 8, 56

Offset: 1

Peter Schorn, Mar 04 2021

nonn

a(n) = 0 iff n is a power of 2.

			a(1) = 0 = 2 mod 1. a(2) = 0 = 2^3 mod 2. a(3) = 2 = 2^3^5 = 2^243 = 2 mod 3.

Cf. A000040.

{ a(n, m=n, s=2) = local(g); if(s==prime(n), return(n%m)); g=s^valuation(m, s); m\=g; lift(chinese(Mod(0, g), Mod(s, m)^a(n, eulerphi(m), nextprime(s+1)) )) }

A334819 Largest quadratic nonresidue modulo n (with n >= 3).

Original entry on oeis.org

2, 3, 3, 5, 6, 7, 8, 8, 10, 11, 11, 13, 14, 15, 14, 17, 18, 19, 20, 21, 22, 23, 23, 24, 26, 27, 27, 29, 30, 31, 32, 31, 34, 35, 35, 37, 38, 39, 38, 41, 42, 43, 44, 45, 46, 47, 48, 48, 50, 51, 51, 53, 54, 55, 56, 56, 58, 59, 59, 61, 62, 63, 63, 65, 66, 67

Offset: 3

Peter Schorn, May 12 2020

The largest nonnegative integer less than n which is not a square modulo n.

If p is a prime congruent 3 modulo 4 then a(p) = p-1 since -1 is not a quadratic residue for such primes.

			The squares modulo 3 are 0 and 1. Therefore a(3) = 2. The nonsquares modulo 4 are 2 and 3 which makes a(4) = 3. Modulo 5 we have 0, 1 and 4 as squares giving a(5) = 3. 0, 1 and 4 are also the squares modulo 6 resulting in a(6) = 5. Since 10007 is a prime of the form 4*k + 3, a(10007) = 10006.

Robert Israel, Table of n, a(n) for n = 3..10000

Cf. A020649, A047210 (the largest square modulo n), A192450 (a(n)=n-1).

f:= proc(n) local k;
  for k from n-1 by -1 do if numtheory:-msqrt(k,n)=FAIL then return k fi
  od
end proc:
map(f, [$3..100]); # Robert Israel, May 14 2020

a[n_] := Module[{r}, For[r = n-1, r >= 1, r--, If[!IntegerQ[Sqrt[Mod[r, n]] ], Return[r]]]];
a /@ Range[3, 100] (* Jean-François Alcover, Aug 15 2020 *)

a(n) = forstep(r = n - 1, 1, -1, if(!issquare(Mod(r, n)), return(r)))

A333669 The smallest nontrivial quadratic residue modulo n.

Original entry on oeis.org

4, 3, 2, 4, 4, 4, 3, 4, 3, 2, 4, 4, 2, 4, 4, 4, 4, 3, 2, 4, 4, 3, 4, 4, 4, 4, 2, 4, 3, 2, 4, 4, 3, 4, 3, 4, 2, 4, 4, 4, 4, 2, 2, 4, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 4, 3, 2, 4, 4, 4, 3, 4, 4, 3, 4, 2, 4, 2, 3, 4, 4, 4, 3, 2, 4, 4, 2, 3, 4, 4, 4, 4, 4

Offset: 5

Peter Schorn, May 07 2020

The values are 2, 3 and 4. If 2 is a square modulo n (see A057126) the value is 2. Otherwise, if 3 is a square modulo n (see A057125) the value is 3. If neither 2 or 3 are a square modulo n the value is 4.

Dedicated to Urs Meyer at the occasion of his 60th birthday.

			The squares modulo 5 are 1 and 4, therefore a(5) = 4.
Modulo 6 the squares are 1, 3 and 4 which makes a(6) = 3.
a(7) = 2 since 2 == 3^2 (mod 7).

Robert Israel, Table of n, a(n) for n = 5..10000

Cf. A057126 for the n where the value is 2 and A057125 for the n where the value is 3 if n was not in A057126.

f:= proc(n) uses numtheory; if quadres(2,n)=1 then 2 elif quadres(3,n)=1 then 3 else 4 fi end proc:
map(f, [$5..100]); # Robert Israel, Sep 15 2020

qrQ[m_, n_] := Module[{k}, Reduce[Mod[m-k^2, n]==0, k, Integers] =!= False];
a[n_] := If[qrQ[2, n], 2, If[qrQ[3, n], 3, 4]];
a /@ Range[5, 100] (* Jean-François Alcover, Oct 25 2020 *)

a(n) = if(issquare(Mod(2,n)),2,issquare(Mod(3,n)),3,4)

cp's OEIS Frontend

User: Peter Schorn

A380816 Number of pairs (x, y) with 0 < x < y < n such that x^y = y^x modulo n.

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A363340 a(n) is the smallest positive integer such that a(n) * n is the sum of two squares.

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A354761 Least number of squares and cubes that add up to n.

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A343923 If n = Product (p_j^k_j) then a(n) = Sum (abs(p_j-k_j)) (a(1) = 0 by convention).

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A342176 Tower of primes modulo n: a(n) = (2^3^5^7^ ... ^prime(n)) mod n.

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A334819 Largest quadratic nonresidue modulo n (with n >= 3).

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A333669 The smallest nontrivial quadratic residue modulo n.

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