A076942 -id:A076942 - cp's OEIS frontend

A074792 Least k > 1 such that k^n == 1 (mod n).

2, 3, 4, 3, 6, 5, 8, 3, 4, 9, 12, 5, 14, 13, 16, 3, 18, 5, 20, 3, 4, 21, 24, 5, 6, 25, 4, 13, 30, 11, 32, 3, 34, 33, 36, 5, 38, 37, 16, 3, 42, 5, 44, 21, 16, 45, 48, 5, 8, 9, 52, 5, 54, 5, 16, 13, 7, 57, 60, 7, 62, 61, 4, 3, 66, 23, 68, 13, 70, 29, 72, 5, 74, 73, 16, 37, 78, 17, 80, 3

Offset: 1

Benoit Cloitre, Sep 07 2002

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000

a(n) = {A076944(n)}^(1/n).

Cf. A076942, A076943, A076944.

Do[k = 2; While[ !IntegerQ[(k^n - 1)/n], k++ ]; Print[k], {n, 1, 80}] (* Robert G. Wilson v *)

a(n)=if(n<0,0,s=2; while((s^n-1)%n>0,s++); s)

a(n)=my(s=2); while(Mod(s,n)^n-1!=0, s++); return(s) \\ Rémy Sigrist, Apr 02 2017

If p is prime a(p)=p+1 and a(2p)=2p-1; if n is in A050384 a(n)=n+1; if n is in A067945 a(n)=3 etc. It seems that sum(k=1, n, a(k)) is asymptotic to c*n^2 with c=0.2...

A067721 Least number k such that k (k + n) is a perfect square, or 0 if impossible.

Original entry on oeis.org

1, 0, 0, 1, 0, 4, 2, 9, 1, 3, 8, 25, 4, 36, 18, 1, 2, 64, 6, 81, 16, 4, 50, 121, 1, 20, 72, 9, 36, 196, 2, 225, 4, 11, 128, 1, 12, 324, 162, 13, 5, 400, 8, 441, 100, 3, 242, 529, 1, 63, 40, 17, 144, 676, 18, 9, 7, 19, 392, 841, 4, 900, 450, 1, 8, 16, 22, 1089, 256, 23, 2, 1225

Offset: 0

Robert G. Wilson v, Feb 05 2002

nonn

Impossible only for 1, 2 and 4. k equals 1 when n is in A005563. k equals 2 when n is in A054000.

Let k*(k+n)= c*c, gcd(n,k,c)=1 . Then primitive triples (n,k,c) are of the form : 1) n is prime. (n,k,c)=( p, (p*p-2*p+1)/4, (p*p-1)/4 ) 2) n=(c/t)*(c/t)- t*t, n is not a prime, t positive integer. (n,k,c)=( (c/t)*(c/t)- t*t, t*t, c ). [Ctibor O. Zizka, May 04 2009]

			a(7) = 9 because 9 (7+9) = 144 = 12^2.

Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms 0..2000 from Carmine Suriano)

Cf. A005563, A007913, A054000, A067632, A076942.

Do[k = 1; While[ !IntegerQ[ Sqrt[ k (k + n)]], k++ ]; Print[k], {n, 5, 75} ]

from itertools import takewhile
from collections import deque
from sympy import divisors
def A067721(n): return ((a:=next(iter(deque((d for d in takewhile(lambda d:d>2) if n else 1 # Chai Wah Wu, Aug 21 2024

A076943 Smallest k > 0 such that n*k + 1 is an n-th power.

Original entry on oeis.org

1, 4, 21, 20, 1555, 2604, 299593, 820, 29127, 348678440, 67546215517, 20345052, 61054982558011, 281241170407092, 76861433640456465, 2690420, 128583032925805678351, 211927625868, 275941052631578947368421, 174339220

Offset: 1

Amarnath Murthy, Oct 19 2002

			For n = 7, 1 + 7*a(7) = 1 + 7*299593 = 2097152 = 2^21 = 8^7.
For n = 10, 1 + 10*a(10) = 1 + 10*348678440 = 3486784401 = 3^20 = 9^10. - _Marius A. Burtea_, Jun 01 2019

Marius A. Burtea, Table of n, a(n) for n = 1..150

Cf. A076942, A076944, A074792.

sol:=[];
for u in [1..20] do
   for k in [2..100] do
      if IsIntegral((k^u-1)/u) then sol[u]:=(k^u-1)/u; break; end if;
   end for;
end for;
sol; // Marius A. Burtea, Jun 01 2019

Do[k = 2; While[ !IntegerQ[(k^n - 1)/n], k++ ]; Print[(k^n - 1)/n], {n, 1, 20}] (* Robert G. Wilson v, Oct 21 2002 *)

a(n) <= ((n+1)^n - 1) / n.
a(p^k) = ((p+1)^(p^k) - 1) / p^k. - Charlie Neder, May 23 2019
a(2*p) = ((2*p-1)^(2*p) - 1) / (2*p). - Charlie Neder, May 23 2019

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

A256008 Self-inverse permutation of positive integers: 4k+1 is swapped with 4k+3, and 4k+2 with 4k+4.

Original entry on oeis.org

3, 4, 1, 2, 7, 8, 5, 6, 11, 12, 9, 10, 15, 16, 13, 14, 19, 20, 17, 18, 23, 24, 21, 22, 27, 28, 25, 26, 31, 32, 29, 30, 35, 36, 33, 34, 39, 40, 37, 38, 43, 44, 41, 42, 47, 48, 45, 46, 51, 52, 49, 50, 55, 56, 53, 54, 59, 60, 57, 58, 63, 64, 61, 62, 67, 68, 65, 66, 71, 72, 69, 70, 75, 76, 73, 74, 79

Offset: 1

Ivan Neretin, May 06 2015

A lexicographically minimal sequence of distinct positive integers such that a(n)*n + 1 is a square. The same condition without the requirement for a(n) to be distinct would produce A076942.

Ivan Neretin, Table of n, a(n) for n = 1..10000
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.
Index entries for sequences that are permutations of the natural numbers
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A004443, A076942, A269526.

[n-2*(-1)^((2*n+1-(-1)^n) div 4): n in [1..100]]; // Wesley Ivan Hurt, Oct 13 2015

I:=[3,4,1,2]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-2)+2*Self(n-3)-Self(n-4): n in [1..80]]; // Vincenzo Librandi, Oct 14 2015

/* By definition: */ &cat[[4*k+3,4*k+4,4*k+1,4*k+2]: k in [0..20]]; // Bruno Berselli, Oct 19 2015

A256008:=n->n-2*(-1)^((2*n+1-(-1)^n)/4): seq(A256008(n), n=1..100); # Wesley Ivan Hurt, Oct 13 2015

Table[BitXor[n - 1, 2] + 1, {n, 77}]
CoefficientList[Series[(3 - 2*x - x^2 + 2*x^3)/((x - 1)^2*(x^2 + 1)), {x, 0, 100}], x] (* Wesley Ivan Hurt, Oct 13 2015 *)
LinearRecurrence[{2, -2, 2, -1}, {3, 4, 1, 2}, 80] (* Vincenzo Librandi, Oct 14 2015 *)

a(n) = bitxor(n-1,2)+1 \\ Charles R Greathouse IV, May 06 2015

Vec(x*(3-2*x-x^2+2*x^3)/((x-1)^2*(x^2+1)) + O(x^100)) \\ Altug Alkan, Oct 13 2015

a(n) = (-1+I)*((-I)^n+I*I^n)+n \\ Colin Barker, Oct 19 2015

def a(n): return ((n-1)^2) + 1
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Mar 21 2023

From Wesley Ivan Hurt, Oct 13 2015: (Start)
G.f.: x*(3-2*x-x^2+2*x^3)/((x-1)^2*(x^2+1)).
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4) for n>4.
a(n) = n-2*(-1)^((2*n+1-(-1)^n)/4). (End)
a(n) = (-1+i)*((-i)^n+i*i^n)+n, where i = sqrt(-1). - Colin Barker, Oct 19 2015
a(n) = 1 + A004443(n-1). - Alois P. Heinz, Jan 23 2022

A215653 a(n) = smallest positive m such that m^2 = 1+k*n with positive k.

Original entry on oeis.org

2, 3, 2, 3, 4, 5, 6, 3, 8, 9, 10, 5, 12, 13, 4, 7, 16, 17, 18, 9, 8, 21, 22, 5, 24, 25, 26, 13, 28, 11, 30, 15, 10, 33, 6, 17, 36, 37, 14, 9, 40, 13, 42, 21, 19, 45, 46, 7, 48, 49, 16, 25, 52, 53, 21, 13, 20, 57, 58, 11, 60, 61, 8, 31, 14, 23, 66, 33, 22, 29

Offset: 1

Zak Seidov, Aug 19 2012

nonn

Apparently a(n) = A070667(n) for n > 2. Note the linear patterns in the graph.

			a(1) = 2, k = 3; a(2) = 3, k = 4; a(3) = 2, k = 1; a(1000) = 249, k = 62.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Zak Seidov)
Dorin Andrica and Vlad Crişan, The Smallest Nontrivial Solution to x^k == 1 (mod n) and Related Sequences, Amer. Math. Monthly, Vol. 126, No. 2 (2019), pp. 173-178.

Cf. A061369, A070667, A076942.

Flatten[{2,Table[Select[Range[2,1000],PowerMod[#,2,k]==1&,1],{k,2,1000}]}] (* first 1000 terms *)

a(n) = {my(m = n + 1); while(!issquare(m), m += n); sqrtint(m);} \\ Amiram Eldar, Mar 16 2025

a(n) = sqrt(1+n*A076942(n)).

a(n) = sqrt(A061369(n)). - Amiram Eldar, Mar 16 2025

A076944 Least number such that n*k+1 is an n-th power.

Original entry on oeis.org

2, 9, 64, 81, 7776, 15625, 2097152, 6561, 262144, 3486784401, 743008370688, 244140625, 793714773254144, 3937376385699289, 1152921504606846976, 43046721, 2185911559738696531968, 3814697265625

Offset: 1

Amarnath Murthy, Oct 19 2002

nonn

a(n) <= {(n+1)^n - 1}/n.

Cf. A076942, A076943, A074792.

Do[k = 2; While[ !IntegerQ[(k^n - 1)/n], k++ ]; Print[k^n], {n, 1, 19}]

a(n) = n*A076943(n) + 1.

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

A215696 a(n)=smallest positive k>n+2 such that k*n+1 is a square.

Original entry on oeis.org

8, 12, 8, 12, 16, 20, 24, 15, 32, 36, 40, 24, 48, 52, 24, 33, 64, 68, 72, 42, 40, 84, 88, 35, 96, 100, 104, 60, 112, 56, 120, 69, 56, 132, 48, 78, 144, 148, 72, 60, 160, 72, 168, 96, 91, 180, 184, 63, 192, 196, 88, 114, 208, 212, 105, 85, 104, 228, 232, 84

Offset: 1

Zak Seidov, Aug 21 2012

nonn

For any n and k=n+2, 1+k*n=(n+1)^2, so here we consider the case k>n+2. Cases kA076942, A215653.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A076942, A215653.

for(n=1,100,k=n+3;while(!issquare(1+k*n),k++);print1(k","))

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A074792 Least k > 1 such that k^n == 1 (mod n).

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A067721 Least number k such that k (k + n) is a perfect square, or 0 if impossible.

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A076943 Smallest k > 0 such that n*k + 1 is an n-th power.

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A256008 Self-inverse permutation of positive integers: 4k+1 is swapped with 4k+3, and 4k+2 with 4k+4.

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A215653 a(n) = smallest positive m such that m^2 = 1+k*n with positive k.

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A076944 Least number such that n*k+1 is an n-th power.

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A215696 a(n)=smallest positive k>n+2 such that k*n+1 is a square.

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