A117950 -id:A117950 - cp's OEIS frontend

A057128 Numbers n such that -3 is a square mod n.

1, 2, 3, 4, 6, 7, 12, 13, 14, 19, 21, 26, 28, 31, 37, 38, 39, 42, 43, 49, 52, 57, 61, 62, 67, 73, 74, 76, 78, 79, 84, 86, 91, 93, 97, 98, 103, 109, 111, 114, 122, 124, 127, 129, 133, 134, 139, 146, 147, 148, 151, 156, 157, 158, 163, 169, 172, 181, 182, 183, 186, 193

Offset: 1

Henry Bottomley, Aug 10 2000

nonn

The fact that there are no numbers in this sequence of the form 6k+5 leads to the result that all prime factors of central polygonal numbers (A002061 of the form n^2-n+1) are either 3 or of the form 6k+1. This in turn leads to there being an infinite number of primes of the form 6k+1, since if P=product[all known primes of form 6k+1] then all the prime factors of 9P^2-3P+1 must be unknown primes of form 6k+1.
Numbers that are not multiples of 8 or 9 and for which all prime factors greater than 3 are congruent to 1 mod 6. - Eric M. Schmidt, Apr 21 2013
Numbers that divide at least some member of A117950. - Robert Israel, Feb 19 2016

			a(7)=13 since -3 mod 13=10 mod 13=6^2 mod 13.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Includes the primes in A045331 and these (primes congruent to {1, 2, 3} mod 6) are the prime factors of the terms in this sequence. Cf. A008784, A057125, A057126, A057127, A057129.

Cf. A117950.

select(t -> numtheory:-quadres(-3,t) = 1, {$1..1000}); # Robert Israel, Feb 19 2016

Select[Range[200], IntegerQ[PowerMod[-3, 1/2, #]]&] // Quiet (* Jean-François Alcover, Mar 05 2019 *)

isok(n) = issquare(Mod(-3,n)); \\ Michel Marcus, Feb 19 2016

def A057128(n) :
    if n%8==0 or n%9==0: return False
    for (p, m) in factor(n) :
        if p % 6 not in [1, 2, 3] : return False
        return True
# Eric M. Schmidt, Apr 21 2013

A189833 a(n) = n^2 + 8.

Original entry on oeis.org

8, 9, 12, 17, 24, 33, 44, 57, 72, 89, 108, 129, 152, 177, 204, 233, 264, 297, 332, 369, 408, 449, 492, 537, 584, 633, 684, 737, 792, 849, 908, 969, 1032, 1097, 1164, 1233, 1304, 1377, 1452, 1529, 1608, 1689, 1772, 1857, 1944, 2033

Offset: 0

Vladimir Joseph Stephan Orlovsky, Apr 28 2011

From César Eliud Lozada, Mar 29 2021: (Start)
Numbers a(n) such that sqrt( a(n) + 4*n*sqrt(2) ) = n + 2*sqrt(2). Examples:
For n=1:  sqrt( 9 +  4*sqrt(2)) = 1 + 2*sqrt(2),
For n=2:  sqrt(12 +  8*sqrt(2)) = 2 + 2*sqrt(2),
For n=3:  sqrt(17 + 12*sqrt(2)) = 3 + 2*sqrt(2). (End)

G. C. Greubel, Table of n, a(n) for n = 0..10000 (terms 0..955 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002522, A059100, A117950, A087475.

[n^2+8: n in [0..50]]; // Vincenzo Librandi, Apr 29 2011

Table[n^2+8,{n,0,100}]
LinearRecurrence[{3,-3,1},{8,9,12},50] (* Harvey P. Dale, Jun 21 2022 *)

a(n)=n^2+8 \\ Charles R Greathouse IV, Jun 17 2017

From G. C. Greubel, Jan 13 2018: (Start)
G.f.: (8 - 15*x + 9*x^2)/(1 - x)^3.
E.g.f.: (8 + x + x^2)*exp(x). (End)
From Amiram Eldar, Jul 04 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + 2*sqrt(2)*Pi*coth(2*sqrt(2)*Pi))/16.
Sum_{n>=0} (-1)^n/a(n) = (1 + 2*sqrt(2)*Pi*cosech(2*sqrt(2)*Pi))/16. (End)
From Amiram Eldar, Feb 05 2024: (Start)
Product_{n>=0} (1 - 1/a(n)) = (sqrt(7/2)/2)*sinh(sqrt(7)*Pi)/sinh(2*sqrt(2)*Pi).
Product_{n>=0} (1 + 1/a(n)) = (3/(2*sqrt(2)))*sinh(3*Pi)/sinh(2*sqrt(2)*Pi). (End)

Offset changed from 1 to 0 by Vincenzo Librandi, Apr 29 2011

A163253 An interspersion: the order array of the odd-numbered columns of the double interspersion at A161179.

Original entry on oeis.org

1, 4, 2, 9, 5, 3, 16, 10, 7, 6, 25, 17, 13, 11, 8, 36, 26, 21, 18, 14, 12, 49, 37, 31, 27, 22, 19, 15, 64, 50, 43, 38, 32, 28, 23, 20, 81, 65, 57, 51, 44, 39, 33, 29, 24, 100, 82, 73, 66, 58, 52, 45, 40, 34, 30, 121, 101, 91, 83, 74, 67, 59, 53, 46, 41, 35

Offset: 1

Clark Kimberling, Jul 23 2009

A permutation of the natural numbers.
Row 1 consists of the squares.
Beginning with row 5, the columns obey a 3rd-order recurrence:
c(n)=c(n-1)+c(n-2)-c(n-3)+1; thus disregarding row 1, the nonsquares are partitioned by this recurrence.
Except for initial terms, the first ten rows match A000290, A002522, A002061, A059100, A014209, A117950, A027688, A087475, A027689, A117951, and the first column, A035106.

			Corner:
1....4....9...16...25
2....5...10...17...26
3....7...13...21...31
6...11...18...27...38
The double interspersion A161179 begins thus:
1....4....7...12...17...24
2....3....8...11...18...23
5....6...13...16...25...30
9...10...19...22...33...38
Expel the even-numbered columns, leaving
1....7...17...
2....8...18...
5...13...25...
9...19...33...
Then replace each of those numbers by its rank when all the numbers are jointly ranked.

Clark Kimberling, Doubly interspersed sequences, double interspersions and fractal sequences, The Fibonacci Quarterly 48 (2010) 13-20.

Cf. A000037, A000290, A161179, A163254, A163255, A163256, A163257, A163258.

Let S(n,k) denote the k-th term in the n-th row. Three cases:
S(1,k)=k^2;
if n is even, then S(n,k)=k^2+(n-2)k+(n^2-2*n+4)/4;
if n>=3 is odd, then S(n,k)=k^2+(n-2)k+(n^2-2*n+1)/4.

Edited and augmented by Clark Kimberling, Jul 24 2009

A189836 a(n) = n^2 + 11.

Original entry on oeis.org

11, 12, 15, 20, 27, 36, 47, 60, 75, 92, 111, 132, 155, 180, 207, 236, 267, 300, 335, 372, 411, 452, 495, 540, 587, 636, 687, 740, 795, 852, 911, 972, 1035, 1100, 1167, 1236, 1307, 1380, 1455, 1532, 1611, 1692, 1775, 1860, 1947

Offset: 0

Vladimir Joseph Stephan Orlovsky, Apr 28 2011

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002522, A059100, A117950, A087475.

[n^2 + 11: n in [0..50]]; // G. C. Greubel, Jan 13 2018

Table[n^2+11,{n,0,100}]
LinearRecurrence[{3,-3,1},{11,12,15},60] (* Harvey P. Dale, Aug 24 2020 *)

a(n)=n^2+11 \\ Charles R Greathouse IV, Jun 17 2017

From G. C. Greubel, Jan 13 2018: (Start)
G.f.: (11 - 21*x + 12*x^2)/(1 - x)^3.
E.g.f.: (11 + x + x^2)*exp(x). (End)
From Amiram Eldar, Nov 02 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(11)*Pi*coth(sqrt(11)*Pi))/22.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(11)*Pi*cosech(sqrt(11)*Pi))/22. (End)
From Amiram Eldar, Feb 12 2024: (Start)
Product_{n>=0} (1 - 1/a(n)) = sqrt(10/11)*sinh(sqrt(10)*Pi)/sinh(sqrt(11)*Pi).
Product_{n>=0} (1 + 1/a(n)) = 2*sqrt(3/11)*sinh(2*sqrt(3)*Pi)/sinh(sqrt(11)*Pi). (End)

A213921 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places clockwise. Table T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 10, 8, 9, 13, 17, 14, 6, 16, 21, 26, 22, 11, 12, 25, 31, 37, 32, 18, 15, 20, 36, 43, 50, 44, 27, 23, 24, 30, 49, 57, 65, 58, 38, 33, 19, 35, 42, 64, 73, 82, 74, 51, 45, 28, 29, 48, 56, 81, 91, 101, 92, 66, 59, 39, 34, 41, 63, 72, 100, 111

Offset: 1

Boris Putievskiy, Mar 05 2013

A permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). Enumeration table T(n,k) is layer by layer. The order of the list:
T(1,1)=1;
T(1,2), T(2,1), T(2,2);
. . .
T(1,n), T(3,n), ... T(n,3), T(n,1), T(2,n), T(4,n), ... T(n,4), T(n,2);
...

			The start of the sequence as table:
   1   2   5  10  17  26 ...
   3   4   8  14  22  32 ...
   7   9   6  11  18  27 ...
  13  16  12  15  23  33 ...
  21  25  20  24  19  28 ...
  31  36  30  35  29  34 ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   2,  3;
   5,  4,  7;
  10,  8,  9, 13;
  17, 14,  6, 16, 21;
  26, 22, 11, 12, 25, 31;
  ...

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060734, A060736; table T(n,k) contains: in rows A002522, A014206, A059100, A027688, A117950, A027689, A087475, A027690, A117951, A027691, A114949, A027692, A117619; in columns A002061, A000290, A002378, A005563, A028387, A008865, A028552, A028872, A014209, A028347, A028875.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if i > j:
   result=i*i-(j%2)*i+2-int((j+2)/2)
else:
   result=j*j-((i%2)+1)*j + int((i+3)/2)

As a table:
T(n,k) = n*n - (k mod 2)*n + 2 - floor((k+2)/2), if n>k;
T(n,k) = k*k - ((n mod 2)+1)*k + floor((n+3)/2), if n<=k.
As a linear sequence:
a(n) = i*i - (j mod 2)*i + 2 - floor((j+2)/2), if i>j;
a(n) = j*j - ((i mod 2)+1)*j + floor((i+3)/2), if i<=j; where i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n, t = floor((-1+sqrt(8*n-7))/2).

A273324 Integers n such that n^2 + 3 is the sum of 4 but no fewer nonzero squares.

Original entry on oeis.org

2, 5, 6, 10, 11, 14, 18, 21, 22, 26, 27, 30, 34, 37, 38, 42, 43, 46, 50, 53, 54, 58, 59, 62, 66, 69, 70, 74, 75, 78, 82, 85, 86, 90, 91, 94, 98, 101, 102, 106, 107, 110, 114, 117, 118, 122, 123, 126, 130, 133, 134, 138, 139, 142, 146, 149, 150, 154, 155, 158, 162, 165, 166, 170

Offset: 1

Altug Alkan, May 20 2016

If n^2 + k is a term of A004215, then the minimum positive value of k is 3, obviously.

See also the first differences (A278536) of this sequence.

			2 is in the sequence because 2^2 + 3 = 7 is a term of A004215.

Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A000196, A004215, A117950, A278491, A278536.

isA004215(n) = {n\4^valuation(n, 4)%8==7}
lista(nn) = for(n=1, nn, if(isA004215(n^2+3), print1(n, ", ")));

a(n) = A000196(1+A278491(n)). - Antti Karttunen, Nov 26 2016

A351468 Irregular triangle read by rows where row n is Newey's sequence containing all permutations of 1..n.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 3, 1, 2, 1, 3, 1, 2, 3, 4, 1, 2, 3, 1, 4, 2, 1, 3, 1, 2, 3, 4, 5, 1, 2, 3, 4, 1, 5, 2, 3, 1, 4, 5, 2, 1, 3, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 1, 6, 2, 3, 4, 1, 5, 6, 2, 3, 1, 4, 5, 6, 2, 1, 3

Offset: 2

Kevin Ryde, Feb 23 2022

Row n contains n^2 - 2*n + 4 = A117950(n-1) terms, numbered as columns k >= 1. Row n contains within it all permutations of 1..n as subsequences. These subsequences need not be consecutive terms (and in general are not).
Newey's construction (section 6) is an initial 1..n then successive term 1 and n-2 terms from a rotating block of 2..n. The effect is term 1 at k=1, k=n+1, then steps n-1 apart, and the rest filled with repeating 2..n.
Koutas and Hu (equation (1)) form the same by blocks D_k(m|n) which start with 1 and then appropriate parts of rotated 2..n like Newey.
Jon E. Schoenfield in A062714 starts from repeated 2..n and inserts 1's.
For n = 3 to 7, Newey shows these sequences are as short as possible (A062714) for any sequence containing all permutations, and noted the "obvious" conjecture that maybe they would be shortest always. But this is not so, since Jon E. Schoenfield gives a shorter n=16 in A062714, and Radomirovic constructs shorter for all n >= 10.

			Triangle begins
  n=2:  1,2,1,2
  n=3:  1,2,3,1,2,1,3
  n=4:  1,2,3,4,1,2,3,1,4,2,1,3
  n=5:  1,2,3,4,5,1,2,3,4,1,5,2,3,1,4,5,2,1,3
For row n=3, the permutations of 1,2,3 are located within the row as follows (some are present in multiple ways too).
   1,2,3,1,2,1,3     row n=3
   1-2-3             \
   1---3---2         | all permutations
           2-1-3     | of 1,2,3 within
     2-3-1           | row n=3
       3-1-2         |
       3---2-1       /
For row n=4, see example in A062714.
For row n=5, the pattern of 1's among repeating 2..5 is
    2,3,4,5,  2,3,4,  5,2,3,  4,5,2,  3
  1,        1,      1,      1,      1,
   \-------/ \-----/ \-----/ \-----/
    5 apart,   thereafter 4 apart

Kevin Ryde, Table of n, a(n) for rows 2 .. 30, flattened
P. J. Koutas and T. C. Hu, Shortest String Containing All Permutations, Discrete Mathematics, volume 11, 1975, pages 125-132.
Malcolm Newey, Notes on a Problem Involving Permutations as Subsequences, Stanford Artificial Intelligence Laboratory, Memo AIM-190, STAN-CS-73-340, 1973.

Cf. A117950 (row lengths), A062714 (shortest possible).

Cf. A351469 (Adelman's sequences).

T(n,k) = if(k<=n,k, my(q,r);[q,r]=divrem(k-2,n-1); if(r==0&&q

row(n) = my(r=1,t=1); vector((n-1)^2+3,i, if(i==1,1, r++>n,r=1+(n>2);1, if(t++>n,t=2, t)));

T(n,k) = k for 1 <= k <= n, otherwise.

T(n,k) = 1 if r=0 and q

T(n,k) = 2 + ((r-q) mod (n-1)),

where division q = floor((k-2)/(n-1)) remainder r = (k-2) mod (n-1). [Adapted from Jon E. Schoenfield in A062714.]

A171746 Let f(n) = n + floor(sqrt(n)). Then a(n) is the smallest number of iterations of f on n such that a perfect square is obtained.

Original entry on oeis.org

3, 2, 1, 5, 2, 4, 1, 3, 7, 2, 4, 6, 1, 3, 5, 9, 2, 4, 6, 8, 1, 3, 5, 7, 11, 2, 4, 6, 8, 10, 1, 3, 5, 7, 9, 13, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 15, 2, 4, 6, 8, 10, 12, 14, 1, 3, 5, 7, 9, 11, 13, 17, 2, 4, 6, 8, 10, 12, 14, 16, 1, 3, 5, 7, 9, 11, 13, 15, 19, 2, 4, 6, 8, 10, 12, 14, 16, 18, 1, 3, 5

Offset: 1

Neven Juric (neven.juric(AT)apis-it.hr), Oct 07 2010

nonn

Iterate A028392, starting with n: a(n) is the number of steps until a square will be reached. - Reinhard Zumkeller, Feb 23 2012

			f(9)=12, f(12)=15, f(15)=18, f(18)=22, f(22)=26, f(26)=31, f(31)=36. The first square number in this sequence 12,15,18,22,26,31,36 is on the seventh place and therefore a(9)=7.

Matematicko-fizicki list 1/144, problem 2-2, page 29, (1985-1986).

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

Cf. A010052, A028392.

a171746 = (+ 1) . length . takeWhile (== 0) .
                           map a010052 . tail . iterate a028392
-- Reinhard Zumkeller, Feb 23 2012, Oct 14 2010

f[n_] := Length@ NestWhileList[ # + Floor@Sqrt@# &, n, ! IntegerQ@Sqrt@# || # == n &] - 1; Array[f, 93] (* Robert G. Wilson v, Oct 08 2010 *)

f(n) = n + sqrtint(n); \\ A028392
a(n) = my(k=1); while (!issquare(n=f(n)), k++); k; \\ Michel Marcus, Nov 06 2022

From Robert G. Wilson v, Oct 08 2010: (Start)
a(k)=1 for A002061(n): n^2 - n + 1 for n>1;
a(k)=2 for A002522(n): n^2 + 1     for n>1;
a(k)=3 for A014206(n): n^2 + n + 2 for n>1;
a(k)=4 for A059100(n): n^2 + 2     for n>1;
a(k)=5 for A027688(n): n^2 + n + 3 for n>2;
a(k)=6 for A117950(n): n^2 + 3     for n>2;
a(k)=7 for A027689(n): n^2 + n + 4 for n>4;
a(k)=8 for A087475(n): n^2 + 4     for n>3;
a(k)=9 for A027690(n): n^2 + n + 5 for n>4; ... (End)
a(n^2) = 2*n + 1: a(A000290(n)) = A005408(n). - Reinhard Zumkeller, Oct 14 2010

A255844 a(n) = 2*n^2 + 6.

Original entry on oeis.org

6, 8, 14, 24, 38, 56, 78, 104, 134, 168, 206, 248, 294, 344, 398, 456, 518, 584, 654, 728, 806, 888, 974, 1064, 1158, 1256, 1358, 1464, 1574, 1688, 1806, 1928, 2054, 2184, 2318, 2456, 2598, 2744, 2894, 3048, 3206, 3368, 3534, 3704, 3878, 4056, 4238, 4424, 4614

Offset: 0

Avi Friedlich, Mar 08 2015

This is the case k=3 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2. Also, it is noted that a(n)*n = (n + 1)^3 + (n - 1)^3.
Equivalently, numbers m such that 2*m-12 is a square.
For n = 0..16, 3*a(n)-1 is prime (see A087370); for n = 0..12, 3*a(n)-5 is prime (see A107303).

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016825 (first differences), A087370, A107303, A114949, A117950.
Cf. A152811: nonnegative numbers of the form 2*m^2-6.
Subsequence of A000378.
Cf. similar sequences listed in A255843.

Magma
```
[2*n^2+6: n in [0..50]];
```
Mathematica
```
Table[2 n^2 + 6, {n, 0, 50}]
```
PARI
```
vector(50, n, n--; 2*n^2+6)
```
Sage
```
[2*n^2+6 for n in (0..50)]
```

G.f.: 2*(3-5*x+4*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*A117950(n).
From Amiram Eldar, Mar 28 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(3)*Pi*coth(sqrt(3)*Pi))/12.
Sum_{n>=0} (-1)^n/a(n) = (1 + (sqrt(3)*Pi)*cosech(sqrt(3)*Pi))/12. (End)
E.g.f.: 2*exp(x)*(3 + x + x^2). - Elmo R. Oliveira, Jan 25 2025

Corrected and extended by Bruno Berselli, Mar 11 2015

A328791 Triangular numbers of the form k^2 + 3.

Original entry on oeis.org

3, 28, 903, 30628, 1040403, 35343028, 1200622503, 40785822028, 1385517326403, 47066803275628, 1598885794044903, 54315050194251028, 1845112820810490003, 62679520857362409028, 2129258596329511416903, 72332112754346025765628, 2457162575051435364614403

Offset: 1

Jon E. Schoenfield, Oct 27 2019

nonn

There exist triangular numbers of the form k^2 + j for j=0 (A001110), j=1 (A164055), j=2 (A214838), and j=3 (this sequence), but not for j=4,7,8,13,16,18,... (A328792).

Cf. A001110, A164055, A214838, A328792.
Intersection of A000217 and A117950.
Cf. A276598 (the k's).

limit = 10**7 # rough limit for k
A000217 = set(k*(k+1)//2 for k in range(14*limit//10))
A117950 = set(k**2 + 3 for k in range(limit))
print(sorted(A000217 & A117950)) # Michael S. Branicky, Mar 28 2021

a(1) = 3, a(2) = 28; for n > 2, a(n) = 34*a(n-1) - a(n-2) - 46.

Previous Showing 11-20 of 33 results. Next

cp's OEIS Frontend

A057128 Numbers n such that -3 is a square mod n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

A189833 a(n) = n^2 + 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A163253 An interspersion: the order array of the odd-numbered columns of the double interspersion at A161179.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A189836 a(n) = n^2 + 11.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A213921 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places clockwise. Table T(n,k) read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Formula

A273324 Integers n such that n^2 + 3 is the sum of 4 but no fewer nonzero squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A351468 Irregular triangle read by rows where row n is Newey's sequence containing all permutations of 1..n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI