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A371969 Perimeters of triangles with integer sides, which can be decomposed into 3 triangles that have a common vertex strictly inside the surrounding triangle and also integer sides.

49, 50, 54, 64, 75, 78, 80, 88, 90, 91, 98, 100, 104, 108, 112, 117, 120, 121, 125, 126, 128, 133, 136, 140, 144, 147, 150, 156, 160, 162, 165, 168, 169, 170, 175, 176, 180, 182, 184, 188, 192, 195, 196, 198, 200, 203, 208, 210, 216, 220, 224, 225, 231, 234, 238, 240

Offset: 1

Klaus Nagel and Hugo Pfoertner, Apr 14 2024

nonn

			a(1) = 49 is the perimeter of the first decomposable triangle with sides of the outer triangle [8, 19, 22], and sides meeting at the 4th "inner" vertex: 17, 6, 4. The 3 inner triangles have sides [8, 4, 6], [19, 17, 4], and [22, 6, 17].

These triangles can be viewed as degenerate tetrahedrons, in which all triangular inequalities for the faces are satisfied, and the Cayley-Menger determinant, which determines whether the 4th vertex yields a valid tetrahedron, takes the value 0.

Klaus Nagel, Illustration of a(1) = 49.
Klaus Nagel, Illustration of a(2) = 50.
Klaus Nagel, Illustration of a(3) = 54.
Klaus Nagel, Illustration of a(4) = 64; another triangle is [20, 21, 23].
Hugo Pfoertner, List of triangles up to perimeter 400; one example of each.
Wikipedia, Cayley-Menger determinant

Cf. A070083, A316841, A342720, A371345, A371973.

H(a,b,c) = {my (s=(a+b+c)/2); s*(s-a)*(s-b)*(s-c)};
CM(w1,w2,w3,v1,v2,v3) = matdet([0,1,1,1,1; 1,0,w3^2,w2^2,v1^2; 1,w3^2,0,w1^2,v2^2; 1,w2^2,w1^2,0,v3^2; 1,v1^2,v2^2,v3^2,0]);
is_a371969(peri) = {forpart (w=peri, my (A=H(w[1],w[2],w[3]), epsA=1e-12); for (v1=1, w[3]-2, for (v2=w[3]-v1+1, w[3]-2, my (A3=H(w[3],v2,v1)); if (A3>=A, next); for (v3=1, w[3]-2, if (v3+v2<=w[1] || v3+v1<=w[2], next); my (A1=H(w[1],v2,v3)); if (A1>=A, next); my (A2=H(w[2],v1,v3)); if (A2>=A, next); my (C=CM(w[1],w[2],w[3],v1,v2,v3)); if (C==0 && abs(sqrt(A)-sqrt(A1)-sqrt(A2)-sqrt(A3)) < epsA,
\\ print (peri," ",Vec(w)," ",[v1,v2,v3]);
return(1))))), [1,(peri-1)\2], [3,3]); 0};
for (n=3, 100, if (is_a371969(n), print1(n,", ")))

A367147 Index of matching grid point in the bijection between two infinite triangular grids with one grid rotated by Pi/6 around the common point (0,0), using an enumeration of the grid points by A307014 and A307016.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 12, 14, 15, 9, 17, 18, 29, 7, 8, 23, 10, 11, 30, 13, 20, 21, 22, 33, 24, 16, 26, 27, 28, 36, 42, 19, 38, 39, 25, 41, 31, 32, 57, 34, 35, 60, 54, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 72, 37, 63, 66, 40, 69, 55, 73, 74, 56, 76, 77, 58, 79, 80, 59

Offset: 0

Klaus Nagel and Hugo Pfoertner, Nov 06 2023

nonn

The methods used to achieve a distance-limited bijection of the points of two square grids (see A307110) are applied here to triangular grids. The two grids, which are rotated by 30 degrees = Pi/6 from each other, are assigned the colors red and blue to distinguish them, which are also used in the illustrations. The blue triangular grid is turned clockwise by 15 degrees = Pi/12, all points are lined up on parallel lines with inclination Pi/12 towards the vertical axis. These are called blue lines. The vertical distance between adjacent points is cos(Pi/12). The same is done for the red grid with a CCW rotation of Pi/12. The whole plane is divided into stripes with a width of cos(Pi/12) ~= 0.9659. Every blue line and every red line contains exactly one grid point of its color in each stripe. The blue and red lines alternately intersect the horizontal centerline of a stripe. The distance between two intersections of the same color is d = sqrt(3)/(2*cos(Pi/12)). The bijection maps the section of a blue line in a stripe to the section of the unique red line, that intersects the centerline less than d/2 away. The grid points on these two line sections are the partners of the tile bijection.
While the method described only finds a minimum of the maximum distance of approximately 0.9659 by assigning the bijection partners using tiles, applying the Hopcroft-Karp algorithm to the bipartite graph corresponding to a sufficiently large section of the two infinite grids achieves significantly lower maximum distances. We conjecture that an upper bound for the maximum distance is sqrt(2)/2~=0.7071. See the corresponding link.
A method that reduces the maximal occurring bijection distance to its conjectured minimum, and only requires local rearrangements, as described for the square grids in A307731, is currently not known in the present case of the triangular grids.

			   n  A307014(n)        Bijection partner
   |  |  A307016(n)     in rotated grid
   |  |  |                          rotated by Pi/6
   |  |  |   x    y     i  j  a(n)   u      v   Distance([x,y],[u,v])
   0  0  0  0.0  0.0    0  0   0    0.0    0.0  0.0
   1  1  0  1.0  0.0    1  0   1    0.866  0.5  0.51764
   2  0  1  0.5  0.866  0  1   2    0.0    1.0  0.51764
   3 -1  1 -0.5  0.866 -1  1   3   -0.866  0.5  0.51764
   4 -1  0 -1.0  0.0   -1  0   4   -0.866 -0.5  0.51764
   5  0 -1 -0.5 -0.866  0 -1   5    0.0   -1.0  0.51764
   6  1 -1  0.5 -0.866  1 -1   6    0.866 -0.5  0.51764
   7  1  1  1.5  0.866  2 -1  12    1.732  0.0  0.89658
   8 -1  2  0.0  1.732  0  2  14    0.0    2.0  0.26795
   9 -2  1 -1.5  0.866 -2  2  15   -1.732  1.0  0.26795
  10 -1 -1 -1.5 -0.866 -2  1   9   -1.732  0.0  0.89658
  11  1 -2  0.0 -1.732  0 -2  17    0.0   -2.0  0.26795
  12  2 -1  1.5 -0.866  2 -2  18    1.732 -1.0  0.26795
  13  2  0  2.0  0.0    3 -2  29    2.598 -0.5  0.77955
  14  0  2  1.0  1.732  1  1   7    0.866  1.5  0.26795
  15 -2  2 -1.0  1.732 -1  2   8   -0.866  1.5  0.26795

Klaus Nagel, Stripes and tiles used for mapping.
Hugo Pfoertner, Numbering of points in red and blue grid.
Hugo Pfoertner, Notes on the minimum achievable bijection distance, Nov 2023.
Hugo Pfoertner, PARI program.
Wikipedia, Hopcroft-Karp algorithm

Cf. A307014, A307016, A307110, A307731, A367148.

\\ See linked file; function call to output data:
a367147(70)

A350090 a(n) is the number of indices i in the range 0 <= i <= n-1 such that A003215(n) - A003215(i) is an oblong number (A002378), where A003215 are the hex numbers.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 2, 3, 1, 1, 1, 3, 1, 1, 3, 3, 1, 3, 3, 3, 3, 5, 1, 1, 1, 5, 1, 1, 3, 1, 3, 1, 7, 1, 3, 3, 1, 1, 3, 7, 1, 1, 3, 3, 1, 3, 3, 1, 1, 3, 3, 1, 3, 7, 1, 3, 7, 1, 7, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 7, 5, 3, 3, 1, 5, 3, 3, 7, 3, 1, 1, 3, 3, 3, 7, 1, 3, 1, 3, 1

Offset: 0

Klaus Purath and Michel Marcus, Dec 14 2021

nonn

There are very few even terms in the data (3 up to 10000). They are obtained for indices coming from A001921. For odd terms see A350120.

a(n) = 1 for n in A111251.

			For n=5, the 5 numbers hex(5)-hex(i), for i=0 to 4, are (90, 84, 72, 54, 30) out of which 90, 72 and 30 are oblong, so a(5) = 3.

Michel Marcus, Table of n, a(n) for n = 0..10000

Cf. A002378, A003215.

Cf. also A001921, A111251, A350120.

obQ[n_] := IntegerQ @ Sqrt[4*n + 1]; hex[n_] := 3*n*(n + 1) + 1; a[n_] := Module[{h = hex[n]}, Count[Range[0, n - 1], ?(obQ[h - hex[#]] &)]]; Array[a, 100, 0] (* _Amiram Eldar, Dec 14 2021 *)

hex(n) = 3*n*(n+1)+1; \\ A003215
isob(n) = my(m=sqrtint(n)); m*(m+1)==n; \\ A002378
a(n) = my(h=hex(n)); sum(k=0, n-1, isob(h - hex(k)));

a(n) = numdiv(3*n*n + 3*n + 1) - 1; \\ Jinyuan Wang, Dec 19 2021

a(n) = A000005(A003215(n)) - 1. - Jinyuan Wang, Dec 19 2021

Edited by N. J. A. Sloane, Dec 25 2021

A296303 Number of minimal nonnegative nonzero solutions of the linear Diophantine equation x_1 + 2x_2 + ... + nx_n = y_1 + 2y_2 + ... + ny_n.

Original entry on oeis.org

1, 4, 13, 34, 99, 210, 559, 1164, 2531, 4940, 10735

Offset: 1

Klaus Pommerening, Dec 10 2017

Every linear Diophantine equation with arbitrary integer coefficients may be reduced to this one.
The minimal nonnegative nonzero solutions form a generating system of the semigroup of all nonnegative solutions.
The asymptotic behavior of a(n) is unknown, it is somewhere between a*exp(b*sqrt(n))/(sqrt(n)) and c*exp(d*n)/n with positive real numbers a,b,c,d.
A096337 contains the number of minimal nonnegative nonzero solutions of the linear congruence x_1 + 2 x_2 + ... + (n-1) x_{n-1} == 0 (mod n). There is an obvious relation with a(n) since every solution (x_1, ..., x_{n-1}) of the linear congruence yields a solution (x_1, ..., x_{n-1}; 0, 0, ..., 0, k) of the linear Diophantine equation.

			The 13 minimal solutions for n=3 are (x-coordinates followed by y-coordinates): (0,0,1;0,0,1), (0,0,1;1,1,0), (0,0,1;3,0,0), (0,0,2;0,3,0), (0,1,0;0,1,0), (0,1,0;2,0,0), (0,2,0;1,0,1), (0,3,0;0,0,2), (1,0,0;1,0,0), (1,0,1;0,2,0), (1,1,0;0,0,1), (2,0,0;0,1,0), (3,0,0;0,0,1).

M. Clausen, A. Fortenbacher, Efficient solution of linear Diophantine equations, J. Symbolic Comput. 8 (1989), 201-216.
D. V. Pasechnik, On computing Hilbert bases via the Elliott-MacMahon algorithm, Theor. Comp. Sc. 263 (2001), 37-46.
K. Pommerening, The indecomposable solutions of linear Diophantine equations

Cf. A096337, A026905, A002894.

Python
```
See Pommerening link.
```

Lower and upper bounds (proved) are a(n) >= 2*A026905(n) for n >= 3 and a(n) <= A002894(n-1).

A192993 Numbers that are in more than one way the concatenation of the decimal representation of two nonzero squares.

Original entry on oeis.org

164, 1441, 1625, 1961, 2564, 4841, 12116, 14449, 16400, 25625, 46241, 48464, 115625, 116641, 144100, 148841, 160025, 162500, 163844, 169169, 184964, 193636, 196100, 256400, 361225, 368649, 466564, 484100, 493025, 961009, 973441, 1166464

Offset: 1

Klaus Brockhaus and Zak Seidov, Jul 14 2011

Subsequence of A191933.
If k is a term, then k followed by two zeros is also a term.
None of the terms < 40000000 is in more than two ways the concatenation of the decimal representation of two nonzero squares.
A038670 is a subsequence. - Reinhard Zumkeller, Jul 15 2011

			2564 is the concatenation of 256 and 4 as well as of 25 and 64; 256, 4, 25, 64 are squares, so 2564 is a term.

Klaus Brockhaus, Table of n, a(n) for n = 1..100 (terms < 40000000)

Cf. A000290, A191933.

import Data.List (findIndices)
a192993 n = a192993_list !! (n-1)
a192993_list = findIndices (> 1) $ map a193095 [0..]
-- Reinhard Zumkeller, Jul 17 2011

SplitToSquares:=function(n); V:=[]; S:=Intseq(n); for j in [1..#S-1] do A:=[ S[k]: k in [1..j] ]; a:=Seqint(A); B:=[ S[k]: k in [j+1..#S] ]; b:=Seqint(B); if a gt 0 and A[#A] gt 0 and IsSquare(a) and IsSquare(b) then Append(~V, []); end if; end for; return V; end function; [ p: p in [1..1200000] | #P gt 1 where P is SplitToSquares(p) ]; /* to obtain the splittings replace " p: " with " : " */

f@n_ := DeleteDuplicates@
  Select[First@# & /@
    Select[Partition[
      Sort@(FromDigits@Flatten@IntegerDigits@# & /@
         Tuples[Range@Sqrt[10^(n - 1) - 1]^2, {2}]), 2, 1],
     Differences@# == {0} &], # <
10^n &]; f@7 (* Hans Rudolf Widmer, Jun 12 2023 *) (* Numbers with at most n digits that are in more than one way the concatenation of the decimal representation of two nonzero squares. *)

A191933 Numbers that are the concatenation of the decimal representation of two nonzero squares.

Original entry on oeis.org

11, 14, 19, 41, 44, 49, 91, 94, 99, 116, 125, 136, 149, 161, 164, 169, 181, 251, 254, 259, 361, 364, 369, 416, 425, 436, 449, 464, 481, 491, 494, 499, 641, 644, 649, 811, 814, 819, 916, 925, 936, 949, 964, 981, 1001, 1004, 1009, 1100, 1121, 1144, 1169, 1196

Offset: 1

Klaus Brockhaus, Jun 19 2011

Complement of A193096; A193095(a(n)) > 0; A038670, A039686, A167535, A192993, A193097 and A193144 are subsequences. [Reinhard Zumkeller, Jul 17 2011]

Klaus Brockhaus, Table of n, a(n) for n = 1..1000

Cf. A000290, A192993.

import Data.List (findIndices)
a191933 n = a191933_list !! (n-1)
a191933_list = findIndices (> 0) $ map a193095 [0..]
-- Reinhard Zumkeller, Jul 17 2011

CheckSplits:=function(n); v:=false; S:=Intseq(n); for j in [1..#S-1] do A:=[ S[k]: k in [1..j] ]; a:=Seqint(A); B:=[ S[k]: k in [j+1..#S] ]; b:=Seqint(B); if a gt 0 and A[#A] gt 0 and IsSquare(a) and IsSquare(b) then v:=true; end if; end for; return v; end function; [ p: p in [1..1200] | CheckSplits(p) ];

Take[Union[Flatten[Table[FromDigits[Flatten[{IntegerDigits[m^2], IntegerDigits[n^2]}]], {m, 20}, {n, 20}]]], 50] (* Alonso del Arte, Aug 11 2011 *)
squareQ[n_] := IntegerQ[Sqrt[n]]; okQ[n_] := MatchQ[IntegerDigits[n], {a__ /; squareQ[FromDigits[{a}]], b__ /; First[{b}] > 0 && squareQ[FromDigits[ {b}]]}]; Select[Range[2000], okQ] (* Jean-François Alcover, Dec 13 2016 *)

A192618 Prime powers p^k with even exponents k > 0 such that (1 + p^k)/2 is prime.

Original entry on oeis.org

9, 25, 81, 121, 361, 625, 841, 2401, 3481, 3721, 5041, 6241, 10201, 14641, 17161, 19321, 28561, 32761, 39601, 73441, 83521, 121801, 143641, 167281, 201601, 212521, 271441, 279841, 323761, 326041, 398161, 410881, 436921, 546121, 564001, 674041

Offset: 1

Klaus Brockhaus, Jul 05 2011

nonn

Subsequence of A056798.
From R. J. Mathar, Jul 11 2011: (Start)
For odd k we first have the case k=1, where (1+p)/2 is either classified as A005383 or A176897.
For odd k >= 3, (1+p^k)/2 is not prime. [Sketch of proof: for p=2 it is not integer. Otherwise for odd k, (1+p^k)/(1+p) = Sum_{j=0..k-1} (-p)^j, an integer, so 1+p^k is a multiple of 1+p. For odd p, (1+p^k)/2 is a multiple of (1+p)/2 and therefore composite.] (End)

Klaus Brockhaus, Table of n, a(n) for n = 1..1000

Cf. A056798.

e:=20; u:=1000; z:=Min(2^e, u^2); S:=[ q: p in PrimesUpTo(u), k in [2..e by 2] | q le z and IsEven(1+q) and IsPrime((1+q) div 2) where q is p^k ]; Sort(~S); S;

Select[Union[Flatten[Table[Prime[n]^k, {n, 142}, {k, 0, 32, 2}]]], PrimeQ[(# + 1)/2] &] (* Alonso del Arte, Jul 05 2011 *)

A191859 The primes created by concatenation of anti-divisors in A191647.

Original entry on oeis.org

2, 3, 23, 347, 349, 311, 391627, 3471331, 384067, 2310175897, 239111323273399, 23167, 3784097136227, 235983149249, 3428116271, 37111677121283, 23293, 3471949133311, 231314228398154359, 378112153101159371, 2379127163381

Offset: 1

Klaus Brockhaus, Jun 18 2011

a(n) is the concatenation of the anti-divisors of A191647(n).

			A191647(6) = 16, the anti-divisors of 16 are 3, 11. Hence a(6) = 311.
A191647(8) = 46, the anti-divisors of 46 are 3, 4, 7, 13, 31. Hence a(8) = 3471331.

Klaus Brockhaus, Table of n, a(n) for n = 1..1000

Cf. A191647, A066272, A130846, A120712.

Antidivisors:=func< n | [ d: d in [2..n-1] | n mod d ne 0 and ( (IsEven(d) and 2*n mod d eq 0) or (IsOdd(d) and ((2*n-1) mod d eq 0 or (2*n+1) mod d eq 0)) ) ] >; CAD:=function(n); A:=Antidivisors(n); S:=[]; for k in [1..#A] do S:= Intseq(A[k]) cat S; end for; p:=Seqint(S); return p; end function; A191859List:=func< m | [ p: n in [1..m] | IsPrime(p) where p is CAD(n) ] >; A191859List(600);

A190758 Primes p such that x^41 = 2 has a solution mod p, and p is congruent to 1 mod 41.

Original entry on oeis.org

17467, 18287, 31817, 42641, 116359, 139483, 163673, 172283, 176383, 181549, 190979, 225829, 226813, 231323, 259531, 288313, 299137, 307009, 352109, 404507, 421891, 445097, 464777, 484621, 528163, 592861, 604997, 609179, 611393, 629843

Offset: 1

Klaus Brockhaus, May 18 2011

nonn

Klaus Brockhaus, Table of n, a(n) for n = 1..5000

Cf. A049573, A059236, A142199.

forprime(p=2, 700000, if(trap(, 0, sqrtn(Mod(2, p), 41); 1), if(p%41==1, print1(p, ", "))));

A186041 Numbers of the form 3k + 2, 5k + 3, or 7*k + 4.

Original entry on oeis.org

2, 3, 4, 5, 8, 11, 13, 14, 17, 18, 20, 23, 25, 26, 28, 29, 32, 33, 35, 38, 39, 41, 43, 44, 46, 47, 48, 50, 53, 56, 58, 59, 60, 62, 63, 65, 67, 68, 71, 73, 74, 77, 78, 80, 81, 83, 86, 88, 89, 92, 93, 95, 98, 101, 102, 103, 104, 107, 108, 109, 110, 113, 116, 118, 119, 122

Offset: 1

Klaus Brockhaus, Feb 11 2011, Mar 09 2011

n is in the sequence iff n is in A016789 or in A016885 or in A017029.
First differences are periodic with period length 57. Least common multiple of 3, 5, 7 is 105; number of terms <= 105 is 57.
Sequence is not essentially the same as A053726: a(n) = A053726(n-3) for 3 < n < 33, a(34)=62, A053726(34-3)=61.
Sequence is not essentially the same as A104275: a(n) = A104275(n-2) for 3 < n < 33, a(34)=62, A104275(34-3)=61.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (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, 0, 0, 0, 0, 0, 1, -1).

Cf. A016789, A016885, A017029, A053726, A104275.

IsA186041:=func< n | exists{ k: k in [0..n div 3] | n in [3*k+2, 5*k+3, 7*k+4] } >; [ n: n in [1..200] | IsA186041(n) ];

Take[With[{no=50},Union[Join[3Range[0,no]+2,5Range[0,no]+3,7Range[0,no]+4]]],70]  (* Harvey P. Dale, Feb 16 2011 *)

a(n) = a(n-57) + 105.
a(n) = a(n-1) + a(n-57) - a(n-58).
G.f.: x*(x^57 + x^56 + x^55 + x^54 + 3*x^53 + 3*x^52 + 2*x^51 + x^50 + 3*x^49 + x^48 + 2*x^47 + 3*x^46 + 2*x^45 + x^44 + 2*x^43 + x^42 + 3*x^41 + x^40 + 2*x^39 + 3*x^38 + x^37 + 2*x^36 + 2*x^35 + x^34 + 2*x^33 + x^32 + x^31 + 2*x^30 + 3*x^29 + 3*x^28 + 2*x^27 + x^26 + x^25 + 2*x^24 + x^23 + 2*x^22 + 2*x^21 + x^20 + 3*x^19 + 2*x^18 + x^17 + 3*x^16 + x^15 + 2*x^14 + x^13 + 2*x^12 + 3*x^11 + 2*x^10 + x^9 + 3*x^8 + x^7 + 2*x^6 + 3*x^5 + 3*x^4 + x^3 + x^2 + x + 2) / ((x - 1)^2*(x^2 + x + 1)*(x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x^36 - x^35 + x^33 - x^32 + x^30 - x^29 + x^27 - x^26 + x^24 - x^23 + x^21 - x^20 + x^18 - x^16 + x^15 - x^13 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1)).

cp's OEIS Frontend

User: Klaus

A371969 Perimeters of triangles with integer sides, which can be decomposed into 3 triangles that have a common vertex strictly inside the surrounding triangle and also integer sides.

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A367147 Index of matching grid point in the bijection between two infinite triangular grids with one grid rotated by Pi/6 around the common point (0,0), using an enumeration of the grid points by A307014 and A307016.

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A350090 a(n) is the number of indices i in the range 0 <= i <= n-1 such that A003215(n) - A003215(i) is an oblong number (A002378), where A003215 are the hex numbers.

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A296303 Number of minimal nonnegative nonzero solutions of the linear Diophantine equation x_1 + 2*x_2 + ... + n*x_n = y_1 + 2*y_2 + ... + n*y_n.

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Formula

A192993 Numbers that are in more than one way the concatenation of the decimal representation of two nonzero squares.

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Haskell

Magma

Mathematica

A191933 Numbers that are the concatenation of the decimal representation of two nonzero squares.

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Haskell

Magma

Mathematica

A192618 Prime powers p^k with even exponents k > 0 such that (1 + p^k)/2 is prime.

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Magma

Mathematica

A296303 Number of minimal nonnegative nonzero solutions of the linear Diophantine equation x_1 + 2x_2 + ... + nx_n = y_1 + 2y_2 + ... + ny_n.

A186041 Numbers of the form 3k + 2, 5k + 3, or 7*k + 4.