A002144 -id:A002144 - cp's OEIS frontend

A030662 Number of combinations of n things from 1 to n at a time, with repeats allowed.

1, 5, 19, 69, 251, 923, 3431, 12869, 48619, 184755, 705431, 2704155, 10400599, 40116599, 155117519, 601080389, 2333606219, 9075135299, 35345263799, 137846528819, 538257874439, 2104098963719, 8233430727599, 32247603683099, 126410606437751, 495918532948103

Offset: 1

Donald Mintz (djmintz(AT)home.com)

Add terms of an increasingly bigger diamond-shaped part of Pascal's triangle:
.......................... 1
............ 1 .......... 1 1
.. 1 ...... 1 1 ........ 1 2 1
. 1 1 =5 . 1 2 1 =19 .. 1 3 3 1 =69
.. 2 ...... 3 3 ........ 4 6 4
............ 6 ......... 10 10
.......................... 20
-  Ralf Stephan, May 17 2004
The prime p divides a((p-1)/2) for p in A002144 (Pythagorean primes). - Alexander Adamchuk, Jul 04 2006
Also, number of square submatrices of a square matrix. - Jono Henshaw (jjono(AT)hotmail.com), Apr 22 2008
Partial sums of A051924. - J. M. Bergot, Jun 22 2013
Number of partitions with Ferrers diagrams that fit in an n X n box (excluding the empty partition of 0). - Michael Somos, Jun 02 2014
Also number of non-descending sequences with length and last number are less or equal to n, and also the number of integer partitions (of any positive integer) with length and largest part are less or equal to n. - Zlatko Damijanic, Dec 06 2024

			G.f. = x + 5*x^2 + 19*x^3 + 69*x^4 + 251*x^5 + 923*x^6 + 3431*x^7 + ...

T. D. Noe, Table of n, a(n) for n = 1..500
Narcisse G. Bell Bogmis, Guy R. Biyogmam, Hesam Safa, and Calvin Tcheka, Upper bounds on the dimension of the Schur Lie-multiplier of Lie-nilpotent Leibniz n-algebras, arXiv:2403.14884 [math.RA], 2024. See p. 7.
Joseph D. Horton and Andrew Kurn, Counting sequences with complete increasing subsequences, Congress Numerantium, 33 (1981), 75-80. MR 681905
Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Raimundas Vidunas, Counting derangements and Nash equilibria, Ann. Comb. 21, No. 1, 131-152 (2017).
Jianqiang Zhao, Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota-Baxter Algebras, arXiv preprint arXiv:1412.8044 [math.NT], 2014.

Cf. A000984, A001700, A002144, A051924, A091908, A144660, A322938.
Column k=2 of A047909.
Central column of triangle A014473.
Right-hand column 2 of triangle A102541.

[(n+1)*Catalan(n)-1: n in [1..40]]; // G. C. Greubel, Apr 07 2024

seq(sum((binomial(n,m))^2,m=1..n),n=1..23); # Zerinvary Lajos, Jun 19 2008
f:=n->add( add( binomial(i+j,i), i=0..n),j=0..n); [seq(f(n),n=0..12)]; # N. J. A. Sloane, Jan 31 2009

Table[Sum[Sum[(2n-i-j)!/(n-i)!/(n-j)!,{i,1,n}],{j,1,n}],{n,1,20}] (* Alexander Adamchuk, Jul 04 2006 *)
a[n_] := 2*(2*n-1)!/(n*(n-1)!^2)-1; Table[a[n], {n, 1, 26}] (* Jean-François Alcover, Oct 11 2012, from first formula *)

a(n)=binomial(2*n,n)-1 \\ Charles R Greathouse IV, Jun 26 2013

from math import comb
def a(n): return comb(2*n, n) - 1
print([a(n) for n in range(1, 27)]) # Michael S. Branicky, Jul 11 2023

def a(n) : return binomial(2*n,n) - 1
[a(n) for n in (1..26)] # Peter Luschny, Apr 21 2012

a(n) = A000984(n) - 1.
a(n) = 2*A001700(n-1) - 1.
a(n) = 2*(2*n-1)!/(n!*(n-1)!)-1.
a(n) = Sum_{k=1..n} binomial(n, k)^2. - Benoit Cloitre, Aug 20 2002
a(n) = Sum_{j=0..n} Sum_{i=j..n+j} binomial(i, j). - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jul 23 2003
a(n) = Sum_{i=0..n-1} Sum_{j=0..n-1} binomial(i+j, i). - N. J. A. Sloane, Jan 31 2009
Also for n>1: a(n)=(2*n)!/(n!)^2-1. - Hugo Pfoertner, Feb 10 2004
a(n) = Sum_{j=1..n} Sum_{i=1..n} (2n-i-j)!/((n-i)!*(n-j)!). - Alexander Adamchuk, Jul 04 2006
a(n) = A115112(n) + 1. - Jono Henshaw (jjono(AT)hotmail.com), Apr 22 2008
G.f.: Q(0)*(1-4*x)/x - 1/x/(1-x), where Q(k)= 1 + 4*(2*k+1)*x/( 1 - 1/(1 + 2*(k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 11 2013
D-finite with recurrence: n*a(n) +2*(-3*n+2)*a(n-1) +(9*n-14)*a(n-2) +2*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Jun 25 2013
0 = a(n)*(+16*a(n+1) - 70*a(n+2) + 68*a(n+3) - 14*a(n+4)) + a(n+1)*(-2*a(n+1) + 61*a(n+2) - 96*a(n+3) + 23*a(n+4)) + a(n+2)*(-6*a(n+2) + 31*a(n+3)  - 10*a(n+4)) + a(n+3)*(-2*a(n+3) + a(n+4)) for all n in Z. - Michael Somos, Jun 02 2014
From Ilya Gutkovskiy, Jan 25 2017: (Start)
O.g.f.: (1 - x - sqrt(1 - 4*x))/((1 - x)*sqrt(1 - 4*x)).
E.g.f.: exp(x)*(exp(x)*BesselI(0,2*x) - 1). (End)
a(n) = 3*n*Sum_{k=1..n} (-1)^(k+1)/(2*n+k)*binomial(2*n+k,n-k). - Vladimir Kruchinin, Jul 29 2025
a(n) = n * binomial(2*n, n) * Sum_{k = 1..n} 1/(k*binomial(n+k, k)). - Peter Bala, Aug 05 2025

A267099 Fully multiplicative involution swapping the positions of 4k+1 and 4k+3 primes: a(1) = 1; a(prime(k)) = A267101(k), a(xy) = a(x)a(y) for x, y > 1.

Original entry on oeis.org

1, 2, 5, 4, 3, 10, 13, 8, 25, 6, 17, 20, 7, 26, 15, 16, 11, 50, 29, 12, 65, 34, 37, 40, 9, 14, 125, 52, 19, 30, 41, 32, 85, 22, 39, 100, 23, 58, 35, 24, 31, 130, 53, 68, 75, 74, 61, 80, 169, 18, 55, 28, 43, 250, 51, 104, 145, 38, 73, 60, 47, 82, 325, 64, 21, 170, 89, 44, 185, 78, 97, 200, 59, 46, 45, 116, 221, 70, 101

Offset: 1

Antti Karttunen, Feb 01 2016

Lexicographically earliest self-inverse permutation of natural numbers where each prime of the form 4k+1 is replaced by a prime of the form 4k+3 and vice versa, with the composite numbers determined by multiplicativity.
Fully multiplicative with a(p_n) = p_{A267100(n)} = A267101(n).
Maps each term of A004613 to some term of A004614, each (nonzero) term of A001481 to some term of A268377 and each term of A004431 to some term of A268378 and vice versa.
Sequences A072202 and A078613 are closed with respect to this permutation.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A000040, A000720, A010051, A020639, A032742, A267100, A267101, A354102 (Möbius transform), A354103 (inverse Möbius transform), A354192 (fixed points).
Cf. A002144, A002145, A005094, A065338, A072202, A078613, A079635.
Cf. A001481, A268377, A004431, A268378, A004613, A004614.
Cf. also A108548.

up_to = 2^16;
A267097list(up_to) = { my(v=vector(up_to),i=0,c=0); forprime(p=2,prime(up_to), if(1==(p%4), c++); i++; v[i] = c); (v); };
v267097 = A267097list(up_to);
A267097(n) = v267097[n];
A267098(n) = ((n-1)-A267097(n));
list_primes_of_the_form(up_to,m,k) = { my(v=vector(up_to),i=0); forprime(p=2,, if(k==(p%m), i++; v[i] = p; if(i==up_to,return(v)))); };
v002144 = list_primes_of_the_form(2*up_to,4,1);
A002144(n) = v002144[n];
v002145 = list_primes_of_the_form(2*up_to,4,3);
A002145(n) = v002145[n];
A267101(n) = if(1==n,2,if(1==(prime(n)%4),A002145(A267097(n)),A002144(A267098(n))));
A267099(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A267101(primepi(f[k,1]))); factorback(f); }; \\ Antti Karttunen, May 18 2022
(Scheme, with memoization-macro definec)
(definec (A267099 n) (cond ((<= n 1) n) ((= 1 (A010051 n)) (A267101 (A000720 n))) (else (* (A267099 (A020639 n)) (A267099 (A032742 n))))))

a(1) = 1; after which, if n is k-th prime [= A000040(k)], then a(n) = A267101(k), otherwise a(A020639(n)) * a(A032742(n)).
Other identities. For all n >= 1:
a(A000040(n)) = A267101(n).
a(2*n) = 2*a(n).
a(3*n) = 5*a(n).
a(5*n) = 3*a(n).
a(7*n) = 13*a(n).
a(11*n) = 17*a(n).
etc. See examples in A267101.
A000035(n) = A000035(a(n)). [Preserves the parity of n.]
A005094(a(n)) = -A005094(n).
A079635(a(n)) = -A079635(n).

Verbal description prefixed to the name by Antti Karttunen, May 19 2022

A084647 Hypotenuses for which there exist exactly 3 distinct integer triangles.

Original entry on oeis.org

125, 250, 375, 500, 750, 875, 1000, 1125, 1375, 1500, 1750, 2000, 2197, 2250, 2375, 2625, 2750, 2875, 3000, 3375, 3500, 3875, 4000, 4125, 4394, 4500, 4750, 4913, 5250, 5375, 5500, 5750, 5875, 6000, 6125, 6591, 6750, 7000, 7125, 7375, 7750

Offset: 1

Eric W. Weisstein, Jun 01 2003

nonn

Numbers whose square is decomposable in 3 different ways into the sum of two nonzero squares: these are those with exactly one prime divisor of the form 4k+1 with multiplicity three. - Jean-Christophe Hervé, Nov 11 2013

			a(1) = 125 = 5^3, and 125^2 = 100^2 + 75^2 = 117^2 + 44^2 = 120^2 + 35^2. - _Jean-Christophe Hervé_, Nov 11 2013

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1140 terms from Jean-Christophe Hervé)
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084646 (2), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

Clear[lst,f,n,i,k] f[n_]:=Module[{i=0,k=0},Do[If[Sqrt[n^2-i^2]==IntegerPart[Sqrt[n^2-i^2]],k++ ],{i,n-1,1,-1}]; k/2]; lst={}; Do[If[f[n]==3,AppendTo[lst,n]],{n,4*5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)

Terms are obtained by the products A004144(k)*A002144(p)^3 for k, p > 0, ordered by increasing values. - Jean-Christophe Hervé, Nov 12 2013

A084648 Hypotenuses for which there exist exactly 4 distinct integer triangles.

Original entry on oeis.org

65, 85, 130, 145, 170, 185, 195, 205, 221, 255, 260, 265, 290, 305, 340, 365, 370, 377, 390, 410, 435, 442, 445, 455, 481, 485, 493, 505, 510, 520, 530, 533, 545, 555, 565, 580, 585, 595, 610, 615, 625, 629, 663, 680, 685, 689, 697, 715, 730, 740, 745

Offset: 1

Eric W. Weisstein, Jun 01 2003

nonn

Numbers whose square is decomposable in 4 different ways into the sum of two nonzero squares: these are those with exactly 2 distinct prime divisors of the form 4k+1, each with multiplicity one, or with only one prime divisor of this form with multiplicity 4. - Jean-Christophe Hervé, Nov 11 2013

If m is a term, then 2*m and p*m are terms where p is any prime of the form 4k+3. - Ray Chandler, Dec 30 2019

			a(1) = 65 = 5*13, and 65^2 = 52^2 + 39^2 = 56^2 + 33^2 = 60^2 + 25^2 = 63^2 + 16^2. - _Jean-Christophe Hervé_, Nov 11 2013

Ray Chandler, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

Clear[lst,f,n,i,k] f[n_]:=Module[{i=0,k=0},Do[If[Sqrt[n^2-i^2]==IntegerPart[Sqrt[n^2-i^2]],k++ ],{i,n-1,1,-1}]; k/2]; lst={}; Do[If[f[n]==4,AppendTo[lst,n]],{n,6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)

A084649 Hypotenuses for which there exist exactly 5 distinct Pythagorean triangles.

Original entry on oeis.org

3125, 6250, 9375, 12500, 18750, 21875, 25000, 28125, 34375, 37500, 43750, 50000, 56250, 59375, 65625, 68750, 71875, 75000, 84375, 87500, 96875, 100000, 103125, 112500, 118750, 131250, 134375, 137500, 143750, 146875, 150000, 153125

Offset: 1

Eric W. Weisstein, Jun 01 2003

nonn

Numbers whose square is decomposable in 5 different ways into the sum of two nonzero squares: these are those with exactly one prime divisor of the form 4k+1 with multiplicity 5. - Jean-Christophe Hervé, Nov 12 2013

			a(1) = 5^5, a(5) = 6*5^5, a(65) = 13^5. - _Jean-Christophe Hervé_, Nov 12 2013

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1019 terms from Jean-Christophe Hervé)
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

Clear[lst,f,n,i,k] f[n_]:=Module[{i=0,k=0},Do[If[Sqrt[n^2-i^2]==IntegerPart[Sqrt[n^2-i^2]],k++ ],{i,n-1,1,-1}]; k/2]; lst={}; Do[If[f[n]==5,AppendTo[lst,n]],{n,3*6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)

Terms are obtained by the products A004144(k)*A002144(p)^5 for k, p > 0 ordered by increasing values. - Jean-Christophe Hervé, Nov 12 2013

A084646 Hypotenuses for which there exist exactly 2 distinct integer triangles.

Original entry on oeis.org

25, 50, 75, 100, 150, 169, 175, 200, 225, 275, 289, 300, 338, 350, 400, 450, 475, 507, 525, 550, 575, 578, 600, 675, 676, 700, 775, 800, 825, 841, 867, 900, 950, 1014, 1050, 1075, 1100, 1150, 1156, 1175, 1183, 1200, 1225, 1350, 1352, 1369, 1400

Offset: 1

Eric W. Weisstein, Jun 01 2003

nonn

Numbers whose square is decomposable in 2 different ways into the sum of two nonzero squares: these are those with exactly one prime divisor of the form 4k+1 with multiplicity two. - Jean-Christophe Hervé, Nov 11 2013

Ray Chandler, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084645 (1), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

Clear[lst,f,n,i,k] f[n_]:=Module[{i=0,k=0},Do[If[Sqrt[n^2-i^2]==IntegerPart[Sqrt[n^2-i^2]],k++ ],{i,n-1,1,-1}]; k/2]; lst={}; Do[If[f[n]==2,AppendTo[lst,n]],{n,4*5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)

Terms are obtained by the products A004144(k)*A002144(p)^2 for k, p > 0, ordered by increasing values. - Jean-Christophe Hervé, Nov 12 2013

A046080(a(n)) = 2, A046109(a(n)) = 20. - Jean-Christophe Hervé, Dec 01 2013

A175647 Decimal expansion of the Product_{primes p == 1 (mod 4)} 1/(1-1/p^2).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 01 2010

The Euler product of the Riemann zeta function at 2 restricted to primes in A002144, which is the inverse of the infinite product (1-1/5^2)*(1-1/13^2)*(1-1/17^2)*(1-1/29^2)*...

There is a complementary Product_{primes p == 3 (mod 4)} 1/(1-1/p^2) = 1.16807558541051428866969673706404040136467... such that (this constant here)*1.16807.../(1-1/2^2) = zeta(2) = A013661.

			1.0561821217268161417379307653162198905...

S. Ettahri, O. Ramare, L. Surel, Fast multi-precision computation of some Euler products, arxiv:1908.06808 (2019), Section 9.
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Cf. A004613, A004614, A013661, A175646.

digits = 105;
LandauRamanujanK = 1/Sqrt[2]*NProduct[((1 - 2^(-2^n))*Zeta[2^n]/  DirichletBeta[2^n])^(1/2^(n+1)), {n, 1, 24}, WorkingPrecision -> digits+5];
RealDigits[1/(4*LandauRamanujanK/Pi)^2, 10, digits][[1]] (* Jean-François Alcover, Jan 12 2021 *)

Equals 1/A088539. - Vaclav Kotesovec, May 05 2020
From Amiram Eldar, Sep 27 2020: (Start)
Equals Sum_{k>=1} 1/A004613(k)^2.
The complementary product equals Sum_{k>=1} 1/A004614(k)^2. (End)

More digits from Vaclav Kotesovec, Jun 27 2020

A002331 Values of x in the solution to p = x^2 + y^2, x <= y, with prime p = A002313(n).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 4, 2, 5, 3, 5, 4, 1, 3, 7, 4, 7, 6, 2, 9, 7, 1, 2, 8, 4, 1, 10, 9, 5, 2, 12, 11, 9, 5, 8, 7, 10, 6, 1, 3, 14, 12, 7, 4, 10, 5, 11, 10, 14, 13, 1, 8, 5, 17, 16, 4, 13, 6, 12, 1, 5, 15, 2, 9, 19, 12, 17, 11, 5, 14, 10, 18, 4, 6, 16, 20, 19, 10, 13, 4, 6, 15, 22, 11, 3, 5

Offset: 1

N. J. A. Sloane

nonn

			The following table shows the relationship between several closely related sequences:
Here p = A002144 = primes == 1 (mod 4), p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................

A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
A. T. Benjamin and D. Zeilberger, Pythagorean primes and palindromic continued fractionsINTEGERS 5(1) (2005) #A30
John Brillhart, Note on representing a prime as a sum of two squares, Math. Comp. 26 (1972), pp. 1011-1013.
A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904. [Annotated scans of selected pages]
K. Matthews, Serret's algorithm Server.
J. Todd, A problem on arc tangent relations, Amer. Math. Monthly, 56 (1949), 517-528.
Eric Weisstein's World of Mathematics, Fermat's 4n Plus 1 Theorem.

Cf. A002330, A002313, A002144, A027862 (locates y=x+1).

See A002330 for Maple program.
# alternative
A002331 := proc(n)
    A363051(A002313(n)) ;
end proc:
seq(A002331(n),n=1..100) ; # R. J. Mathar, Feb 01 2024

pmax = 1000; x[p_] := Module[{x, y}, x /. ToRules[Reduce[0 <= x <= y && x^2 + y^2 == p, {x, y}, Integers]]]; For[n=1; p=2, pJean-François Alcover, Feb 26 2016 *)

f(p)=my(s=lift(sqrt(Mod(-1,p))),x=p,t);if(s>p/2,s=p-s); while(s^2>p,t=s;s=x%s;x=t);s
forprime(p=2,1e3,if(p%4-3,print1(sqrtint(p-f(p)^2)", ")))
\\ Charles R Greathouse IV, Apr 24 2012

do(p)=qfbsolve(Qfb(1,0,1),p)[2]
forprime(p=2,1e3,if(p%4-3,print1(do(p)", "))) \\ Charles R Greathouse IV, Sep 26 2013

a(n) = A096029(n) - A096030(n) for n > 1. - Lekraj Beedassy, Jul 16 2004
a(n+1) = Min(A002972(n), 2*A002973(n)). - Reinhard Zumkeller, Feb 16 2010
a(n) = A363051(A002313(n)). - R. J. Mathar, Jan 31 2024

A048861 a(n) = n^n - 1.

Original entry on oeis.org

0, 3, 26, 255, 3124, 46655, 823542, 16777215, 387420488, 9999999999, 285311670610, 8916100448255, 302875106592252, 11112006825558015, 437893890380859374, 18446744073709551615, 827240261886336764176, 39346408075296537575423, 1978419655660313589123978

Offset: 1

Charles T. Le (charlestle(AT)yahoo.com)

From Alexander Adamchuk, Jan 22 2007: (Start)
a(n) is divisible by (n-1).
Corresponding quotients are a(n)/(n-1) = {1,3,13,85,781,9331, ...} = A023037(n).
p divides a(p-1) for prime p.
p divides a((p-1)/2) for prime p = {3,11,17,19,41,43,59,67,73,83,89,97,...} = A033200 Primes congruent to {1, 3} mod 8; or, odd primes of form x^2+2*y^2.
p divides a((p-1)/3) for prime p = {61,67,73,103,151,193,271,307,367,...} = A014753 3 and -3 are both cubes (one implies other) mod these primes p=1 mod 6.
p divides a((p-1)/4) for prime p = {5,13,17,29,37,41,53,61,73,...} = A002144 Pythagorean primes: primes of form 4n+1.
p divides a((p-1)/5) for prime p = {31,191,251,271,601,641,761,1091,...}.
p divides a((p-1)/6) for prime p = {7,241,313,337,409,439,607,631,727,751,919,937,...}. (End)
For n > 1, a(n) is largest number that can be represented using n digits in the base-n number system. - Chinmaya Dash, Mar 31 2022

			For n=3, a(n) = 3^3 - 1 = 27 - 1 = 26. - _Michael B. Porter_, Nov 12 2017

M. Le, Primes in the sequences n^n+1 and n^n-1, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, 156-157.

G. C. Greubel, Table of n, a(n) for n = 1..385
F. Smarandache, Only Problems, Not Solutions!

Cf. A000312, A048860, A023037, A033200, A014753, A002144.

[ n^n-1: n in [1..25]]; // Vincenzo Librandi, Dec 29 2010

Table[n^n - 1, {n, 1, 50}] (* G. C. Greubel, Nov 10 2017 *)

a(n)=n^n-1 \\ Charles R Greathouse IV, Feb 24 2012

E.g.f.: 1/(1+LambertW(-x)) - exp(x). - Vaclav Kotesovec, Dec 20 2014

Extended (and corrected) by Patrick De Geest, Jul 15 1999

A089120 Smallest prime factor of n^2 + 1.

Original entry on oeis.org

2, 5, 2, 17, 2, 37, 2, 5, 2, 101, 2, 5, 2, 197, 2, 257, 2, 5, 2, 401, 2, 5, 2, 577, 2, 677, 2, 5, 2, 17, 2, 5, 2, 13, 2, 1297, 2, 5, 2, 1601, 2, 5, 2, 13, 2, 29, 2, 5, 2, 41, 2, 5, 2, 2917, 2, 3137, 2, 5, 2, 13, 2, 5, 2, 17, 2, 4357, 2, 5, 2, 13, 2, 5, 2, 5477, 2, 53, 2, 5, 2, 37, 2, 5, 2

Offset: 1

Cino Hilliard, Dec 05 2003

This includes A002496, primes that are of the form n^2+1.

Note that a(n) is the smallest prime p such that n^(p+1) == -1 (mod p). - Thomas Ordowski, Nov 08 2019

H. Rademacher, Lectures on Elementary Number Theory, pp. 33-38.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A002496, A014442, A085722, A144255, A193432.

[Min(PrimeDivisors(n^2+1)):n in [1..100]]; // Marius A. Burtea, Nov 13 2019

Array[FactorInteger[#^2 + 1][[1, 1]] &, {83}] (* Michael De Vlieger, Sep 08 2015 *)

smallasqp1(m) = { for(a=1,m, y=a^2 + 1; f = factor(y); v = component(f,1); v1 = v[length(v)]; print1(v[1]",") ) }

A089120(n)=factor(n^2+1)[1,1]  \\ M. F. Hasler, Mar 11 2012

a(2k+1)=2; a(10k +/- 2)=5, else a(26k +/- 8)=13, else a(34k +/- 4)=17, else a(58k +/- 12)=29, else a(74k +/- 6)=37,... - M. F. Hasler, Mar 11 2012

A089120(n) = 2 if n is odd, else A089120(n) = min { A002144(k) | n = +/- A209874(k) (mod 2*A002144(k)) }.

cp's OEIS Frontend

A030662 Number of combinations of n things from 1 to n at a time, with repeats allowed.

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A267099 Fully multiplicative involution swapping the positions of 4k+1 and 4k+3 primes: a(1) = 1; a(prime(k)) = A267101(k), a(x*y) = a(x)*a(y) for x, y > 1.

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A084647 Hypotenuses for which there exist exactly 3 distinct integer triangles.

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A084648 Hypotenuses for which there exist exactly 4 distinct integer triangles.

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A084649 Hypotenuses for which there exist exactly 5 distinct Pythagorean triangles.

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A084646 Hypotenuses for which there exist exactly 2 distinct integer triangles.

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A175647 Decimal expansion of the Product_{primes p == 1 (mod 4)} 1/(1-1/p^2).

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A267099 Fully multiplicative involution swapping the positions of 4k+1 and 4k+3 primes: a(1) = 1; a(prime(k)) = A267101(k), a(xy) = a(x)a(y) for x, y > 1.