A067756 -id:A067756 - cp's OEIS frontend

A048161 Primes p such that q = (p^2 + 1)/2 is also a prime.

3, 5, 11, 19, 29, 59, 61, 71, 79, 101, 131, 139, 181, 199, 271, 349, 379, 409, 449, 461, 521, 569, 571, 631, 641, 661, 739, 751, 821, 881, 929, 991, 1031, 1039, 1051, 1069, 1091, 1129, 1151, 1171, 1181, 1361, 1439, 1459, 1489, 1499, 1531, 1709, 1741, 1811, 1831, 1901

Offset: 1

Harvey Dubner (harvey(AT)dubner.com)

Primes which are a leg of an integral right triangle whose hypotenuse is also prime.
It is conjectured that there are an infinite number of such triangles.
The Pythagorean triple {p, (p^2 - 1)/2, (p^2 + 1)/2} corresponds to {a(n), A067755(n), A067756(n)}. - Lekraj Beedassy, Oct 27 2003
There is no Pythagorean triangle all of whose sides are prime numbers. Still there are Pythagorean triangles of which the hypotenuse and one side are prime numbers, for example, the triangles (3,4,5), (11,60,61), (19,180,181), (61,1860,1861), (71,2520,2521), (79,3120,3121). [Sierpiński]
We can always write p=(Y+1)^2-Y^2, with Y=(p-1)/2, therefore q=(Y+1)^2+Y^2. - Vincenzo Librandi, Nov 19 2010
p^2 and p^2+1 are semiprimes; p^2 are squares in A070552 Numbers n such that n and n+1 are products of two primes. - Zak Seidov, Mar 21 2011

			For p=11, (p^2+1)/2=61; p=61, (p^2+1)/2=1861.
For p(1)=3, the right triangle 3, 4, 5 is the smallest where 5=(3*3+1)/2.
For p(10)=101, the right triangle is 101, 5100, 5101 where 5101=(101*101+1)/2.

Wacław Sierpiński, Pythagorean triangles, Dover Publications, Inc., Mineola, NY, 2003, p. 6. MR2002669

T. D. Noe, Table of n, a(n) for n = 1..10000
H. Dubner and T. Forbes, Prime Pythagorean triangles, Journal of Integer Sequences, Vol. 4(2001), #01.2.3.

Cf. A067755, A067756. Complement in primes of A094516.
Cf. A017281, A154428, A010051, A065091, A005383.
Cf. A048270, A048295, A308635, A308636. Primes contained in A002731.

a048161 n = a048161_list !! (n-1)
a048161_list = [p | p <- a065091_list, a010051 ((p^2 + 1) `div` 2) == 1]
-- Reinhard Zumkeller, Aug 26 2012

[p: p in PrimesInInterval(3, 2000) | IsPrime((p^2+1) div 2)]; // Vincenzo Librandi, Dec 31 2013

a := proc (n) if isprime(n) = true and type((1/2)*n^2+1/2, integer) = true and isprime((1/2)*n^2+1/2) = true then n else end if end proc: seq(a(n), n = 1 .. 2000) # Emeric Deutsch, Jan 18 2009

Select[Prime[Range[200]], PrimeQ[(#^2 + 1)/2] &] (* Stefan Steinerberger, Apr 07 2006 *)
a[ n_] := Module[{p}, If[ n < 1, 0, p = a[n - 1]; While[ (p = NextPrime[p]) > 0, If[ PrimeQ[(p*p + 1)/2], Break[]]]; p]]; (* Michael Somos, Nov 24 2018 *)

{a(n) = my(p); if( n<1, 0, p = a(n-1) + (n==1); while(p = nextprime(p+2), if( isprime((p*p+1)/2), break)); p)}; /* Michael Somos, Mar 03 2004 */

from sympy import isprime, nextprime; p = 2
while p < 1901: p = nextprime(p); print(p, end = ', ') if isprime((p*p+1)//2) else None # Ya-Ping Lu, Apr 24 2025

A000035(a(n))*A010051(a(n))*A010051((a(n)^2+1)/2) = 1. - Reinhard Zumkeller, Aug 26 2012

More terms from David W. Wilson

A067755 Even legs of Pythagorean triangles whose other leg and hypotenuse are both prime.

Original entry on oeis.org

4, 12, 60, 180, 420, 1740, 1860, 2520, 3120, 5100, 8580, 9660, 16380, 19800, 36720, 60900, 71820, 83640, 100800, 106260, 135720, 161880, 163020, 199080, 205440, 218460, 273060, 282000, 337020, 388080, 431520, 491040, 531480, 539760, 552300

Offset: 1

Henry Bottomley, Jan 31 2002

nonn

Apart from the first two terms, every term is divisible by 60 and is of the form 450*k^2 +/- 30*k or 450*k^2 +/- 330*k + 60 for some k.

In such a triangle, this even leg is always the longer leg, and the hypotenuse = a(n) + 1. The Pythagorean triples are (A048161(n), a(n), A067756(n)), so, for a(2) = 12, the corresponding Pythagorean triple is (5, 12, 13). - Bernard Schott, Apr 12 2023

			4 is a term: in the right triangle (3, 4, 5), 3 and 5 are prime.
5100 is a term: in the right triangle (101, 5100, 5101), 101 and 5101 are prime.

Ray Chandler, Table of n, a(n) for n = 1..10000
H. Dubner and T. Forbes, Prime Pythagorean triangles, J. Integer Seqs., Vol. 4 (2001), #01.2.3.

Cf. A048161, A067756. Contains every value of A051858.

lst={}; Do[q=(Prime[n]^2+1)/2; If[PrimeQ[q], AppendTo[lst, (Prime[n]^2-1)/2]], {n, 200}]; lst (* Frank M Jackson, Nov 02 2013 *)

a(n) = (A048161(n)^2 - 1)/2 = A067756(n) - 1.

A068501 Values m such that the consecutive pair parameters(m,m+1) generate Pythagorean triples whose odd terms are both prime.

Original entry on oeis.org

1, 2, 5, 9, 14, 29, 30, 35, 39, 50, 65, 69, 90, 99, 135, 174, 189, 204, 224, 230, 260, 284, 285, 315, 320, 330, 369, 375, 410, 440, 464, 495, 515, 519, 525, 534, 545, 564, 575, 585, 590, 680, 719, 729, 744, 749, 765, 854, 870, 905, 915, 950, 974, 1080, 1119

Offset: 1

Lekraj Beedassy, Mar 25 2002

nonn

Setting u=m; v=m+1, triples (a,b,c) with a=u+v, b=2*u*v, c = u^2+v^2 = (a^2+1)/2 correspond to (A048161, A067755, A067756), a and c being both prime.

Zak Seidov, Table of n, a(n) for n = 1..10000
Robert Simms, Deriving Pythagorean Triples (web archive)

Cf. A051892.

lst={};Do[If[PrimeQ[(n+1)^2-n^2]&&PrimeQ[(n+1)^2+n^2],AppendTo[lst,n]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 01 2010 *)
Reap[Do[a=Prime[k];If[PrimeQ[(a^2+1)/2],Sow[(a-1)/2]],{k,2,10^5}]][[2,1]](* Zak Seidov, Apr 16 2011 *)

More terms from Larry Reeves (larryr(AT)acm.org), Jun 19 2002

A051859 Values of C (the hypotenuse) of a Pythagorean triangle with A (the short leg) and C both prime and part of a twin prime.

Original entry on oeis.org

5, 13, 61, 181, 421, 5101, 60901, 135721, 161881, 163021, 218461, 595141, 1108561, 2574181, 2740141, 3248701, 3535141, 3723721, 3729181, 8197201, 13933921, 20218441, 23605321, 28569241, 33874681, 47248921, 68667481, 69372421

Offset: 1

Stuart M. Ellerstein (ellerstein(AT)aol.com), Dec 14 1999

nonn

All terms of the sequence must be part of a Pythagorean triple of the form (2u-1), 2u*(u-1), (2u^2 - 2u + 1). - Joshua Zucker, May 12 2006

Harvey P. Dale, Table of n, a(n) for n = 1..1000

See A051642 for the A's and A051858 for the B's.

Subset of A067756.

tppQ[{a_,b_,c_}]:=AllTrue[{a,c},PrimeQ]&&AnyTrue[a+{2,-2},PrimeQ] && AnyTrue[ c+{2,-2},PrimeQ]; Select[Table[{2n-1,2n(n-1),2n^2-2n+1},{n,2,10000}],tppQ][[All,3]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 27 2021 *)

A051858 + 1, or 2*A051891^2 - 2*A051891 + 1, or 2*A051892^2 + 2*A051892 + 1. - Joshua Zucker, May 12 2006

More terms from Joshua Zucker, May 12 2006

A292989 Triangular numbers having exactly 6 divisors.

Original entry on oeis.org

28, 45, 153, 171, 325, 4753, 7381, 29161, 56953, 65341, 166753, 354061, 5649841, 6060421, 6835753, 6924781, 12708361, 19478161, 24231241, 52035301, 56791153, 147258541, 186660181, 282304441, 326081953, 520273153, 536657941, 704531953, 784139401, 1215121753

Offset: 1

Jon E. Schoenfield, Dec 08 2017

nonn

Intersection of A000217 (triangular numbers) and A030515 (numbers with exactly 6 divisors).
This sequence is also the union of
(1) numbers of the form p*(2p-1) where p is prime and 2p-1 is the square of a prime (this sequence begins 45, 325, 7381, 65341, 354061, ...),
(2) numbers of the form p^2*(2p^2 - 1) where both p and 2p^2 - 1 are prime (this sequence begins 28, 153, 4753, 29161, ...), and
(3) numbers of the form p^2*(2p^2 + 1) where both p and 2p^2 + 1 are prime (the only such number is 171).

			28 = 2^2 * 7, so it has 6 divisors: {1, 2, 4, 7, 14, 28};
45 = 3^2 * 5, so it has 6 divisors: {1, 3, 5, 9, 15, 45};
171 = 3^2 * 19, so it has 6 divisors: {1, 3, 9, 19, 57, 171}.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Cf. A000217 (triangular numbers), A030515 (numbers with exactly 6 divisors).
Cf. A067756 (primes p such that 2p-1 is the square of a prime), A106483 (primes p such that 2p^2 - 1 is prime).
Cf. A263951.

Select[Array[PolygonalNumber, 10^5], DivisorSigma[0, #] == 6 &] (* Michael De Vlieger, Dec 09 2017 *)

A286257 Compound filter: a(n) = P(A046523(n), A046523(2n-1)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 5, 14, 12, 27, 5, 86, 14, 27, 23, 90, 12, 84, 27, 152, 23, 148, 5, 148, 27, 27, 80, 324, 25, 61, 44, 148, 23, 495, 5, 935, 61, 27, 61, 702, 5, 142, 61, 324, 138, 495, 23, 148, 90, 61, 23, 1426, 14, 265, 27, 90, 467, 324, 27, 430, 27, 61, 80, 2140, 12, 61, 183, 2144, 61, 495, 23, 607, 27, 495, 23, 2998, 23, 142, 90, 90, 142, 625, 5, 1426, 226, 27, 467

Offset: 1

Antti Karttunen, May 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000027, A046523, A278223, A286255, A286256, A286258.

Cf. A005382 (gives the positions of 5's), A067756 (of 12's), A234098 (of 23's).

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A286257(n) = (1/2)*(2 + ((A046523(n)+A046523((2*n)-1))^2) - A046523(n) - 3*A046523((2*n)-1));
for(n=1, 10000, write("b286257.txt", n, " ", A286257(n)));

from sympy import factorint
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a(n): return T(a046523(n), a046523(2*n - 1)) # Indranil Ghosh, May 07 2017

(define (A286257 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A046523 (+ -1 n n))) 2) (- (A046523 n)) (- (* 3 (A046523 (+ -1 n n)))) 2)))

a(n) = (1/2)*(2 + ((A046523(n)+A046523((2*n)-1))^2) - A046523(n) - 3*A046523((2*n)-1)).

a(n) = (1/2)*(2 + ((A046523(n)+A278223(n))^2) - A046523(n) - 3*A278223(n)).

A068485 One-sixtieth of the even leg of Pythagorean triangles whose other sides are both primes (other than 3, 5 or 13).

Original entry on oeis.org

1, 3, 7, 29, 31, 42, 52, 85, 143, 161, 273, 330, 612, 1015, 1197, 1394, 1680, 1771, 2262, 2698, 2717, 3318, 3424, 3641, 4551, 4700, 5617, 6468, 7192, 8184, 8858, 8996, 9205, 9523, 9919, 10622, 11040, 11427, 11623, 15436, 17256, 17739, 18476, 18725, 19533

Offset: 1

Lekraj Beedassy, Mar 11 2002

nonn

The (primitive) Pythagorean triple is {A048161(n), A067755(n), A067756(n)}.

Cf. A048161, A067755, A067756, A263951.

a068485[n_] := (Select[Map[Prime[#]^2&, Range[4, n]], PrimeQ[(#+1)/2]&]-1)/120
a068485[250] (* data - Hartmut F. W. Hoft, Aug 06 2020 *)

From Hartmut F. W. Hoft, Aug 06 2020: (Start)
a(n) = A067755(n+2)/60, n>=1.
a(n) = (A263951(n+2) - 1)/120, n>=1. (End)

More terms from Sascha Kurz, Mar 26 2002

a(34)-a(45) from Ray Chandler, Apr 12 2010

A382872 For n >= 1, a(n) is the number of divisors of the Pillai's arithmetical function: Sum_{k=1..n} gcd(k, n) (A018804).

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 2, 6, 4, 4, 4, 8, 3, 4, 6, 10, 4, 6, 2, 12, 4, 6, 6, 9, 4, 6, 5, 8, 4, 8, 2, 10, 8, 6, 6, 16, 2, 4, 4, 18, 5, 8, 4, 16, 8, 8, 4, 20, 4, 8, 8, 12, 8, 6, 8, 12, 4, 6, 6, 24, 3, 4, 8, 9, 9, 12, 4, 16, 9, 8, 4, 24, 4, 4, 6, 8, 8, 8, 2, 20

Offset: 1

Ctibor O. Zizka, Apr 07 2025

nonn

a(n) is from A005408 for n from {1, 5, 13, 24, 27, 41, 61, 64, 65, 69, 99, 113, ...}.

a(n) is from A065091 for n from {5, 13, 27, 41, 61, 135, 181, 205, 313, 421, ...}.

			For n = 5, a(5) = A000005(A018804(5)) = A000005(9) = 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A005382, A005408, A018804, A065091, A067756, A277201.

f:= proc(n) local i; numtheory:-tau(add(igcd(i,n),i=1..n)) end proc:
map(f, [$1..100]); # Robert Israel, May 07 2025

f[p_, e_] := (e*(p - 1)/p + 1)*p^e; a[n_] := DivisorSigma[0, Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Apr 07 2025 *)

a(n) = numdiv(sumdiv(n, d, n*eulerphi(d)/d)); \\ Michel Marcus, Apr 07 2025

a(n) = A000005(A018804(n)).
a(A005382(n)) = 2.
a(A067756(n)) = 3.
a(A277201(n)) = 5.

A140391 Pythagorean triangle side lengths triples with two lengths prime.

Original entry on oeis.org

4, 3, 5, 12, 5, 13, 60, 11, 61, 180, 19, 181, 420, 29, 421, 1740, 59, 1741, 1860, 61, 1861, 2520, 71, 2521, 3120, 79, 3121, 5100, 101, 5101, 8580, 131, 8581, 9660, 139, 9661, 16380, 181, 16381, 19800, 199, 19801, 36720, 271, 36721, 60900, 349, 60901, 71820

Offset: 1

Juri-Stepan Gerasimov, Jun 13 2008

Or: consecutive triples of x=A067755(j), y=A048161(j), z=A067756(j), j>=1.

			Contains (x,y,z) = (4,3,5) with 4^2+3^2=5^2 and 3 and 5 prime, then (12,5,13) with 12^5+5^2=13^2 and 5 and 13 prime, then (60,11,61) with 60^2+11^2=61^2 etc. x^2+y^2=z^2

Edited and extended by R. J. Mathar, Jun 17 2008

A230444 Primes of the form (p^k + k - 1)/k for prime p and some k > 1.

Original entry on oeis.org

5, 13, 61, 157, 181, 421, 601, 733, 821, 1741, 1861, 2287, 2521, 3121, 5101, 8581, 9661, 9931, 16381, 19609, 19801, 36721, 60901, 71821, 83641, 100801, 106261, 135721, 161881, 163021, 199081, 205441, 218461, 273061, 282001, 337021, 388081, 431521, 491041

Offset: 1

Irina Gerasimova, Oct 18 2013

nonn

			601 is a term because (7^4 + 4 - 1)/4 = 601 where 7, 601 are both prime,
733 is a term because (13^3 + 3 -1)/3 = 733 where 13, 733 are both prime,
821 is a term because (3^8 + 8 - 1)/8 = 821 where 3, 821 are both prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A048161, A067756.

N:= 10^6: # for terms <= N
S:= {}: p:= 1:
do
  p:= nextprime(p);
  if p^2/2 > N then break fi;
  for k from 2 do
    v:= (p^k + k - 1)/k;
    if v > N then break fi;
    if v::integer and isprime(v) then  S:= S union {v} fi;
od od:
sort(convert(S,list)); # Robert Israel, Jun 22 2023

isA230444(n) = {isprime(n) || return(0); my(k = 2, v, p); while (1, v = k*n+1-k; if (ispower(v, k, &p) && isprime(p), return(1)); if (v < 2^k, return(0)); k++;);} \\ Michel Marcus, Oct 19 2013

More terms from Michel Marcus, Oct 19 2013

cp's OEIS Frontend

A048161 Primes p such that q = (p^2 + 1)/2 is also a prime.

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A067755 Even legs of Pythagorean triangles whose other leg and hypotenuse are both prime.

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A068501 Values m such that the consecutive pair parameters(m,m+1) generate Pythagorean triples whose odd terms are both prime.

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A051859 Values of C (the hypotenuse) of a Pythagorean triangle with A (the short leg) and C both prime and part of a twin prime.

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A292989 Triangular numbers having exactly 6 divisors.

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A286257 Compound filter: a(n) = P(A046523(n), A046523(2n-1)), where P(n,k) is sequence A000027 used as a pairing function.

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A068485 One-sixtieth of the even leg of Pythagorean triangles whose other sides are both primes (other than 3, 5 or 13).

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