A048161 -id:A048161 - cp's OEIS frontend

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

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

A094516 Primes p such that q=(p^2+1)/2 is not a prime.

Original entry on oeis.org

2, 7, 13, 17, 23, 31, 37, 41, 43, 47, 53, 67, 73, 83, 89, 97, 103, 107, 109, 113, 127, 137, 149, 151, 157, 163, 167, 173, 179, 191, 193, 197, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373

Offset: 1

N. J. A. Sloane, following a suggestion by R. K. Guy, Jun 05 2004

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Complement in primes of A048161.

Select[Prime[Range[100]],!PrimeQ[(#^2+1)/2]&] (* Harvey P. Dale, Apr 26 2017 *)

from sympy import isprime
def ok(n): return (isprime(n) and (n == 2 or not isprime((n*n + 1)//2)))
print(*list(filter(ok, range(2, 360))), sep = ', ') # Ya-Ping Lu, May 01 2025

A164511 Least prime p such that p^2+1 is the product of n distinct primes.

Original entry on oeis.org

2, 3, 13, 47, 463, 2917, 30103, 241727, 3202337, 26066087, 455081827, 7349346113, 122872146223, 2523038248697, 28435279521433, 119919330795347

Offset: 1

T. D. Noe, Aug 14 2009

For n>1, there appear to be an infinite number of primes q for which q^2+1 is the product of n distinct primes (and thus has 2^n divisors). This sequence gives the smallest such prime for each n. See A048161 for primes q such that q^2+1 has two prime factors. Note that all prime factors of p^2+1 must be 2 or primes of the form 4k+1.

			1+2^2 = 5
1+3^2 = 2*5
1+13^2 = 2*5*17
1+47^2 = 2*5*13*17
1+463^2 = 2*5*13*17*97
1+2917^2 = 2*5*13*29*37*61
1+30103^2 = 2*5*13*17*41*73*137
1+241727^2 = 2*5*13*17*29*37*41*601
1+3202337^2 = 2*5*13*17*29*41*73*193*277
1+26066087^2 = 2*5*13*17*29*37*41*89*233*337
1+455081827^2 = 2*5*13*17*37*53*61*73*97*317*349

Cf. A180278.

nn=8; t=Table[0,{nn}]; p=1; While[Times@@t==0, While[p=NextPrime[p]; {q,e}=Transpose[FactorInteger[p^2+1]]; !(Union[e]=={1} && Length[e]<=nn && t[[Length[e]]]==0)]; t[[Length[e]]]=p]; t

generate(A, B, n) = A=max(A, vecprod(primes(n))); (f(m, p, j) = my(list=List()); my(s=sqrtnint(B\m, j)); if(j==1, forprime(q=max(p, ceil(A/m)), s, if(q%4 == 3, next); my(v=m*q); if(issquare(v-1) && isprime(sqrtint(v-1)), listput(list, sqrtint(v-1)))), forprime(q=p, s, if(q%4 == 3, next); list=concat(list, f(m*q, q+1, j-1)))); list); vecsort(Vec(f(1, 2, n)));
a(n) = my(x=vecprod(primes(n)), y=2*x); while(1, my(v=generate(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Feb 20 2023

a(n) >= A180278(n). - Daniel Suteu, Feb 20 2023

a(12)-a(13) from Donovan Johnson, Oct 09 2009

a(14)-a(16) from Daniel Suteu, Feb 20 2023

A179502 Numbers k with the property that k^2, k^2+1 and k^2+2 are all semiprimes.

Original entry on oeis.org

11, 29, 79, 271, 379, 461, 521, 631, 739, 881, 929, 1459, 1531, 1709, 2161, 2239, 2341, 2729, 3049, 3491, 3709, 4021, 4349, 4561, 4691, 5021, 5281, 5851, 5879, 6301, 6329, 6829, 7559, 8009, 9151, 10069, 10099, 10151, 10529, 10891, 11719, 11959, 11969, 13799, 14051, 14159

Offset: 1

Zak Seidov, Jan 08 2011

nonn

From the first 10^6 primes, 6680 are terms of the sequence.
Also, all numbers k^2+1 are twice prime, and k^2+2 are thrice prime.
The number of terms less than 10^m beginning with m = 1: 0, 3, 11, 35, 160, 759, 4668, 30319, 204439, ..., .
The number of terms less than the (10^m)-th prime beginning with m = 1: 2, 7, 33, 165, 941, 6680, 48977, 373627, ..., .

Zak Seidov, Table of n, a(n) for n = 1..6680

n^2 are squares in A070552, which is a subsequence of A056809 (m and m+1 are semiprimes) and A001358 (semiprimes).

The sequence is a subsequence of A048161.

fQ[n_] := PrimeQ[(n^2 + 1)/2] && PrimeQ[(n^2 + 2)/3]; Select[ Prime@ Range@ 1667, fQ] (* Robert G. Wilson v, Feb 26 2011 *)
Select[Range[15000],PrimeOmega[#^2+{0,1,2}]=={2,2,2}&] (* Harvey P. Dale, May 12 2025 *)

{n=10;for(i=1,10^4,n=nextprime(n+1);n2=n^2;if(2==bigomega(n2+1)&&2==bigomega(n2+2),print1(n,",")))}

A281505 Numbers of the form y^2 - x^2 such that x^2 + y^2 is a prime and 0 < x < y.

Original entry on oeis.org

3, 5, 9, 11, 15, 19, 21, 25, 29, 35, 39, 45, 49, 51, 55, 59, 61, 65, 69, 71, 75, 79, 85, 91, 95, 99, 101, 105, 115, 121, 129, 131, 139, 141, 145, 159, 165, 169, 171, 175, 181, 189, 195, 199, 201, 205, 209, 215, 219, 221

Offset: 1

Thomas Ordowski, Jan 23 2017

nonn

What is the natural density of this set of these numbers?
There are 204 terms up to 10^3, 1849 up to 10^4, 16881 up to 10^5, 160194 up to 10^6, 1531730 up to 10^7, and 14766494 up to 10^8. - Charles R Greathouse IV, Jan 23 2017
Numbers of the form s*t where 0 < s < t and (s^2 + t^2)/2 is prime. - Robert Israel, Jan 23 2017

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Sam Chow and Carl Pomerance, Triangles with prime hypotenuse, arXiv:1703.10953 [math.NT], 2017.
Cihan Sabuncu, Right-angled triangles with almost prime hypotenuse, arXiv:2401.16334 [math.NT], 2024. Mentions this sequence.

Cf. A002144, A048161 is a subsequence, A070079 contains the same numbers.

filter:= proc(n)
  ormap(s -> isprime((s^2 + (n/s)^2)/2), select(s -> s^2Robert Israel, Jan 23 2017

filter[n_] := AnyTrue[Select[Divisors[n], #^2 < n & ], PrimeQ[(#^2 + (n/#)^2)/2] & ];
Select[Range[1, 1000, 2], filter] (* Jean-François Alcover, Nov 27 2017, after Robert Israel *)

list(lim)=my(v=List()); for(a=1,sqrtint(lim\=1), for(x=1,(lim-a^2)\2\a, if(isprime((x+a)^2+x^2), listput(v,(x+a)^2-x^2)))); Set(v) \\ Charles R Greathouse IV, Jan 23 2017

a(n) = n(log n)^c /(log log n)^O(1), where c = 1 - (1 + log log 2)/log 2 = 0.086... Cf. A027424. - Conjectured by Carl Pomerance, Jan 25 2017

More terms from Altug Alkan, Jan 23 2017

a(17)-a(50) from Charles R Greathouse IV, Jan 23 2017

A284036 Positive integers n such that (n^2 - 3)/2 and (n^2 + 1)/2 are twin primes.

Original entry on oeis.org

3, 5, 11, 19, 25, 29, 65, 79, 101, 205, 209, 221, 245, 275, 289, 299, 349, 371, 409, 415, 449, 521, 535, 569, 571, 575, 595, 649, 661, 695, 739, 781, 791, 935, 949, 991, 1081, 1091, 1099, 1129, 1181, 1225, 1241, 1285, 1345, 1349, 1459, 1489, 1531, 1541, 1615

Offset: 1

Giuseppe Coppoletta, Mar 27 2017

nonn

All terms are obviously odd.

			25 is a term because (25^2 - 3)/2 = 311 and (25^2 + 1)/2 = 313 are twin primes.

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

Cf. A002731, A109358, A048161, A110589, A284034.

filter:= n -> isprime((n^2-3)/2) and isprime((n^2+1)/2):
select(filter, [seq(i,i=1..2000,2)]); # Robert Israel, Apr 24 2017

Select[Range[1, 1285, 2], Times @@ Boole@ Map[PrimeQ, (#^2 + {-3, 1})/2] == 1 &] (* Michael De Vlieger, Mar 28 2017 *)

isok(n) = isprime((n^2 - 3)/2) && isprime((n^2 + 1)/2); \\ Michel Marcus, Apr 04 2017

from sympy import isprime
print([n for n in range(3, 1700, 2) if isprime((n**2 - 3)//2) and isprime((n**2 + 1)//2)]) # Indranil Ghosh, Apr 04 2017

[n for n in range(3,1700,2) if is_prime((n^2 - 3)//2) and is_prime((n^2 + 1)//2)]

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

A143822 Primes p such that sigma_0((p*p + 1)/2) = 4.

Original entry on oeis.org

13, 17, 23, 31, 37, 53, 67, 89, 97, 103, 109, 113, 127, 137, 149, 151, 163, 167, 179, 197, 211, 223, 227, 229, 241, 263, 269, 277, 281, 283, 311, 331, 347, 359, 367, 373, 383, 389, 397, 419, 431, 433, 439, 479, 491, 503, 509, 541, 547, 587, 601, 617, 619, 653

Offset: 1

Ctibor O. Zizka, Sep 02 2008

A048161 are primes p such that sigma_0((p*p+1)/2)= 2. Primes p such that sigma_0((p*p+1)/2)= 3 gives all RMS numbers (A140480) with 2 divisors (prime RMS numbers, prime NSW numbers (A088165)) and all RMS numbers with 4 divisors as those are a multiple of two nonequal RMS prime numbers. In general we look after primes p such that sigma_0((p*p+1)/2) equals some given integer k. RMS numbers n=p_1*...*p_t have k=2^t divisors (p_i prime, t integer >=1) and sigma_2(p_1*...*p_t)=(2^t)* (q_1^r_1 *...* q_t^r_t), q_j prime, r_t integer >=1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005 (sigma_0), A048161, A088165, A140480, A002315.

A066885 := proc(n) local p; p :=ithprime(n) ; (p^2+1)/2 ; end: A000005 := proc(n) numtheory[tau](n) ; end: for n from 2 to 300 do if A000005(A066885(n)) = 4 then printf("%d,",ithprime(n)) ; fi; od: # R. J. Mathar, Sep 04 2008

Select[Range[650], PrimeQ[#] && DivisorSigma[0, (#^2 + 1)/2] == 4 &] (* Amiram Eldar, Mar 11 2020 *)
Select[Prime[Range[150]],DivisorSigma[0,(#^2+1)/2]==4&] (* Harvey P. Dale, Sep 22 2022 *)

97 inserted and extended by R. J. Mathar, Sep 04 2008

A173875 Primes p of the form a^2-b^2 and p*a-b is also prime (with b=prime and a=b+1).

Original entry on oeis.org

5, 11, 59, 1439, 2459, 2819, 3119, 4079, 4799, 5399, 5879, 6899, 7559, 12539, 13799, 14159, 16139, 19379, 25919, 27239, 28019, 28499, 29759, 39119, 40739, 41519, 42179, 44159, 44939, 46919, 53759, 57119, 60539, 63599, 64019, 65579, 66359

Offset: 1

Vincenzo Librandi, Mar 06 2010

nonn

			p=11 is in this sequence because 6^2-5^2=11 and 11*6-5=61.
4799 is in this sequence because 2400^2-2399^2=4799 and 4799*2400-2399=11515201.

Vincenzo Librandi, Table of n, a(n) for n = 1..300

[p: p in PrimesUpTo(10^5) | IsPrime((p-1) div 2) and IsPrime((p^2+1) div 2)]; // Vincenzo Librandi Aug 21 2014

Select[Prime[Range[10000]], PrimeQ[(# - 1)/2] &&PrimeQ[ (#^2 + 1)/2] &] (* Vincenzo Librandi, Aug 21 2014 *)

A005385 INTERSECT A048161. [R. J. Mathar, Mar 29 2010]

Missing terms inserted and sequence extended by R. J. Mathar, Mar 29 2010

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

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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A094516 Primes p such that q=(p^2+1)/2 is not a prime.

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A164511 Least prime p such that p^2+1 is the product of n distinct primes.

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A179502 Numbers k with the property that k^2, k^2+1 and k^2+2 are all semiprimes.

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A281505 Numbers of the form y^2 - x^2 such that x^2 + y^2 is a prime and 0 < x < y.

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A284036 Positive integers n such that (n^2 - 3)/2 and (n^2 + 1)/2 are twin primes.

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A140391 Pythagorean triangle side lengths triples with two lengths prime.

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A143822 Primes p such that sigma_0((p*p + 1)/2) = 4.

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