A067756 -id:A067756 - cp's OEIS frontend

A284035 Upper twin primes which correspond to the hypotenuse in a Pythagorean triple whose short leg is also a prime.

5, 13, 61, 181, 421, 3121, 5101, 60901, 83641, 100801, 135721, 161881, 163021, 218461, 273061, 491041, 595141, 637321, 697381, 1064341, 1108561, 1171981, 1806901, 2574181, 2601481, 2740141, 2763601, 2853661, 3248701, 3535141, 3567121, 3696481, 3723721, 3729181, 4832941

Offset: 1

Giuseppe Coppoletta, Mar 19 2017

nonn

A284034 gives the corresponding short leg primes in the definition.

			The prime q = 3121 is in the sequence because q - 2 = 3119 is prime and {79, 3120, 3121} is a Pythagorean triple with prime short leg (see example in A284034).

Cf. A051859, A067756, A284034.

lista(nn) = forprime(p=3, nn, if (isprime(p) && isprime((p^2-3)/2) && isprime(q=(p^2+1)/2), print1(q, ", "))); \\ Michel Marcus, Apr 01 2017

A284034(n)^2 + (a(n) - 1)^2 = a(n)^2, i.e., a(n) = (A284034(n)^2 + 1)/2.

More terms from Michel Marcus, Apr 01 2017

A308442 Primes of the form (p^k+1)/2 where p is prime and k > 1.

Original entry on oeis.org

5, 13, 41, 61, 181, 313, 421, 1201, 1741, 1861, 2521, 3121, 5101, 7321, 8581, 9661, 14281, 16381, 19801, 36721, 41761, 60901, 71821, 83641, 100801, 106261, 135721, 139921, 161881, 163021, 199081, 205441, 218461, 273061, 282001, 337021, 353641, 388081, 431521, 491041, 531481, 539761, 552301, 571381

Offset: 1

J. M. Bergot and Robert Israel, May 27 2019

nonn

The only primes of the form (p^k-1)/2 are A076481, since (p^k-1)/2 is divisible by (p-1)/2.

k must be a power of 2, since if k has an odd divisor d>1, (p^k+1)/2 is divisible by (p^(k/d)+1)/2.

			a(3) = 41 is in the sequence because 41 = (3^4 + 1)/2.

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

Cf. A076481.

Contains A067756.

N:= 10^6: # to get terms <= N
p:= 2:
Res:= NULL:
do
  p:= nextprime(p);
  if p^2 >= 2*N then break fi;
  pk:= p;
  do
    pk:= pk^2;
    if pk >= 2*N then break fi;
    v:= (pk+1)/2;
    if isprime(v) then Res:= Res, v;
    fi;
  od
od:
sort([Res]); # Robert Israel, May 27 2019

A328058 Primes p such that 2*p-1 is a semiprime.

Original entry on oeis.org

5, 11, 13, 17, 29, 43, 47, 61, 67, 71, 73, 89, 101, 103, 107, 109, 127, 151, 181, 191, 197, 223, 227, 241, 251, 269, 277, 283, 317, 349, 359, 373, 397, 409, 421, 433, 457, 461, 467, 487, 521, 541, 569, 571, 631, 643, 647, 659, 673, 701, 709, 719, 733, 739, 751, 757, 769, 821, 857, 859, 881, 883

Offset: 1

J. M. Bergot and Robert Israel, Oct 03 2019

nonn

			a(3)=13 is in the sequence because it is prime and 2*13-1=5^2 is a semiprime.

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

Cf. A000040, A001358. Includes A067756 and A162336.

Cf. A086006, A234095.

[p: p in PrimesUpTo(1000)| &+[d[2]: d in Factorization(2*p-1)] eq 2]; // Marius A. Burtea, Oct 03 2019

select(t -> isprime(t) and numtheory:-bigomega(2*t-1)=2, [2,seq(i,i=3..10000,2)]);

Select[Prime@ Range@ 153, PrimeOmega[2 # - 1] == 2 &] (* Michael De Vlieger, Oct 03 2019 *)

isok(p) = isprime(p) && (bigomega(2*p-1) == 2); \\ Michel Marcus, Oct 04 2019

A334802 Positive integers of the form x^4 - y^4 that have exactly 4 divisors.

Original entry on oeis.org

15, 65, 671, 3439, 12209, 102719, 113521, 178991, 246559, 515201, 1124111, 1342879, 2964961, 3940399, 9951391, 21254449, 27220159, 34209169, 45259649, 48986321, 70710641, 92110289, 93084991, 125620111, 131687681, 144402721, 201792079, 211782751, 276694241

Offset: 1

C. Kenneth Fan, May 12 2020

nonn

If a(n) = pq, where p > q are both prime, then p is the hypotenuse and q is a leg of a primitive Pythagorean triple. (x^4-y^4 = (x^2+y^2)(x+y)(x-y), hence x-y=1 and x^2+y^2 and x+y are both prime. Note that x^2+y^2 can never be (x+y)^2 so a(n) is never the cube of a prime.)

			2^4 - 1^4 = 15 = 3*5 and (3, 4, 5) is a Pythagorean triple, so 15 is a term.
6^4 - 5^4 = 671 = 11*61 and (11, 60, 61) is a Pythagorean triple, so 671 is a term.

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

Cf. A068501.

Intersection of A030513 and A147857.

f:= proc(y) if isprime(2*y+1) and isprime(2*y^2 + 2*y+1) then (2*y+1)*(2*y^2+2*y+1) fi end proc:
map(f, [$1..1000]); # Robert Israel, Jun 16 2020

Select[(#^4 - (#-1)^4) & /@ Range[420], DivisorSigma[0, #] == 4 &] (* Giovanni Resta, May 12 2020 *)

a(n) = (b(n)+1)^4 - b(n)^4 with b(n)=A068501(n).

a(n) = A048161(n)*A067756(n).

A342583 Numbers k such that prime(k) is the hypotenuse of a Pythagorean triple where one leg is also prime.

Original entry on oeis.org

3, 6, 18, 42, 82, 271, 284, 369, 445, 682, 1069, 1193, 1900, 2241, 3894, 6137, 7108, 8164, 9658, 10126, 12645, 14842, 14936, 17913, 18420, 19480, 23893, 24605, 28959, 32913, 36279, 40847, 43936, 44559, 45500

Offset: 1

Ivan N. Ianakiev, Mar 16 2021

nonn

In such a triangle, the leg that is not prime is always the largest one and is equal to prime(k)-1; these even legs are in A067755. E.g. for a(2) = 6, prime(6) = 13 and the corresponding Pythagorean triple is (5, 12, 13). - Bernard Schott, Apr 03 2021

			a(1) = 3, since prime(3) = 5 is the hypotenuse of the triple (3,4,5).

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A067756 (the hypotenuses).

Cf. A009000, A046083, A046084, A067755.

R:= NULL: count:= 0:
p:= 2:
while count < 100 do
  p:= nextprime(p); n:= (p-1)/2; q:= 2*n^2+2*n+1;
  if isprime(q) then
    count:= count+1; r:= numtheory:-pi(q); R:= R, r;
  fi
od:
R; # Robert Israel, Mar 22 2021

PrimePi[Take[Cases[Import["https://oeis.org/A067756/b067756.txt","Table"],{,}][[All,2]],100]]

A087939 Prime hypotenuse of primitive Pythagorean triangles with nonprime odd short leg.

Original entry on oeis.org

41, 89, 97, 113, 149, 157, 193, 233, 269, 277, 313, 317, 337, 353, 389, 433, 461, 521, 541, 557, 569, 613, 617, 653, 673, 709, 761, 769, 773, 797, 821, 853, 881, 929, 937, 953, 1009, 1013, 1069, 1097, 1109, 1117, 1129, 1201, 1213, 1217, 1249, 1301, 1361

Offset: 1

Lekraj Beedassy, Oct 27 2003

nonn

For prime hypotenuse of primitive Pythagorean triangle with prime (short) leg see A067756.

Corrected and extended by Ray Chandler, Oct 28 2003

A174885 Prime hypotenuses c with concatenation p = c//a//b a prime number.

Original entry on oeis.org

29, 409, 461, 661, 929, 1249, 1289, 1381, 1801, 1901, 2081, 2609, 2621, 2749, 3041, 3301, 3881, 5309, 5701, 6421, 6481, 6521, 6529, 7349, 7489, 7789, 8641, 8849, 9349, 9629, 9649, 9689, 9829, 10321, 10709, 10861, 12841, 14321, 14561, 15061, 16661

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 01 2010

See comments in A174825

c is the prime hypotenuse c, i. e. c of a primitive Pythagorean triple: a^2 + b^2 = c^2

			p = c//a//b: 292021, 409120391, 461380261, 661300589, 929920129, 1249960799, 12895601161,
13811020931, 18011680649, 19011820549, 208116401281, 260918801809, 262111002379,
27492580949, 30414403009, 330129401501, 388123603081, 53095300309, 570122205251
29^2=20^2+21^2, 409^2=120^2+391^2, 461^2=380^2+261^2,
661^2=300^2+589^2, 929^2=920^2+129^2, 1249^2=960^2+799^2,
1289^2=560^2+1161^2,1381^2=1020^2+931^2, 1801^2=1680^2+649^2,
1901^2=1820^2+549^2, 2081^2=1640^2+1281^2, 2609^2=1880^2+1809^2,
2621^2=1100^2+2379^2, 2749^2=2580^2+949^2, 3041^2=440^2+3009^2,
3301^2=2940^2+1501^2, 3881^2=2360^2+3081^2, 5309^2=5300^2+309^2,
5701^2=2220^2+5251^2

W. W. R. Ball, H. S. M. Coxeter: Mathematical Recreations and Essays, New York: Dover, 1987
L. E. Dickson: "Rational Right Triangles", ch. 4 in History of the Theory of numbers, vol. II, Dover Publications 2005
W. Sierpinski: Pythagorean Triangles, Mineola, NY, Dover Publications, Inc, 2003

A008846, A020882, A048161, A067756, A138044, A174825

More terms from Zak Seidov, Apr 04 2010

A266965 Primes of the form p = a^2 + b^2 where |a^2 - b^2| is composite.

Original entry on oeis.org

17, 29, 37, 41, 53, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 433, 449, 457, 461, 509, 521, 541, 557, 569, 577, 593, 601, 613, 617, 641, 653, 661, 673, 677, 701, 709

Offset: 1

Altug Alkan, Jan 07 2016

nonn

Inspired by A266954.
A266954 is a subsequence.
A067756 lists primes of the form p = a^2 + b^2 where |a^2 - b^2| is prime. So union of 2, A067756 and this sequence gives A002313. 2 is an exception because 1^2 - 1^2 = 0 is not prime or composite.

			17 is a term because 4^2 + 1^2 = 17 is prime and 4^2 - 1^2 = 15 is composite.
29 is a term because 5^2 + 2^2 = 29 is prime and 5^2 - 2^2 = 21 is composite.
37 is a term because 6^2 + 1^2 = 37 is prime and 6^2 - 1^2 = 35 is composite.

Cf. A002313, A067756, A266954.

lim = 50; Take[Select[Union@ Flatten@ Table[If[CompositeQ[Abs[a^2 - b^2]], a^2 + b^2, Nothing], {a, lim}, {b, lim}], PrimeQ], 56] (* Michael De Vlieger, Jan 07 2016 *)

is(n) = {for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))}
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=3, 1e3, if(is(p) && !isprime(2*f(p)^2-p), print1(p, ", ")));

list(lim) = my(v=List(), t); lim\=1; for(x=2, sqrtint(lim), for(y=1, min(sqrtint(lim-x^2), x), if(isprime(t=x^2+y^2) && !isprime(x^2-y^2), listput(v, t)))); vecsort(Vec(v), , 8)

cp's OEIS Frontend

A284035 Upper twin primes which correspond to the hypotenuse in a Pythagorean triple whose short leg is also a prime.

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A308442 Primes of the form (p^k+1)/2 where p is prime and k > 1.

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Maple

A328058 Primes p such that 2*p-1 is a semiprime.

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Magma

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PARI

A334802 Positive integers of the form x^4 - y^4 that have exactly 4 divisors.

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A342583 Numbers k such that prime(k) is the hypotenuse of a Pythagorean triple where one leg is also prime.

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Maple

Mathematica

A087939 Prime hypotenuse of primitive Pythagorean triangles with nonprime odd short leg.

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A174885 Prime hypotenuses c with concatenation p = c//a//b a prime number.

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A266965 Primes of the form p = a^2 + b^2 where |a^2 - b^2| is composite.

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