A068501 -id:A068501 - cp's OEIS frontend

A128780 Numbers n such that n^k+(n+1)^k is prime for k = 1, 2, 4.

1, 2, 9, 14, 189, 204, 230, 320, 680, 765, 1080, 1190, 1359, 1364, 1500, 1764, 1850, 2049, 2115, 2360, 2379, 2919, 3050, 3110, 3179, 3579, 3794, 4164, 4215, 4470, 5355, 5619, 5630, 5664, 5810, 5889, 5979, 6035, 6150, 6269, 6824, 6960, 7275, 8045, 8259

Offset: 1

Zak Seidov, Mar 28 2007

nonn

n^k+(n+1)^k is prime only for k = power of 2.
There are 1242 terms < 10^6.
All terms > 2 are congruent to 0 or 4 (mod 5). - Robert Israel, Mar 29 2017

			{2+1, 2^2+3^2,2^4+3^4} = {3,13,97} all prime,
{9+10, 9^2+10^2,9^4+10^4} = {19,181,16561} all prime.

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

Subset of A068501.

select(n -> isprime(2*n+1) and isprime(2*n^2+2*n+1) and isprime(n^4+(n+1)^4),
[1,2,seq(seq(5*i+j,j=[0,4]),i=1..10000)]); # Robert Israel, Mar 29 2017

pnQ[n_]:=And@@PrimeQ/@(n^{1,2,4}+(n+1)^{1,2,4}); Select[Range[9000], pnQ]  (* Harvey P. Dale, Apr 06 2011 *)

A178659 Numbers n such that n^2 +- (n-1)^2 are primes.

Original entry on oeis.org

2, 3, 6, 10, 15, 30, 31, 36, 40, 51, 66, 70, 91, 100, 136, 175, 190, 205, 225, 231, 261, 285, 286, 316, 321, 331, 370, 376, 411, 441, 465, 496, 516, 520, 526, 535, 546, 565, 576, 586, 591, 681, 720, 730, 745, 750, 766, 855, 871, 906, 916, 951, 975, 1081, 1120

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 01 2010

nonn

			2 is in the sequence because 2^2 + 1^2 = 5 and 2^2 - 1^2 = 3 are both prime.
3 is in the sequence because 3^2 + 2^2 = 13 and 3^2 - 2^2 = 5 are both prime.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A068501.

lst={};Do[If[PrimeQ[n^2-(n-1)^2]&&PrimeQ[n^2+(n-1)^2],AppendTo[lst,n]],{n,7!}];lst
Select[Range[8!], PrimeQ[#^2 -(#-1)^2] && PrimeQ[#^2 +(#-1)^2] &] (* G. C. Greubel, Jan 28 2019 *)

A178659()={my(maxx=1000);n=2;ptr=0;
while(n<=maxx,q1=n^2-(n-1)^2;q2=n^2+(n-1)^2;
if(isprime(q1)&&isprime(q2),ptr++;write("b178659.txt",ptr,"  ",n));n++); } \\ Bill McEachen, Jun 13 2014

a(n) = 1 + A068501(n). - Zak Seidov, Feb 10 2015

A127206 Numbers k such that k^j + (k+1)^j is prime for j = 1, 2, 4, 8.

Original entry on oeis.org

1, 765, 39269, 70260, 71399, 85764, 100079, 167789, 218229, 307020, 388449, 468945, 514760, 553400, 568904, 782595, 826284, 1160199, 1220430, 1403775, 1633020, 1714739, 1727930, 1788144, 1932900, 1958705, 2023119, 2037450, 2178804, 2185520, 2193969, 2238474, 2264774

Offset: 1

Zak Seidov, Mar 28 2007

nonn

k^j + (k+1)^j is prime only for j = power of 2.

Subset of A128780 which is a subset of A068501.

			{765 + 766, 765^2 + 766^2, 765^4 + 766^4, 765^8 + 766^8} = {1531, 1171981, 686770904161, 235828747162526935093921}, all prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..331

Cf. A068501, A128780.

[n: n in [1..3*10^6]| IsPrime(2*n+1) and IsPrime(n^2+(n+1)^2) and IsPrime(n^4+(n+1)^4) and IsPrime(n^8+(n+1)^8)]; // Vincenzo Librandi, Nov 18 2018

Do[If[PrimeQ[2n + 1] && PrimeQ[n^2 + (n+1)^2] && PrimeQ[n^4 + (n+1)^4] && PrimeQ[n^8 + (n+1)^8], Print[n]], {n, 5*10^6}] (* Ryan Propper, Mar 30 2007 *)

More terms from Ryan Propper, Mar 30 2007

A178660 Numbers k such that k^3 +- (k+5)^2 are primes.

Original entry on oeis.org

7, 12, 13, 18, 58, 142, 187, 502, 597, 657, 702, 907, 912, 942, 943, 972, 1057, 1168, 1248, 1357, 1453, 1533, 1663, 1938, 2013, 2088, 2272, 2373, 2478, 2608, 2848, 2968, 3003, 3028, 3108, 3247, 3423, 3478, 3583, 3817, 3927, 3957, 4132, 4212, 4632, 4668

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 01 2010

nonn

			7 is a term since: 7^3 +- 12^2 -> (199,487) which are primes.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A068501, A137475, A178659.

Select[Range[8! ],PrimeQ[ #^3-(#+5)^2]&&PrimeQ[ #^3+(#+5)^2]&]
Select[Range[4700],AllTrue[#^3+{(#+5)^2,-(#+5)^2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 19 2018 *)

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).

A173415 Numbers n such that both the difference and the sum of (n-th prime+1)^2 and (n-th prime)^2 are prime.

Original entry on oeis.org

1, 3, 10, 128, 201, 223, 246, 309, 357, 393, 424, 482, 526, 815, 887, 909, 1014, 1196, 1543, 1610, 1653, 1674, 1743, 2219, 2302, 2339, 2371, 2475, 2513, 2611, 2948, 3107, 3273, 3419, 3434, 3516, 3555, 3593, 4070, 4203, 4288, 4332, 4389, 4428, 4724, 4793

Offset: 1

Juri-Stepan Gerasimov, Mar 01 2010

nonn

			a(1)=1 because (1st prime+1)^2 - (1st prime)^2=5 is prime and (1st prime+1)^2 + (1st prime)^2=13 is prime;
a(2)=3 because (3rd prime+1)^2 - (3rd prime)^2=11 is prime and (3rd prime+1)^2 + (3rd prime)^2=61 is prime;
a(3)=10 because (10th prime+1)^2 - (10th prime)^2=59 is prime and (10th prime+1)^2 + (10th prime)^2=1741 is prime;
a(4)=128 because (128th prime+1)^2 - (128th prime)^2=1439 is prime and (128th prime+1)^2 + (128th prime)^2=1035361 is prime.

Cf. A000040, A068501.

npsQ[n_]:=Module[{np=Prime[n],a,b},a=np^2;b=(np+1)^2;And@@PrimeQ[ {a+b,b-a}]]; Select[Range[5000],npsQ] (* Harvey P. Dale, Sep 11 2011 *)

a(n) = Pi(A098717(n)) = A049084(A098717(n)). - R. J. Mathar, Mar 09 2010

Extended beyond a(4) by R. J. Mathar, Mar 09 2010

A254886 a(n) = least k>0 such that n-k^2 and n+k^2 are both primes.

Original entry on oeis.org

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

Offset: 1

Zak Seidov, Feb 10 2015

nonn

If n is a square then a(n)=sqrt(n)-1 or 0.
Also if n is a square and a(n)=sqrt(n)-1 then sqrt(n) is a term in A178659.
First appearances of k for k=1..58 are at n = 4, 7, 14, 21, 36, 43, 90, 117, 86, 111, 210, 149, 768, 201, 236, 285, 468, 329, 366, 411, 446, 1137, 534, 647, 654, 807, 770, 885, 900, 911, 3090, 1665, 1192, 2415, 1296, 1313, 4212, 2163, 1600, 1671, 5448, 1769, 2040, 1941, 2054, 3207, 2214, 2333, 5340, 2601, 2792, 7725, 2814, 3095, 3054, 5913, 3442, 4377.
Among the first 10000 terms, the first missing values are 59, 79, 82, 83, 89, 91, 92, 94, 97, 98, 100.

Cf. A068501, A082467, A178659.

PARI
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k=1;while(k^2Derek Orr, Feb 11 2015
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cp's OEIS Frontend

A128780 Numbers n such that n^k+(n+1)^k is prime for k = 1, 2, 4.

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A178659 Numbers n such that n^2 +- (n-1)^2 are primes.

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A127206 Numbers k such that k^j + (k+1)^j is prime for j = 1, 2, 4, 8.

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Magma

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A178660 Numbers k such that k^3 +- (k+5)^2 are primes.

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Mathematica

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

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Maple

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A173415 Numbers n such that both the difference and the sum of (n-th prime+1)^2 and (n-th prime)^2 are prime.

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A254886 a(n) = least k>0 such that n-k^2 and n+k^2 are both primes.

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PARI