A027861 -id:A027861 - cp's OEIS frontend

A215433 Numbers n such that n^512 + (n+1)^512 is a prime.

59, 864, 1455, 1723, 2118, 2172, 2460, 2851, 2916, 2971, 3193, 3476, 3747, 3782, 3795

Offset: 1

Vincenzo Librandi, Aug 31 2012

Cf. A027861, A153504, A154535, A155211, A174156, A174157, A215431, A215432.

Select[Range[0, 4000], PrimeQ[#^512 + (# + 1)^512] &]
Position[Partition[Range[3800]^512,2,1],?(PrimeQ[Total[#]]&)]//Flatten (* _Harvey P. Dale, Aug 03 2024 *)

A218207 Number of n-digit primes of the form (k-1)^2 + k^2.

Original entry on oeis.org

1, 3, 6, 16, 42, 107, 286, 764, 2124, 5917, 17250, 49818, 145429

Offset: 1

Martin Renner, Oct 23 2012

Cf. A027861, A027862, A218208, A218209, A218211.

n = 0; Table[cnt = 0; While[n++; p = 2*n^2 - 2*n + 1; p < 10^e, If[PrimeQ[p], cnt++]]; n--; cnt, {e, 10}] (* T. D. Noe, Oct 23 2012 *)

a(n) = A218208(n) - A218208(n-1)

A218208 Number of primes up to 10^n that are of the form (k-1)^2 + k^2.

Original entry on oeis.org

1, 4, 10, 26, 68, 175, 461, 1225, 3349, 9266, 26516, 76334, 221763

Offset: 1

Martin Renner, Oct 23 2012

Cf. A027861, A027862, A218207, A218210, A218212.

n = 0; cnt = 0; Table[While[n++; p = 2*n^2 - 2*n + 1; p < 10^e, If[PrimeQ[p], cnt++]]; n--; cnt, {e, 10}] (* T. D. Noe, Oct 23 2012 *)

a(n) = sum(A218207(k), k=1..n)

A376992 a(n) is the least n-digit prime of the form j^2 + (j+1)^2.

Original entry on oeis.org

5, 13, 113, 1013, 10513, 100801, 1006781, 10030721, 100040513, 1001057513, 10000515313, 100016728501, 1000078402181, 10000013617661, 100000472012281, 1000000064846161, 10000005481873013, 100000002459693601, 1000000116852093013, 10000000062611784481, 100000001234170737761

Offset: 1

Stefano Spezia, Oct 11 2024

Cf. A001844, A027861, A027862, A376907, A376993.

f:= proc(n) local j,x;
     for j from ceil((sqrt(2*10^(n-1)-1)-1)/2) do
       x:= j^2 + (j+1)^2;
       if isprime(x) then return x fi
     od
end proc:
map(f, [$1..40]); # Robert Israel, Oct 13 2024

a[n_]:=Module[{k=1}, While[!PrimeQ[m=2k^2+2k+1]||IntegerLength[m]

from math import isqrt
from itertools import count
from sympy import prime
def A376992(n):
    for k in count(isqrt(((a:=10**(n-1))<<1)-1>>2)):
        m = 2*k*(k+1)+1
        if m >= a and isprime(m):
            return m # Chai Wah Wu, Oct 13 2024

Conjecture: a(n+1)/a(n) ~ 10.

A050236 Indices of consecutive squares palindromic primes; x such that x^2 + (x+1)^2 is palindromic and prime.

Original entry on oeis.org

1, 9, 12, 1262

Offset: 1

Eric W. Weisstein

No other terms < 20000000000. - Patrick De Geest, Aug 15 1999

Patrick De Geest, Palindromic Sums of Squares of Consecutive Integers
Carlos Rivera, Puzzle 14. Pal-Primes and sum of powers, The Prime Puzzles & Problems Connection.
Eric Weisstein's World of Mathematics, Palindromic Prime

Cf. A050239.

Intersection of A027861 and A027571.

Flatten[Position[Total/@Partition[Range[1300]^2,2,1],?(PrimeQ[#] && IntegerDigits[#]==Reverse[IntegerDigits[#]]&)]] (* _Harvey P. Dale, Dec 09 2014 *)

isok(m) = my(p=m^2+(m+1)^2); if (isprime(p), my(d=digits(p)); (d == Vecrev(d))); \\ Michel Marcus, Jan 05 2019

A154428 Primes of the form 50n^2 + 10n + 1.

Original entry on oeis.org

61, 1301, 1861, 2521, 5101, 7321, 8581, 9941, 14621, 16381, 20201, 24421, 26681, 34061, 36721, 51521, 68821, 76441, 97241, 101701, 106261, 110921, 135721, 163021, 168781, 199081, 205441, 218461, 252761, 282001, 304981, 312841, 337021, 353641

Offset: 1

Vincenzo Librandi, Jan 09 2009

Subsequence of A027862 associated with the values of A027861 that are multiples of 5. [R. J. Mathar, Jan 12 2009]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A017281, A048161.

Filtered(List([1..100],n->50*n^2+10*n+1),IsPrime); # Muniru A Asiru, Apr 25 2019

[a: n in [0..100] | IsPrime(a) where a is 50*n^2 + 10*n + 1]; // Vincenzo Librandi, Jul 23 2012

select(isprime,[50*n^2+10*n+1$n=1..100])[]; # Muniru A Asiru, Apr 25 2019

Select[Table[50n^2+10n+1,{n,0,200}],PrimeQ] (* Vincenzo Librandi, Jul 23 2012 *)

for (n=0, 100, if (isprime (k=50*n^2+10*n+1), print1 (k, ", "))); \\ Vincenzo Librandi, Jul 23 2012

Replaced 13721 by 135721 - R. J. Mathar, Jan 12 2009

A274234 Numbers n such that n^1024 + (n+1)^1024 is prime.

Original entry on oeis.org

1078, 2020, 2471, 3255, 4200, 5135, 5185, 6218, 6823, 7220, 8416, 9003, 9008, 9267, 9396, 9689, 10316, 11150, 11250, 11543, 11652, 12960, 14021, 14201, 16523, 16751, 17006, 17054, 17747, 17874, 18157, 18640, 18834, 20478, 20481, 20794, 21147, 22166, 22608, 22638, 24450, 24677, 24894, 25709

Offset: 1

Tim Johannes Ohrtmann, Jun 15 2016

nonn

The first five terms are certified primes, according to: factordb/certoverview.php. The others are probable primes. - Lewis Baxter, Jan 05 2021

Richard Fischer, Primzahlen mit der Form (B+1)^N+B^N.
Henri Lifchitz & Renaud Lifchitz, Search for : x^1024+y^1024.

Cf. A005097, A027861, A155211, A153504, A154535, A174156, A174157, A215431, A215432, A215433, A274235, A274236, A274237.

[n: n in [1..10000] |IsPrime(n^1024 + (n+1)^1024)]

Select[Range[1, 10000], PrimeQ[#^1024 + (#+1)^1024] &]

for(n=1, 10000, if(isprime(n^1024 + (n+1)^1024), print1(n, ", ")))

A274235 Numbers n such that n^2048 + (n+1)^2048 is prime.

Original entry on oeis.org

754, 1289, 1368, 1813, 3159, 3280, 3301, 4976, 6204, 6283, 6723, 6904, 7141, 10246, 11417, 13268, 15456, 19428, 19683, 19698, 20298, 21484, 22543, 23702, 23815, 24747, 27010, 32319, 34133, 36201, 37030, 39438, 41292, 44472, 47623, 50198, 51031, 51370, 51521, 52628, 53073, 53309, 53767, 55911, 56630, 59424

Offset: 1

Tim Johannes Ohrtmann, Jun 15 2016

nonn

The terms correspond only to probable primes.

Richard Fischer, Primzahlen mit der Form (B+1)^N+B^N.
Henri Lifchitz & Renaud Lifchitz, Search for : x^2048+y^2048.

Cf. A005097, A027861, A155211, A153504, A154535, A174156, A174157, A215431, A215432, A215433, A274234, A274236, A274237.

[n: n in [1..10000] |IsPrime(n^2048 + (n+1)^2048)];

Select[Range[1, 10000], PrimeQ[#^2048 + (#+1)^2048] &]

for(n=1, 10000, if(isprime(n^2048 + (n+1)^2048), print1(n, ", ")))

A274236 Numbers k such that k^4096 + (k+1)^4096 is prime.

Original entry on oeis.org

311, 2741, 3582, 5293, 6289, 12080, 14082, 16886, 17971, 19936, 21454, 21486, 26652, 26904, 28314, 34693, 35778, 36292, 40868, 43819, 46356, 46467, 49653, 53996, 57150, 58169, 64937, 67398, 77383, 82577, 86031, 86102, 87352, 87684, 89030, 93340, 95346, 97320, 98191, 111483, 113947, 118052, 125442, 125836, 126157, 127832, 130794

Offset: 1

Tim Johannes Ohrtmann, Jun 15 2016

nonn

The terms correspond only to probable primes.

Richard Fischer, Primzahlen mit der Form (B+1)^N+B^N.
Henri Lifchitz and Renaud Lifchitz, Search for : x^4096+y^4096.

Cf. A005097, A027861, A155211, A153504, A154535, A174156, A174157, A215431, A215432, A215433, A274234, A274235, A274237.

[n: n in [1..10000] |IsPrime(n^4096 + (n+1)^4096)];

Select[Range[1, 10000], PrimeQ[#^4096 + (#+1)^4096] &]

for(n=1, 10000, if(isprime(n^4096 + (n+1)^4096), print1(n, ", ")))

A274237 Numbers k such that k^8192 + (k+1)^8192 is prime.

Original entry on oeis.org

3508, 5209, 13428, 15347, 16339, 17779, 22548, 37726, 40408

Offset: 1

Tim Johannes Ohrtmann, Jun 15 2016

The terms correspond only to probable primes.

Richard Fischer, Primzahlen mit der Form (B+1)^N+B^N.
Henri Lifchitz and Renaud Lifchitz, Search for: x^8192+y^8192.

Cf. A005097, A027861, A155211, A153504, A154535, A174156, A174157, A215431, A215432, A215433, A274234, A274235, A274236.

[n: n in [1..10000] |IsPrime(n^8192 + (n+1)^8192)]

Select[Range[1, 10000], PrimeQ[#^8192 + (#+1)^8192] &]

for(n=1, 10000, if(isprime(n^8192 + (n+1)^8192), print1(n, ", ")))

cp's OEIS Frontend

A215433 Numbers n such that n^512 + (n+1)^512 is a prime.

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A218207 Number of n-digit primes of the form (k-1)^2 + k^2.

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Formula

A218208 Number of primes up to 10^n that are of the form (k-1)^2 + k^2.

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Mathematica

Formula

A376992 a(n) is the least n-digit prime of the form j^2 + (j+1)^2.

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A050236 Indices of consecutive squares palindromic primes; x such that x^2 + (x+1)^2 is palindromic and prime.

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PARI

A154428 Primes of the form 50n^2 + 10n + 1.

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A274234 Numbers n such that n^1024 + (n+1)^1024 is prime.

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Magma

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A274235 Numbers n such that n^2048 + (n+1)^2048 is prime.

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Magma

Mathematica

PARI