A002496 -id:A002496 - cp's OEIS frontend

A209874 Least m > 0 such that the prime p=A002313(n+1) divides m^2+1.

1, 2, 8, 4, 12, 6, 32, 30, 50, 46, 34, 22, 10, 76, 98, 100, 44, 28, 80, 162, 112, 14, 122, 144, 64, 16, 82, 60, 228, 138, 288, 114, 148, 136, 42, 104, 274, 334, 20, 266, 392, 254, 382, 348, 48, 208, 286, 52, 118, 86, 24, 516, 476, 578, 194, 154, 504, 106, 58, 26, 566, 96, 380, 670, 722, 62, 456, 582, 318, 526, 246, 520, 650, 726, 494, 324

Offset: 0

M. F. Hasler, Mar 11 2012

nonn

This yields the prime factors of numbers of the form N^2+1, cf. formula in A089120: For n=0,1,2,... check whether N = +/- a(n) [mod 2*A002313(n+1)], if so, then A002313(n+1) is a prime factor of N^2+1.
Obviously, p then divides (2kp +/- a(n))^2+1 for all k >=0 ; in particular it will be the least prime factor of such numbers if there is no earlier match.
Alternatively one could deal separately with the case of odd N, for which p=2 divides N^2+1, and even N, for which only Pythagorean primes A002144(n)=A002313(n+1) can be prime factors of N^2+1.

Cf. A002496, A014442, A085722, A144255, A089120, A193432.

Cf. A002314, A177979.

A209874(n)=if( n, 2*lift(sqrt(Mod(-1, A002144[n])/4)), 1)

/* for illustrative purpose: a(n) is the smaller of the 2 possible remainders mod 2*p of numbers N such that N^2+1 has p as smallest prime factor */ forprime( p=1,199, p>2 & p%4 != 1 & next; my(c=[]); for(i=1,9e9, factor(i^2+1)[1,1]==p |next; c=vecsort(concat(c,i%(2*p)),,8); #c==1 || print1(","c[1]) || break))

For n>0, A209874(n) = 2*sqrt(-1/4 mod A002144(n)), where sqrt(a mod p) stands for the positive x < p/2 such that x^2=a in Z/pZ.

A209874(n) = A209877(n)*2 for n>0.

A237040 Semiprimes of the form k^3 + 1.

Original entry on oeis.org

9, 65, 217, 4097, 5833, 10649, 21953, 74089, 195113, 216001, 343001, 373249, 474553, 1000001, 1061209, 1191017, 1404929, 3241793, 3796417, 4251529, 6859001, 9261001, 12487169, 21952001, 29791001, 35937001, 43614209, 45882713, 55742969, 62099137, 89915393, 94818817, 117649001

Offset: 1

Jonathan Sondow, Feb 02 2014

nonn

k^3 + 1 is a term iff k + 1 and k^2 - k + 1 are both prime.
Is the sequence infinite? This is an analog of Landau's 4th problem, namely, are there infinitely many primes of the form k^2 + 1?
In other words: are there infinitely many primes p such that p^2 - 3*p + 3 is also prime? - Charles R Greathouse IV, Jul 02 2017

			9 = 3*3 = 2^3 + 1 is the first semiprime of the form n^3 + 1, so a(1) = 9.

Vincenzo Librandi, Table of n, a(n) for n = 1..1400
Eric Weisstein's World of Mathematics, Semiprime
Wikipedia, Semiprime
Wikipedia, Landau's problems

Cf. A001358, A002383, A002496, A046315, A081256, A096173, A096174, A237037, A237038, A237039.

Cf. A242262 (semiprimes of the form k^3 - 1).

IsSemiprime:= func; [s: n in [1..500] | IsSemiprime(s) where s is n^3 + 1]; // Vincenzo Librandi, Jul 02 2017

L = Select[Range[500], PrimeQ[# + 1] && PrimeQ[#^2 - # + 1] &]; L^3 + 1
Select[Range[50]^3 + 1, PrimeOmega[#] == 2 &] (* Zak Seidov, Jun 26 2017 *)

lista(nn) = for (n=1, nn, if (bigomega(sp=n^3+1) == 2, print1(sp, ", "));); \\ Michel Marcus, Jun 27 2017

list(lim)=my(v=List(),n,t); forprime(p=3,sqrtnint(lim\1-1,3)+1, if(isprime(t=p^2-3*p+3), listput(v,t*p))); Vec(v) \\ Charles R Greathouse IV, Jul 02 2017

a(n) = A096173(n)^3 + 1 = 8*A237037(n)^3 + 1.

A256775 Primes of the form n^2 + 81.

Original entry on oeis.org

97, 181, 277, 337, 757, 1237, 2017, 3217, 4177, 5557, 5857, 6481, 7477, 11317, 13537, 16981, 19681, 21397, 33937, 37717, 48481, 51157, 52981, 59617, 62581, 65617, 80737, 84181, 87697, 96181, 102481, 106357, 111637, 119797, 144481, 149077, 155317, 160081

Offset: 1

Reinhard Zumkeller, Apr 11 2015

nonn

Conjecture: sequence is infinite.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

subsequence of A045349.
Cf. A010051, A000290; subsequence of A028916.
Primes of form n^2+b^4, b fixed: A002496 (b=1), A243451 (b=2), A256776 (b=4), A256777 (b=5), A256834 (b=6), A256835 (b=7), A256836 (b=8), A256837 (b=9), A256838 (b=10), A256839 (b=11), A256840 (b=12), A256841 (b=13).

a256775 n = a256775_list !! (n-1)
a256775_list = [x | x <- map (+ 81) a000290_list, a010051' x == 1]

[p: p in PrimesUpTo(200000)| IsSquare(p-81)]; // Vincenzo Librandi, Apr 20 2015

Select[Range[400]^2 + 81, PrimeQ] (* Michael De Vlieger, Apr 19 2015 *)

for(n=1,10^3,if(isprime(p=n^2+81),print1(p,", "))) \\ Derek Orr, Apr 24 2015

A256776 Primes of form n^2 + 256.

Original entry on oeis.org

257, 281, 337, 617, 881, 1097, 1217, 1481, 1777, 2281, 2657, 2857, 4481, 5297, 5881, 7481, 8537, 9281, 10457, 12577, 14897, 15881, 17417, 18481, 19577, 23057, 24281, 25537, 26177, 27481, 28817, 30881, 32297, 35977, 38281, 39857, 42281, 44777, 52697, 53617

Offset: 1

Reinhard Zumkeller, Apr 11 2015

nonn

Conjecture: sequence is infinite.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A010051, A000290; subsequence of A028916.

Primes of form n^2+b^4, b fixed: A002496 (b=1), A243451 (b=2), A256775 (b=3), A256777 (b=5), A256834 (b=6), A256835 (b=7), A256836 (b=8), A256837 (b=9), A256838 (b=10), A256839 (b=11), A256840 (b=12), A256841 (b=13).

a256776 n = a256776_list !! (n-1)
a256776_list = [x | x <- map (+ 256) a000290_list, a010051' x == 1]

list(lim)=if(lim<257,return([])); my(v=List(),t); forstep(n=1, sqrtint(lim\1-256), 2, if(isprime(t=n^2+256), listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Aug 18 2017

A256777 Primes of form n^2 + 625.

Original entry on oeis.org

641, 661, 769, 821, 881, 1109, 1201, 1301, 1409, 2069, 2389, 2741, 3329, 3541, 3761, 3989, 4721, 6101, 6709, 7349, 7681, 8369, 9461, 12289, 14081, 14549, 16001, 18049, 19121, 20789, 25589, 28181, 31601, 32309, 33749, 35221, 35969, 37489, 38261, 39041, 39829

Offset: 1

Reinhard Zumkeller, Apr 11 2015

Conjecture: sequence is infinite.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A010051, A000290; subsequence of A028916.

Primes of form n^2+b^4, b fixed: A002496 (b=1), A243451 (b=2), A256775 (b=3), A256776 (b=4), A256834 (b=6), A256835 (b=7), A256836 (b=8), A256837 (b=9), A256838 (b=10), A256839 (b=11), A256840 (b=12), A256841 (b=13).

a256777 n = a256777_list !! (n-1)
a256777_list = [x | x <- map (+ 625) a000290_list, a010051' x == 1]

Select[Range[200]^2+625,PrimeQ] (* Harvey P. Dale, Aug 27 2025 *)

list(lim)=if(lim<641,return([])); my(v=List(),t); forstep(n=4, sqrtint(lim\1-625), 2, if(isprime(t=n^2+625), listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Aug 18 2017

A256834 Primes of form n^2 + 1296.

Original entry on oeis.org

1297, 1321, 1657, 2137, 2521, 3697, 5521, 6337, 7537, 8521, 10321, 11497, 13177, 15937, 16921, 18457, 23497, 24097, 25321, 34057, 35521, 40897, 43321, 45817, 47521, 58417, 59377, 88321, 90697, 94321, 98017, 106921, 109537, 117577, 127321, 131617, 138937

Offset: 1

Reinhard Zumkeller, Apr 11 2015

nonn

Conjecture: sequence is infinite.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A010051, A000290; subsequence of A028916.

Primes of form n^2+b^4, b fixed: A002496 (b=1), A243451 (b=2), A256775 (b=3), A256776 (b=4), A256777 (b=5), A256835 (b=7), A256836 (b=8), A256837 (b=9), A256838 (b=10), A256839 (b=11), A256840 (b=12), A256841 (b=13).

a256834 n = a256834_list !! (n-1)
a256834_list = [x | x <- map (+ 1296) a000290_list, a010051' x == 1]

Select[Range[1,401,2]^2+1296,PrimeQ] (* Harvey P. Dale, Sep 18 2018 *)

A256835 Primes of form n^2 + 2401.

Original entry on oeis.org

2417, 2437, 2657, 2801, 3301, 3557, 3697, 4001, 4337, 4517, 7877, 10501, 11617, 12401, 13217, 19301, 20357, 20897, 26737, 28001, 29297, 33377, 36997, 38501, 40037, 44017, 48197, 49057, 64901, 70001, 77477, 78577, 86501, 90017, 92401, 104801, 107377, 108677

Offset: 1

Reinhard Zumkeller, Apr 11 2015

nonn

Conjecture: sequence is infinite.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A010051, A000290; subsequence of A028916.

Primes of form n^2+b^4, b fixed: A002496 (b=1), A243451 (b=2), A256775 (b=3), A256776 (b=4), A256777 (b=5), A256834 (b=6), A256836 (b=8), A256837 (b=9), A256838 (b=10), A256839 (b=11), A256840 (b=12), A256841 (b=13).

a256835 n = a256835_list !! (n-1)
a256835_list = [x | x <- map (+ 2401) a000290_list, a010051' x == 1]

A256836 Primes of form n^2 + 4096.

Original entry on oeis.org

4177, 4217, 4457, 4721, 4937, 6121, 7121, 7577, 7817, 9137, 9721, 10337, 10657, 11321, 12377, 13121, 15121, 16417, 17321, 18257, 23417, 23977, 25121, 26297, 31321, 34721, 36137, 36857, 38321, 40577, 44497, 47777, 50321, 52057, 52937, 54721, 57457, 81937

Offset: 1

Reinhard Zumkeller, Apr 11 2015

nonn

Conjecture: sequence is infinite.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A010051, A000290; subsequence of A028916.

Primes of form n^2+b^4, b fixed: A002496 (b=1), A243451 (b=2), A256775 (b=3), A256776 (b=4), A256777 (b=5), A256834 (b=6), A256835 (b=7), A256837 (b=9), A256838 (b=10), A256839 (b=11), A256840 (b=12), A256841 (b=13).

a256836 n = a256836_list !! (n-1)
a256836_list = [x | x <- map (+ 4096) a000290_list, a010051' x == 1]

Select[Range[1,300,2]^2+4096,PrimeQ] (* Harvey P. Dale, May 26 2025 *)

A256837 Primes of form n^2 + 6561.

Original entry on oeis.org

6577, 6661, 6961, 7237, 7717, 8161, 8677, 9697, 10657, 12037, 16561, 17377, 18661, 21937, 24517, 25057, 26161, 33457, 35461, 37537, 56737, 57637, 69061, 74161, 77317, 81637, 84961, 106417, 108961, 124897, 129061, 143461, 146437, 147937, 150961, 155557

Offset: 1

Reinhard Zumkeller, Apr 11 2015

nonn

Conjecture: sequence is infinite.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A010051, A000290; subsequence of A028916.

Primes of form n^2+b^4, b fixed: A002496 (b=1), A243451 (b=2), A256775 (b=3), A256776 (b=4), A256777 (b=5), A256834 (b=6), A256835 (b=7), A256836 (b=8), A256838 (b=10), A256839 (b=11), A256840 (b=12), A256841 (b=13)

a256837 n = a256837_list !! (n-1)
a256837_list = [x | x <- map (+ 6561) a000290_list, a010051' x == 1]

A256838 Primes of form n^2 + 10000.

Original entry on oeis.org

10009, 10169, 10289, 10529, 10729, 11369, 11681, 12401, 12601, 12809, 13249, 13721, 14489, 15329, 16561, 16889, 17569, 17921, 19801, 20201, 21881, 22769, 23689, 26641, 27689, 29881, 30449, 32801, 33409, 34649, 35281, 37889, 38561, 39241, 39929, 48809, 53681

Offset: 1

Reinhard Zumkeller, Apr 11 2015

nonn

Conjecture: sequence is infinite.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A010051, A000290; subsequence of A028916.

Primes of form n^2+b^4, b fixed: A002496 (b=1), A243451 (b=2), A256775 (b=3), A256776 (b=4), A256777 (b=5), A256834 (b=6), A256835 (b=7), A256836 (b=8), A256837 (b=9), A256839 (b=11), A256840 (b=12), A256841 (b=13).

a256838 n = a256838_list !! (n-1)
a256838_list = [x | x <- map (+ 10000) a000290_list, a010051' x == 1]

cp's OEIS Frontend

A209874 Least m > 0 such that the prime p=A002313(n+1) divides m^2+1.

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A237040 Semiprimes of the form k^3 + 1.

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A256775 Primes of the form n^2 + 81.

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A256776 Primes of form n^2 + 256.

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A256777 Primes of form n^2 + 625.

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A256834 Primes of form n^2 + 1296.

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A256835 Primes of form n^2 + 2401.

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A256836 Primes of form n^2 + 4096.

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