A089001 -id:A089001 - cp's OEIS frontend

A090698 Primes of the form 2*n^2+1.

3, 19, 73, 163, 883, 1153, 1459, 1801, 2179, 2593, 3529, 4051, 8713, 10369, 11251, 15139, 17299, 18433, 19603, 20809, 22051, 30259, 34849, 36451, 46819, 48673, 52489, 62659, 69193, 71443, 80803, 83233, 95923, 103969, 112339, 115201, 130051

Offset: 1

Kurmang. Aziz. Rashid, Dec 20 2003

A prime p can be expressed as either the sum of two squares or the sum of two squares - 1, p = X^2 + Y^2 or p = X^2 + Y^2 - 1, if and only if p is of the form 2*(m^2)+1 where m is either 1 or a multiple of 3.
Conjecture: 2^(a(n)-1) - 3 is not prime. - Vincenzo Librandi, Feb 04 2013.
Primes in A058331. - Vincenzo Librandi, Apr 10 2015

			19 = 2^2 + 4^2 - 1 = 2*(3^2)+1
73 = 5^2 + 7^2 - 1 = 2*(6^2)+1
163= 8^2 + 10^2 -1 = 2*(9^2)+1
883= 10^2+ 28^2 -1 = 2*(21^2)+1

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A058331, A089001, A089008, A090612.

[a: n in [0..400] | IsPrime(a) where a is 2*n^2+1];// Vincenzo Librandi, Dec 02 2011

Select[Table[2n^2+1,{n,0,900}],PrimeQ] (* Vincenzo Librandi, Dec 02 2011 *)

is(n)=isprime(2*n^2+1) \\ Charles R Greathouse IV, Jan 05 2013

a(n)=2*A089001(n)^2+1 = A000040(A090612(n)).

Extended by Ray Chandler, Dec 21 2003

A278981 a(n) is the first composite number having the same base-n digits as its prime factors (with multiplicity), excluding zero digits (or 0 if no such composite number exists).

Original entry on oeis.org

15, 399, 85, 318, 57, 906, 85, 1670, 1111, 18193, 185, 7205205, 4119, 63791, 4369, 1548502, 489, 258099, 451, 408166, 13315, 1012985, 679, 25841526, 26533, 2884373, 985, 49101338, 1057, 5362755, 1285, 2447558, 179503, 3091422, 1387, 5830693854, 82311, 149338, 2005

Offset: 2

Ely Golden, Dec 02 2016

For an alternate program that only checks a single base at a time, use the code from "#the actual function (alternate)" instead of "#the actual function".
The computation of a(n) is exceedingly inefficient, requiring the checking of all natural values less than a(n). A more efficient way to compute a(n) is very desirable. - Ely Golden, Dec 25 2016
There is a lower bound on a(n), if not 0, of n^2 + n + 1. As well, a(n) must have 3 or more nonzero digits in base n (if n is odd, this lower bound is n^3 + n^2 + n + 1, and a(n) must have 4 or more nonzero digits in base n). This does not significantly improve the computation of a(n), however. - Ely Golden, Dec 30 2016
The pattern in the magnitude of a(n) is unclear. For some values of n, a(n) is much larger than for other values. For example, a(65) is 2460678262, whereas a(64) is only 4369 and a(66) is 4577. It seems as though even values of n typically have smaller values of a(n). - Ely Golden, Dec 30 2016
It is known that a(n) > 0 for any nonzero member of this sequence, as well as any n >= 2 of the form A280270(m), A070689(m), A279480(m), 2*A089001(m), 2*A115104(m), and 2*A280273(m)-1. It is likely, but not known, that a(n) > 0 for all n >= 2. - Ely Golden, Dec 30 2016

			a(2) = 15, as 15 is the first composite number whose base-2 nonzero digits (1111) are the same as the base-2 nonzero digits of its prime factors (11_2 and 101_2).

Ely Golden, Table of n, a(n) for n = 2..72 (terms a(67), a(69), and a(71) computed by Chai Wah Wu)
Ely Golden, Table of n, a(n) for n = 2..11584 (a-file, contains every value of a(n) <= 2^27)
Ely Golden, Proofs regarding the lower bound of A278981(n)

a(10) = A176670(1); a(2) = A278909(1).

g[n_] := g[n] = Flatten[ Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]];
f[b_] := Block[{c = b^2}, While[ PrimeQ@ c || DeleteCases[ Sort[ IntegerDigits[c, b]], 0] != DeleteCases[ Sort[ Flatten[ IntegerDigits[g[c], b]]], 0], c++]; c]; Array[f, 39, 2] (* Robert G. Wilson v, Dec 30 2016 *)

def nonZeroDigits(x,n):
    if(x<=0|n<2):
        return []
    li=[]
    while(x>0):
        d=divmod(x,n)
        if(d[1]!=0):
            li.append(d[1])
        x=d[0]
    li.sort()
    return li;
def nonZeroFactorDigits(x,n):
    if(x<=0|n<2):
        return []
    li=[]
    f=list(factor(x))
    #ensures inequality of nonZeroFactorDigits(x,n) and nonZeroDigits(x,n) if x is prime
    if((len(f)==1)&(f[0][1]==1)):
        return [];
    for c in range(len(f)):
        for d in range(f[c][1]):
            ld=nonZeroDigits(f[c][0],n)
            li+=ld
    li.sort()
    return li;
#the actual function
def a(n):
    c=2
    while(nonZeroFactorDigits(c,n)!=nonZeroDigits(c,n)):
        c+=1;
    return c;
index=2
while(index<=100):
    print(str(index)+" "+str(a(index)))
    index+=1
print("complete")
#the actual function (alternate)
def a(n):
    c=2
    while(nonZeroFactorDigits(c,n)!=nonZeroDigits(c,n)):
        c+=1;
        if(c%1000000==1):
            print("checked up to "+str(c-1))
    return c;
x=3 # 
print(str(x)+" "+str(a(x)))
print("complete")

A090612 Numbers k such that the k-th prime is of the form 2*j^2 + 1.

Original entry on oeis.org

2, 8, 21, 38, 153, 191, 232, 279, 327, 378, 493, 559, 1086, 1272, 1360, 1769, 1989, 2111, 2224, 2344, 2471, 3272, 3721, 3863, 4838, 5006, 5359, 6291, 6871, 7077, 7909, 8127, 9245, 9928, 10654, 10889, 12164, 12957, 13764, 14881, 16034, 16343, 16944

Offset: 1

Ray Chandler, Dec 21 2003

nonn

A090698 indexed by A000040.

			From _Jon E. Schoenfield_, Jan 24 2018: (Start)
prime(8) = 19 = 2*3^2 + 1, so 8 is in the sequence.
prime(21) = 73 = 2*6^2 + 1, so 21 is in the sequence.
prime(33) = 137 = 2*68 + 1, and 68 is not a square, so 33 is not in the sequence. (End)

Robert Israel, Table of n, a(n) for n = 1..1525

Cf. A089001, A089008, A090698.

N:= 1000; # to get all entries corresponding to primes <= 2*N^2+1.
R:= select(isprime,[seq(2*k^2+1,k=1..N)]):
A090612:= map(numtheory[pi],R); # Robert Israel, May 09 2014

Select[Range[18000],IntegerQ[Sqrt[(Prime[#]-1)/2]]&] (* Harvey P. Dale, Apr 25 2016 *)

a(n)=k such that A000040(k) = A090698(n) = 2*A089001(n)^2 + 1.

A116954 Numbers k such that 3*k^3 + 1 is prime.

Original entry on oeis.org

4, 10, 14, 16, 18, 20, 30, 36, 38, 48, 54, 58, 64, 70, 74, 86, 96, 106, 120, 140, 150, 154, 166, 170, 174, 176, 180, 200, 230, 234, 244, 260, 266, 268, 288, 296, 300, 304, 306, 308, 324, 330, 338, 340, 346, 348, 368, 384, 388, 394, 396, 406, 408, 434, 438, 440

Offset: 1

Parthasarathy Nambi, Apr 20 2006

All terms are even. - Robert Israel, Jan 24 2018

			If k=48 then 3*k^3 + 1 = 331777 (prime).

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

Cf. A089001.

[ n: n in [0..500] | IsPrime(3*n^3 + 1) ]; // Vincenzo Librandi, Jan 31 2011

select(n -> isprime(3*n^3+1), [seq(i,i=2..1000,2)]); # Robert Israel, Jan 24 2018

Select[Range[2000], PrimeQ[3#^3 + 1] &] (* Stefan Steinerberger, Apr 22 2006 *)
Select[Range[2, 500], PrimeQ[3#^3 + 1] &] (* Stefan Steinerberger, Jul 22 2006 *)

is(n)=isprime(3*n^3+1) \\ Charles R Greathouse IV, Feb 20 2017

More terms from Stefan Steinerberger, Apr 22 2006, Jul 22 2006

A239874 Integers k such that 2k^2 + 1 and 2k^3 + 1 are prime.

Original entry on oeis.org

1, 6, 9, 21, 27, 30, 72, 96, 99, 162, 186, 204, 237, 264, 297, 321, 357, 360, 375, 492, 537, 621, 759, 819, 834, 897, 936, 1065, 1242, 1326, 1329, 1359, 1419, 1494, 1506, 1596, 1662, 1704, 1740, 1749, 1761, 1842, 1869, 2157, 2175, 2250, 2274, 2451, 2547

Offset: 1

Zak Seidov, Mar 28 2014

nonn

All terms > 1 are multiples of 3. Also, no term is congruent to 3 modulo 5.

Zak Seidov, Table of n, a(n) for n = 1..1367 [Duplicate terms removed by _Georg Fischer_, Nov 03 2024]

Intersection of A089001 and A168550.

select(t -> isprime(2*t^2+1) and isprime(2*t^3+1), [$1..6000]); # Robert Israel, Nov 03 2024

s={1};Do[If[PrimeQ [2k^2+1]&&PrimeQ[2k^3+1],AppendTo[s,k]],{k,3,10^3,3}];s
Select[Range[3500], PrimeQ[2 #^2 + 1] && PrimeQ[2 #^3 + 1]&] (* Vincenzo Librandi, Mar 29 2014 *)

s=[]; for(n=1, 4000, if(isprime(2*n^2+1) && isprime(2*n^3+1), s=concat(s, n))); s \\ Colin Barker, Mar 28 2014

A089008 Numbers k such that 18*k^2 + 1 is prime.

Original entry on oeis.org

1, 2, 3, 7, 8, 9, 10, 11, 12, 14, 15, 22, 24, 25, 29, 31, 32, 33, 34, 35, 41, 44, 45, 51, 52, 54, 59, 62, 63, 67, 68, 73, 76, 79, 80, 85, 88, 91, 95, 99, 100, 102, 107, 108, 109, 117, 119, 120, 122, 125, 129, 131, 133, 135, 139, 141, 142, 143, 147, 150, 152, 154, 156

Offset: 1

N. J. A. Sloane, Dec 20 2003

nonn

There are 8 consecutive terms at n=13537 and n=105819293 for n < 10^9. - Jean C. Lambry, Oct 19 2015

Since 18*k^2 + 1 is divisible by 17 for k == 4, 13 (mod 17), the maximum possible number of consecutive terms is 8, in which case the first term must be congruent to 5 modulo 17 and 7 or 8 modulo 11. - Jianing Song, Nov 14 2021

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A089001, A090612, A090698.

Select[Range[200],PrimeQ[18#^2+1]&]  (* Harvey P. Dale, Apr 25 2011 *)

for(n=0, 1e3, if(isprime(k=(18*n^2 + 1)), print1(n", "))) \\ Altug Alkan, Oct 19 2015

a(n) = A089001(n+1)/3.

A239666 a(n) = least number k such that n*k^n+1 is prime, or 0 if no such number exists.

Original entry on oeis.org

1, 1, 4, 1, 8, 1, 4, 3, 10, 1, 42, 1, 60, 15, 22, 1, 8, 1, 198, 42, 10, 1, 8, 115, 34, 21, 0, 1, 54, 1, 130, 3, 4, 7, 72, 1, 778, 204, 30, 1, 108, 1, 178, 15, 14, 1, 924, 28, 234, 63, 1376, 1, 44, 3, 16, 27, 256, 1, 180, 1, 706, 51, 98, 0, 546, 1, 4, 153, 150, 1, 170

Offset: 1

Derek Orr, Mar 29 2014

nonn

a(n) = 1 iff n+1 is prime.
If a(n) = 0, then n is in A097792. Note that the converse is not true: a(4) = 1, not 0.
If n is in A097792 and n > 4, then a(n) = 0. For a sketch of this proof, either n = 4b^4 for some positive integer b > 2 or n = (bp)^p for some prime p > 2 and some positive integer b. In the first case, n*k^n+1 can be factored by Sophie Germain's identity into two trinomials where neither can equal 1 since b > 2, so n*k^n+1 must be composite. In the second case, (bpk^{b^p p^(p-1)}+1) is a factor of n*k^n+1 since p is odd. - William Dean, Oct 23 2024

			3*1^3+1 = 4 is not prime. 3*2^3+1 = 25 is not prime. 3*3^3+1 = 82 is not prime. 3*4^3+1 = 193 is prime. Thus, a(3) = 4.

William Dean, Table of n, a(n) for n = 1..1000

Cf. A097792, A116954, A089001.

is_A097792(n)={my(p,t); n%4==0 && ispower(n\4, 4) || ((2 < p = ispower(n, , &t)) && if(isprime(p), t%p==0, foreach(factor(p)[, 1], q, q%2 && n%q==0 && return(1))))}
a(n) = if(n!=4 && is_A097792(n), 0, for(k=1,oo,if(ispseudoprime(n*k^n+1),return(k)))); \\ [corrected by Andrew Howroyd, Oct 25 2024]

A239925 Integers n such that 2n^2+1, 2n^3+1, 2n^4+1 and 2n^5+1 are prime.

Original entry on oeis.org

1, 30, 8025, 44250, 49335, 49599, 155061, 218196, 255975, 293754, 324684, 333405, 336045, 367839, 381804, 416796, 476814, 514005, 529650, 558291, 668856, 682716, 747810, 893190, 930336, 933576, 1004004, 1246266, 1270860, 1383126, 1392111, 1427211, 1491645, 1497024, 1745904, 1786551

Offset: 1

Zak Seidov, Mar 29 2014

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..134

Subsequence of A239920. Cf. A089001, A168550, A239874.

Select[Range[0, 2000000], PrimeQ[2 #^2 + 1] && PrimeQ[2 #^3 + 1] && PrimeQ[2 #^4 + 1] && PrimeQ[2 #^5 + 1] &] (* Vincenzo Librandi, Mar 29 2014 *)
Select[Range[179*10^4], AllTrue[2 #^Range[2, 5] + 1, PrimeQ] &] (* Harvey P. Dale, Sep 24 2021 *)

s=[]; for(n=1, 2000000, if(isprime(2*n^2+1) && isprime(2*n^3+1) && isprime(2*n^4+1) && isprime(2*n^5+1), s=concat(s, n))); s \\ Colin Barker, Mar 29 2014

A240234 Least number k such that k*n^k + 1 is prime, or 0 if no such number exists.

Original entry on oeis.org

1, 1, 2, 1, 1242, 1, 34, 5, 2, 1, 10, 1

Offset: 1

Derek Orr, Apr 02 2014

a(n) = 1 if and only if n + 1 is prime.

Next term a(13), if it exists, is greater than 1000000. Other unknown terms are at index: 25, 29, 41, 47, 49, 53, 55, 69, 73, 79. (See links). - Jeppe Stig Nielsen, Aug 08 2014

			1*3^1 + 1 = 4 is not prime. 2*3^2 + 1 = 19 is prime. Thus, a(3) = 2.

Guenter Loeh, Generalized Cullen primes n b^n + 1
Jeppe Stig Nielsen, Table of a(n) for n = 1..100, with unknown entries denoted by question marks

Cf. A089001, A116954.

a(n) = for(k=1,4000,if(ispseudoprime(k*n^k+1),return(k)))
n=1;while(n<100,print1(a(n),", ");n++)

A188549 Numbers k such that 8*k^2+1 is a prime.

Original entry on oeis.org

3, 12, 15, 18, 21, 33, 36, 48, 51, 66, 78, 81, 93, 102, 114, 120, 132, 150, 153, 162, 180, 183, 213, 225, 228, 231, 234, 237, 243, 246, 252, 279, 282, 285, 294, 303, 318, 324, 333, 375, 378, 381, 384, 393, 396, 399, 417, 432, 450, 459, 468, 480, 489, 495, 510

Offset: 1

Vincenzo Librandi, Apr 04 2011

Half of the even terms of A089001. - R. J. Mathar, Apr 09 2011

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

Cf. A090685 (Primes of the form 8*n^2 + 1), A089001.

[n: n in [1..1800]|IsPrime(8*n^2 + 1)]; // Vincenzo Librandi, Apr 04 2011

Select[Range[600],PrimeQ[8#^2+1]&] (* Harvey P. Dale, Apr 11 2012 *)

cp's OEIS Frontend

A090698 Primes of the form 2*n^2+1.

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A278981 a(n) is the first composite number having the same base-n digits as its prime factors (with multiplicity), excluding zero digits (or 0 if no such composite number exists).

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A090612 Numbers k such that the k-th prime is of the form 2*j^2 + 1.

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A116954 Numbers k such that 3*k^3 + 1 is prime.

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A239874 Integers k such that 2*k^2 + 1 and 2*k^3 + 1 are prime.

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A089008 Numbers k such that 18*k^2 + 1 is prime.

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A239666 a(n) = least number k such that n*k^n+1 is prime, or 0 if no such number exists.

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A239874 Integers k such that 2k^2 + 1 and 2k^3 + 1 are prime.