A115104 -id:A115104 - cp's OEIS frontend

A199307 Primes of the form 4n^3 + 1.

5, 109, 257, 1373, 2917, 4001, 27437, 62501, 157217, 202613, 237277, 296353, 470597, 629857, 665501, 1492993, 1556069, 1898209, 2456501, 2634013, 3217429, 3322337, 4244833, 5038849, 5180117, 6572129, 10512289, 11453153, 12706093

Offset: 1

N. J. A. Sloane, Nov 05 2011

Dirichlet's theorem on primes in arithmetic progressions tells us, for example, that there are infinitely many primes of the form 4n+1. For primes represented by polynomials of degree greater than 1, the Bateman-Horn paper gives a conjecture on the density.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
P. Bateman and R. A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Mathematics of Computation, 16 (1962), 363-367.
Wikipedia, Bateman-Horn Conjecture

Cf. A115104, A001912, A002496, A005574.

[ a: n in [0..200] | IsPrime(a) where a is 4*n^3+1 ]; // Vincenzo Librandi, Nov 08 2011

Select[Table[4n^3+1,{n,1100}],PrimeQ] (* Vincenzo Librandi, Aug 01 2012 *)

for(n=1,1e3,if(isprime(t=4*n^3+1),print1(t", "))) \\ Charles R Greathouse IV, Nov 21 2011

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")

A115349 Numbers k such that (4*k^5 + 1) is prime.

Original entry on oeis.org

1, 12, 15, 19, 27, 40, 49, 57, 60, 90, 93, 102, 132, 133, 147, 148, 153, 177, 190, 219, 240, 249, 258, 265, 274, 277, 280, 294, 313, 324, 337, 342, 363, 382, 394, 435, 448, 453, 462, 483, 489, 502, 522, 534, 538, 550, 580, 588, 609, 613, 634, 643, 648, 649

Offset: 1

Parthasarathy Nambi, Mar 07 2006

			If k=90 then (4*90^5 + 1) = 23619600001, which is prime.
If k=133 then (4*133^5 + 1) = 166463183573, which is prime.

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

Cf. A115104, A001912.

[n: n in [0..1500] | IsPrime(4*n^5 + 1)]; // Vincenzo Librandi, Jan 31 2011

select(t -> isprime(4*t^5+1), [$1..1000]); # Robert Israel, Jun 19 2018

Do[If[PrimeQ[4*n^5 + 1], Print[n]], {n, 0, 1000}]
Select[Range[700],PrimeQ[4#^5+1]&] (* Harvey P. Dale, Aug 25 2023 *)

is(n)=isprime((4*n^5+1)) \\ Charles R Greathouse IV, Jun 13 2017

More terms from Craig Baribault (csb166(AT)psu.edu), Mar 14 2006 and Jessica M. Cornwall (jmc510(AT)psu.edu), Mar 22 2006

A207838 Numbers k such that 5*k^4 + 1 is prime.

Original entry on oeis.org

6, 12, 114, 120, 324, 336, 390, 420, 498, 504, 540, 672, 756, 768, 840, 852, 876, 1014, 1062, 1092, 1122, 1170, 1188, 1248, 1266, 1314, 1344, 1398, 1440, 1470, 1524, 1758, 1770, 1818, 1860, 1968, 2028, 2046, 2088, 2184, 2190, 2232, 2262, 2268, 2304, 2382, 2430

Offset: 1

Bruno Berselli, Feb 21 2012

All terms are multiples of 6.

Vincenzo Librandi, Table of n, a(n) for n = 1..3000

Cf. A207837 (primes of the form 5*k^4+1).

Cf. A005097, A111051, A115104.

[6*n: n in [1..406] | IsPrime(6480*n^4+1)];

Select[Range[2440], PrimeQ[5 #^4 + 1] &] (* by definition *)

for(n=1, 406, r=6480*n^4+1; if(isprime(r), print1(6*n", ")));

a(n) = ((A207837(n) - 1)/5)^1/4. - Paul F. Marrero Romero, Dec 07 2023

A115449 Numbers n such that 4*n^5 - 1 is prime.

Original entry on oeis.org

1, 2, 3, 8, 12, 23, 27, 42, 68, 75, 86, 96, 113, 117, 125, 135, 140, 146, 168, 182, 185, 188, 191, 198, 233, 245, 255, 267, 281, 287, 297, 306, 311, 318, 327, 360, 362, 366, 377, 390, 392, 395, 408, 416, 423, 432, 447, 456, 465, 486, 488, 497, 516, 531, 555

Offset: 1

Parthasarathy Nambi, Mar 08 2006

nonn

			If n=96 then (4*n^5 - 1) = 32614907903 (prime).

Cf. A115104, A001912.

Select[Range[500], PrimeQ[4*#^5 - 1] &] (* Stefan Steinerberger, Mar 09 2006 *)

for(i=1,2000,if(isprime(4*i^5-1),print1(i,","))) \\ Matthew Conroy, Mar 12 2006

for(i=1,2000,if(isprime(4*i^5-1),print1(i,","))) \\ Matthew Conroy, Mar 12 2006

More terms from Stefan Steinerberger, Zak Seidov and Matthew Conroy, Mar 12 2006

More terms from Matthew Conroy, Mar 12 2006

A116947 Numbers k such that 4*k^6 + 1 is prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 17, 20, 22, 28, 30, 40, 45, 67, 68, 70, 75, 82, 85, 87, 88, 95, 108, 123, 125, 140, 150, 153, 163, 172, 190, 197, 200, 210, 217, 220, 223, 232, 237, 248, 268, 270, 282, 283, 287, 303, 310, 320, 333, 340, 358, 367, 403, 405, 407, 423, 438, 445, 447

Offset: 1

Parthasarathy Nambi, Apr 03 2006

			If k=197 then (4*k^6 + 1) is a prime with 15 digits.

Cf. A115104, A001912, A115349.

Select[Range[500], PrimeQ[(4*#^6 + 1)] &] (* Stefan Steinerberger, Apr 06 2006 *)

is(n)=isprime(4*n^6+1) \\ Charles R Greathouse IV, Jun 13 2017

More terms from Stefan Steinerberger, Apr 06 2006

A274794 Numbers n such that n^3 is the sum of two triangular numbers in exactly one way.

Original entry on oeis.org

0, 1, 3, 4, 7, 9, 10, 19, 24, 25, 34, 37, 39, 42, 49, 54, 55, 72, 73, 78, 85, 87, 93, 94, 102, 108, 109, 118, 138, 142, 147, 157, 160, 165, 168, 175, 192, 195, 202, 210, 214, 220, 228, 232, 243, 247, 249, 250, 252, 253, 258, 267, 273, 274, 279, 289, 297, 312, 333

Offset: 1

Altug Alkan, Jul 07 2016

nonn

A115104 is a subsequence. Terms such that 4*n^3 + 1 is not prime are 24, 337, 457, 750, 840, 1015, ...

			3 is a term because 3^3 = 27 = 6 + 21.

Cf. A000217, A000578, A020756, A052343, A115104, A119345.

Select[Range@ 333, Length[PowersRepresentations[4 #^3 + 1, 2, 2]] == 1 &] (* after Ant King at A052343, or *)
nn = 20; t = (#^2 + #)/2 & /@ Range[0, nn^3]; Select[Range[0, nn], Function[n, Count[Transpose@ {#, n^3 - #} &@ Range[0, Floor[n^3/2]], k_ /; Times @@ Boole@ Map[MemberQ[t, #] &, k] == 1] == 1]] (* Michael De Vlieger, Jul 07 2016 *)

a052343(n) = sum(i=0, (sqrtint(4*n + 1) - 1)\2, issquare(n - i - i^2));
lista(nn) = for(n=0, nn, if(a052343(n^3) == 1, print1(n, ", ")));

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A199307 Primes of the form 4n^3 + 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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A115349 Numbers k such that (4*k^5 + 1) is prime.

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A207838 Numbers k such that 5*k^4 + 1 is prime.

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A115449 Numbers n such that 4*n^5 - 1 is prime.

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A116947 Numbers k such that 4*k^6 + 1 is prime.

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A274794 Numbers n such that n^3 is the sum of two triangular numbers in exactly one way.

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