A145292 -id:A145292 - cp's OEIS frontend

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A202018 a(n) = n^2 + n + 41.

41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601, 1681, 1763, 1847, 1933, 2021, 2111, 2203, 2297, 2393

Offset: 0

Reinhard Zumkeller, Dec 08 2011

Euler's famous prime-generating polynomial; a(0) through a(39) are all prime.

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 225.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 138-139, 145.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A060566, A010051, A000040, A002808.
Cf. A002378, A005846, A145292.
Cf. A056561.

List([0..50], n -> n^2 +n+41); # G. C. Greubel, Dec 04 2018

Haskell
```
a202018 = (+ 41) . a002378
```

[n^2 + n + 41 : n in [0..50]]; // Wesley Ivan Hurt, Sep 28 2014

A202018:=n->n^2+n+41: seq(A202018(n), n=0..50); # Wesley Ivan Hurt, Sep 28 2014

Table[n^2 + n + 41, {n, 0, 49}] (* Alonso del Arte, Dec 08 2011 *)

a(n)=n^2+n+41 \\ Charles R Greathouse IV, Dec 08 2011

[n^2+n+41 for n in range(50)] # G. C. Greubel, Dec 04 2018

(0 to 49).map((n: Int) => n * n + n + 41) // Alonso del Arte, Nov 29 2018

a(n) = A005846(n) for n < 41, a(41) = A145292(1);
Union of A005846 (primes) and A145292 (composites);
a(n) = A002378(n) + 41.
a(a(n) + n) = a(n)*a(n+1). - Vladimir Shevelev, Jul 16 2012 (This identity holds for all sequences of the form n^2 + n + c, Joerg Arndt, Jul 17 2012).
a(0) = 41 and for n > 0, a(n) = a(n-1) + 2*n. - Jean-Christophe Hervé, Sep 27 2014
From Colin Barker, Sep 28 2014: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (41*x^2 - 80*x + 41) / (1-x)^3. (End)
a(n) = 2*a(n-1) - a(n-2) + 2. - Vincenzo Librandi, Mar 04 2016
E.g.f.: (x^2 + 2*x + 41)*exp(x). - Robert Israel, Mar 10 2016
Sum_{n>=0} 1/a(n) = tanh(sqrt(163)*Pi/2)*Pi/sqrt(163). - Amiram Eldar, May 12 2025

A145293 a(n) is the smallest nonnegative x such that the Euler polynomial x^2 + x + 41 has exactly n distinct prime proper divisors.

Original entry on oeis.org

0, 41, 420, 2911, 38913, 707864, 6618260, 78776990, 725005500

Offset: 1

Artur Jasinski, Oct 07 2008

The Euler polynomial gives primes for consecutive x from 0 to 39.
For numbers x for which x^2 + x + 41 is not prime, see A007634.
For composite numbers of the form x^2 + x + 41, see A145292.

			a(1)=0 because when x=0 then x^2+x+41=41 (1 distinct prime divisor);
a(2)=41 because when x=41 then x^2+x+41=1763=41*43 (2 distinct prime divisors);
a(3)=420 because when x=420 then x^2+x+41=176861=47*53*71 (3 distinct prime divisors);
a(4)=2911 because when x=2911 then x^2+x+41=8476873=41*47*53*83 (4 distinct prime divisors);
a(5)=38913 because when x=38913 then x^2+x+41=1514260523=43*47*61*71*173 (5 distinct prime divisors);
a(6)=707864 because when x=707864 then x^2+x+41=501072150401=41*43*47*53*71*1607 (6 distinct prime divisors);
a(7)=6618260 because when x=6618260 then x^2+x+41=43801372045901=41*43*47*61*83*131*797 (7 distinct prime divisors);
a(8)=78776990 because when x=78776990 then x^2+x+41=6205814232237131=41*43*61*71*97*131*167*383 (8 distinct prime divisors).
a(9)=725005500: a(9)^2 + a(9) + 41 = 525632975755255541 = 41*43*47*53*61*71*151*397*461. - _Hugo Pfoertner_, Mar 05 2018

Cf. A005846, A007634, A097822, A145292, A145294, A145295.

Cf. A228122. - Zak Seidov, Feb 03 2016

a = {}; Do[x = 1; While[Length[FactorInteger[x^2 + x + 41]] < k - 1, x++ ]; AppendTo[a, x]; Print[x], {k, 2, 10}]; a

Corrected and edited, a(8) added by Zak Seidov, Jan 31 2016
Example for a(8) corrected by Hugo Pfoertner, Mar 02 2018
a(9) from Hugo Pfoertner, Mar 05 2018

A228183 Semiprimes generated by the Euler polynomial x^2 + x + 41.

Original entry on oeis.org

1681, 1763, 2021, 2491, 3233, 4331, 5893, 6683, 6847, 7181, 7697, 8051, 8413, 9353, 10547, 10961, 12031, 13847, 14803, 15047, 15293, 16043, 16297, 17071, 18673, 19223, 19781, 20633, 21797, 24221, 25481, 26123, 26447, 26773, 27101, 29111, 29797, 30143

Offset: 1

Shyam Sunder Gupta, Aug 15 2013

This is a subsequence of A145292. The first numbers in A145292 but not in here are 176861, 186233, 241613, 242597, ...

			The semiprime 1763 = 41^2 + 41 + 41 = 41*43 is in the sequence.

Shyam Sunder Gupta and K. D. Bajpai, Table of n, a(n) for n = 1..10000 (first 4760 terms from Shyam Sunder Gupta)

Cf. A001358, A145292.

IsSemiprime:=func; [s: x in [2..200] | IsSemiprime(s) where s is x^2+x+41]; // Bruno Berselli, Aug 15 2013

Mathematica

a = {}; Do[If[PrimeOmega[x^2 + x + 41] == 2, AppendTo[a, x^2 + x + 41]], {x, 1, 200}]; a (* For the b-file: *) n = 0; Do[t = k^2 + k + 41; If[PrimeOmega[t] == 2, n++; Print[n, " ", t]], {k, 30000}] (* K. D. Bajpai, Apr 22 2014 *)

A145294 Smallest x >= 0 such that the Euler polynomial x^2 + x + 41 has a prime divisor of multiplicity n.

Original entry on oeis.org

0, 40, 1721, 14144, 2294005, 326924482, 6386359423, 1341160319494, 149759650255065, 1167478867440605, 243422399538851918, 9662500171353620019, 122479951673184550424, 12148820281768361731597, 177497315692809432279207, 11767210525408975519141638

Offset: 1

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Author

Artur Jasinski, Oct 07 2008

Keywords

nonn

Comments

The Euler polynomial gives primes for consecutive x from 0 to 39.

For numbers x for which x^2 + x + 41 is not prime, see A007634.

For composite numbers of the form x^2 + x + 41, see A145292.

For the smallest x such that polynomial x^2 + x + 41 has exactly n distinct prime divisors, see A145293.

Sequence interpreted as a(n)^2 + a(n) + 41 having a prime divisor with multiplicity that is exactly n. - Bert Dobbelaere, Jan 22 2019

Examples

a(2)=40 because when x=40 then x^2 + x + 41 = 1681 = 41^2; a(3)=1721 because when x=1721 then x^2 + x + 41 = 2963603 = 43*41^3; a(4)=14144 because when x=14144 then x^2 + x + 41 = 200066921 = 41*47^4; a(5)=2294005 because when x=2294005 then x^2 + x + 41 = 5262461234071 = 35797*43^5. a(6)=326924482: a(6)^2 + a(6) + 41 = 106879617257892847 = 9915343 * 47^6. - _Hugo Pfoertner_, Mar 08 2018

Links

Bert Dobbelaere, Table of n, a(n) for n = 1..100

Bert Dobbelaere, Python program

Crossrefs

Cf. A005846, A007634, A145292, A145293, A145295.

Extensions

Title changed, a(1) and a(6) from Hugo Pfoertner, Mar 08 2018

More terms from Bert Dobbelaere, Jan 22 2019

A228122 Smallest nonnegative number x such that x^2 + x + 41 has exactly n prime factors counting multiplicities.

Original entry on oeis.org

0, 40, 420, 1721, 14144, 139563, 3019035, 24304266, 206583092, 3838101265

Offset: 1

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Shyam Sunder Gupta, Aug 11 2013

Keywords

nonn
more

Examples

a(1) = 0 because if x = 0 then x^2 + x + 41 = 41, which has 1 prime factor. a(2) = 40 because if x = 40 then x^2 + x + 41 = 1681 = 41*41, which has 2 prime factors, counting multiplicities. a(3) = 420 because if x = 420 then x^2 + x + 41 = 176861 = 47*53*71, which has 3 prime factors.

Crossrefs

Cf. A005846, A007634, A145292, A145293, A056561.

Programs

Mathematica

a = {}; Do[x = 0; While[PrimeOmega[x^2 + x + 41] != k, x++]; AppendTo[a, x], {k, 9}]; a

PARI

a(n) = {my(m=0); while (bigomega(m^2+m+41) != n, m++); m;} \\ Michel Marcus, Jan 31 2016

Python

from sympy import factorint def A228122(n): k = 0 while sum(factorint(k*(k+1)+41).values()) != n: k += 1 return k # Chai Wah Wu, Sep 07 2018

Extensions

a(9) from Zak Seidov, Feb 01 2016

a(10) from Giovanni Resta, Sep 08 2018

A147520 a(n) = Smallest number x such that Euler Polynomial x^2 + x + 41 is divisible by 41^n.

Original entry on oeis.org

0, 40, 1721, 139563, 14268368, 1636255182, 6386359423, 1953929098233, 149759650255065, 1814531956108700, 243422399538851918, 9662500171353620019, 122479951673184550424, 12148820281768361731597, 177497315692809432279207, 14173382150616650630276616, 1225594969529024683212496795

Offset: 1

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Artur Jasinski, Nov 06 2008

Keywords

nonn

Comments

For values of x^2 + x + 41, see A147521. For values (x^2 + x + 41)/(41^n), see A147522.

By Hensel's lemma, x^2 + x + 41 has two roots mod 41^n; their sum == -1 mod 41^n. Thus 0 <= a(n) < 41^n/2. - Robert Israel, Apr 09 2018

Links

Robert Israel, Table of n, a(n) for n = 1..620

Crossrefs

Cf. A145292, A145293, A145294, A147520, A147521, A147522.

Programs

Maple

f:= n -> min(map(t -> rhs(op(t)), [msolve(x^2+x+41, 41^n)])): map(f, [$1..30]); # Robert Israel, Apr 09 2018

Mathematica

a = {}; Do[x = 0; While[Mod[x^2 + x + 41, 41^n] != 0, x++ ]; AppendTo[a,x];Print[{n, x, x^2 + x + 41, (x^2 + x + 41)/41^n}], {n, 1, 6}];a (* Artur Jasinski *)

Extensions

More terms from Robert Israel, Apr 09 2018

A147521 a(n) is the smallest value of Euler polynomial x^2 + x + 41 which is divisible by 41^n.

Original entry on oeis.org

41, 1681, 2963603, 19477970573, 203586339651833, 2677331022258108347, 40785586686127252393, 3817838920923578492820563, 22427952844519540079208409331, 3292526219739666997778171798741, 59254464597252454704406353071130683, 93363909561408736238900593787191180421

Offset: 1

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Artur Jasinski, Nov 06 2008

Keywords

nonn

Comments

For values x see A147520 For values (x^2 + x + 41)/(41^n) see A147522.

Crossrefs

Cf. A145292, A145293, A145294, A147520, A147521, A147522.

Programs

Mathematica

a = {}; Do[x = 0; While[Mod[x^2 + x + 41, 41^n] != 0, x++ ]; AppendTo[a,x^2 + x + 41];Print[{n, x, x^2 + x + 41, (x^2 + x + 41)/41^n}], {n, 1, 6}];a

Extensions

a(7)-a(12) from Hugo Pfoertner, Jan 12 2019

A147522 Minimal number such that a(n)*41^n is of the form x^2 + x + 41.

Original entry on oeis.org

1, 1, 43, 6893, 1757233, 563636267, 209420753, 478130829203, 68506995922171, 245296117830341, 107670977147002963, 4137830996824703141, 16215855859012574521, 3891295582697539371727, 20259360308241533953897, 3150687184319350782852593, 574603643148719626677940181

Offset: 1

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Artur Jasinski, Nov 06 2008

Keywords

nonn

Comments

For values x see A147520. For x^2 + x + 41 values see A147521.

Crossrefs

Cf. A145292, A145293, A145294, A147520, A147521, A147522.

Programs

Mathematica

a = {}; Do[x = 0; While[Mod[x^2 + x + 41, 41^n] != 0, x++ ]; AppendTo[a,(x^2 + x + 41)/41^n];Print[{n, x, x^2 + x + 41, (x^2 + x + 41)/41^n}], {n, 1, 6}]; a

Extensions

a(7)-a(17) from Hugo Pfoertner, Jan 12 2019

A241554 Semiprimes generated by the polynomial 2 * n^2 + 29.

Original entry on oeis.org

1711, 1829, 2077, 2479, 3071, 3901, 5029, 6527, 6757, 7471, 7967, 8479, 10397, 10981, 11581, 14141, 15167, 15517, 15871, 16591, 16957, 17701, 18079, 18847, 19631, 20837, 22927, 23791, 25567, 26941, 27877, 28829, 29797, 30287, 31279, 31781, 32287, 35941, 38117

Offset: 1

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K. D. Bajpai, Apr 25 2014

Keywords

nonn

Comments

2 * n^2 + 29 is a well-known Legendre prime-producing polynomial which generates 29 distinct primes for n = 0, 1, ..., 28. For n = 29, it yields the first semiprime, 1711 = 29 * 59.

The number n = 185 is the least positive integer for which 2*n^2 + 29 = 68479 = 31 * 47 * 47 is not squarefree.

Examples

2 * 30^2 + 29 = 1829 = 31 * 59, which is a semiprime and is a term. 2 * 35^2 + 29 = 2479 = 37 * 67, which is a semiprime and is a term.

Links

K. D. Bajpai, Table of n, a(n) for n = 1..10000

R. A. Mollin, Prime-Producing Quadratics, The American Mathematical Monthly, Vol. 104, No. 6 (Jun. - Jul., 1997), pp. 529-544.

Crossrefs

Cf. A007641 (for primes).

Cf. A001358, A072381, A082919, A145292, A228183, A237627, A353004, A353388.

Programs

Maple

with(numtheory):A241554:= proc() local k; k:=2*x^2+29;if bigomega(k)=2 then RETURN (k); fi; end: seq(A241554(), x=0..500);

Mathematica

A241554 = {}; Do[k = 2 * n^2 + 29; If[PrimeOmega[k] == 2, AppendTo[A241554, k]], {n,200}]; A241554

PARI

s=[]; for(n=1, 200, t=2*n^2+29; if(bigomega(t)==2, s=concat(s, t))); s \\ Colin Barker, Apr 26 2014

A241716 Primes p such that p^3 - 2 is semiprime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 41, 43, 47, 61, 79, 89, 101, 107, 139, 157, 181, 199, 239, 271, 307, 311, 331, 337, 347, 349, 379, 397, 409, 421, 431, 479, 487, 499, 521, 523, 541, 571, 607, 613, 641, 643, 661, 673, 701, 719, 761, 769, 811, 823, 829, 839, 877, 881, 883

Offset: 1

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K. D. Bajpai, Apr 27 2014

Keywords

nonn

Examples

11 is prime and appears in the sequence because 11^3 - 2 = 1329 = 3 * 443, which is a semiprime. 17 is prime and appears in the sequence because 17^3 - 2 = 4911 = 3 * 1637, which is a semiprime. 23 is prime but does not appear in the sequence because 23^3 - 2 = 12165 = 3 * 5 * 811, which is not a semiprime.

Links

K. D. Bajpai, Table of n, a(n) for n = 1..9380

Crossrefs

Cf. A001358, A063637, A063638, A072381, A082919, A145292, A228183, A237627, A241483, A241493, A241659.

Programs

Maple

with(numtheory):A241716:= proc() local k; k:=ithprime(x); if bigomega(k^3-2)=2 then RETURN (k); fi; end: seq(A241716(), x=1..500);

Mathematica

A241716 = {}; Do[t = Prime[n]; If[PrimeOmega[t^3 - 2] == 2, AppendTo[A241716, t]], {n, 500}]; A241716 Select[Prime[Range[200]],PrimeOmega[#^3-2]==2&] (* Harvey P. Dale, Dec 09 2018 *)

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A202018 a(n) = n^2 + n + 41.

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A145293 a(n) is the smallest nonnegative x such that the Euler polynomial x^2 + x + 41 has exactly n distinct prime proper divisors.

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A228183 Semiprimes generated by the Euler polynomial x^2 + x + 41.

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A145294 Smallest x >= 0 such that the Euler polynomial x^2 + x + 41 has a prime divisor of multiplicity n.

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A228122 Smallest nonnegative number x such that x^2 + x + 41 has exactly n prime factors counting multiplicities.

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Python

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A147520 a(n) = Smallest number x such that Euler Polynomial x^2 + x + 41 is divisible by 41^n.

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Maple

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A147521 a(n) is the smallest value of Euler polynomial x^2 + x + 41 which is divisible by 41^n.

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Extensions