A076605 -id:A076605 - cp's OEIS frontend

A037464 Bisection of A076605.

3, 5, 7, 7, 11, 13, 13, 17, 19, 19, 23, 23, 5, 29, 31, 31, 11, 37, 37, 41, 43, 43, 47, 47, 17, 53, 53, 19, 59, 61, 61, 13, 67, 67, 71, 73, 73, 11, 79, 79, 83, 83, 29, 89, 89, 31, 31, 97, 97, 101, 103, 103, 107, 109, 109, 113, 113, 23, 17, 17, 41, 41, 127, 127, 131, 131, 19

Offset: 1

N. J. A. Sloane, Nov 29 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A076605, A074399.

Table[ Last[ Table[ # [[1]]] & /@ FactorInteger[n^2 - 1]], {n, 2, 150, 2}]

a(n) = my(p = factor(4*n^2-1)[, 1]); p[#p]; \\ Amiram Eldar, Nov 03 2024

a(n) = max(A006530(2n-1), A006530(2n+1)).

a(n) = A076605(2*n). - Amiram Eldar, Nov 03 2024

Extended by Robert G. Wilson v, Dec 02 2002

A014442 Largest prime factor of n^2 + 1.

Original entry on oeis.org

2, 5, 5, 17, 13, 37, 5, 13, 41, 101, 61, 29, 17, 197, 113, 257, 29, 13, 181, 401, 17, 97, 53, 577, 313, 677, 73, 157, 421, 53, 37, 41, 109, 89, 613, 1297, 137, 17, 761, 1601, 29, 353, 37, 149, 1013, 73, 17, 461, 1201, 61, 1301, 541, 281, 2917, 89, 3137, 13, 673

Offset: 1

Glen Burch (gburch(AT)erols.com)

All a(n) except for a(1) = 2 are the Pythagorean primes, i.e., primes of form 4k+1. Conjecture: every Pythagorean prime appears in a(n) at least once.
Problem 11831 [Ozols 2015] is to prove that lim inf a(n)/n is zero. - Michael Somos, May 11 2015
From Michael Kaltman, Jun 10 2015: (Start)
For all numbers k in A256011, a(k) < k.
Conjecture: every Pythagorean prime p appears exactly two times among the first p integers of the sequence. Further: if a(i) = a(j) = p and both i and j are less than p (and i is not equal to j), then i + j = p and ij == 1 (mod p). [If a(k) = p as well, then k > p; in fact, k is in A256011.] Two examples: a(2) = a(3) = 5, with 2+3 = 5 and 2*3 = 6 == 1 (mod 5); a(4) = a(13) = 17, with 4+13 = 17 and 4*13 = 52 == 1 (mod 17).
(End)
The conjecture is true. If p is a Pythagorean prime, -1 is a quadratic residue mod p. Then -1 has exactly two square roots mod p, i.e., there are exactly two integers x,y with 1 <= x,y <= p-1 such that x^2 == y^2 == -1 (mod p), i.e., p divides x^2+1 and y^2+1, and moreover y == -x (mod p) so x + y = p, and x*y == -x^2 == 1 (mod p). Any other prime factor q of x^2 + 1 must divide (x^2+1)/p, and since x^2+1 < p^2 we have q < p, so a(x) = p and similarly a(y) = p. - Robert Israel, Jun 11 2015
Conjecture: if n is even and a(n) > n, then n+a(n) is in A256011. Examples: 2+a(2) = 2+5 = 7, 4+a(4) = 4+17 = 21, 6+a(6) = 6+37 = 43, and so on. Note that 18+a(18) is NOT in A256011, but 18 itself is. - Michael Kaltman, Jun 13 2015
This is also true. Suppose A = a(n) > n. n^2+1 is odd so A is an odd prime; n^2 + 1 = A *B with B < A also odd. Then (A+n)^2 + 1 = A*(A+2*n+B) and A+2*n+B is even. The largest prime factor of A+2*n+B is thus at most (A+2*n+B)/2 < A + n, while A < A + n as well. - Robert Israel, Jun 17 2015
Størmer shows that a(n) tends to infinity with n. Chowla shows that a(n) >> log log n.  Schinzel shows that lim inf a(n)/log log n >= 4 and, using lower bounds for linear forms of logarithms, this inequality can be generalized for general quadratic polynomials, with 2 replaced by 4/7 for irreducible ones and 2/7 for reducible ones. - Tomohiro Yamada, Apr 15 2017
According to Hooley, an unpublished manuscript of Chebyshev contains the result that a(n)/n is unbounded which was first published and fully proved by Markov. - Charles R Greathouse IV, Oct 27 2018
Note that a(n) is the largest prime p such that n^(p+1) == -1 (mod p). - Thomas Ordowski, Nov 08 2019

A. A. Markov, Über die Primteiler der Zahlen von der Form 1+4x^2, Bulletin de l'Académie impériale des sciences de St.-Pétersbourg 3 (1895), pp. 55-59.
H. Rademacher, Lectures on Elementary Number Theory, pp. 33-38.

T. D. Noe, Table of n, a(n) for n = 1..10000
S. Chowla, The greatest prime factor of x^2 + 1, J. London Math. Soc. 10 (1935), 117--120.
J. M. Deshouillers and H. Iwaniec, On the greatest prime factor of n^2+1, Annales de l'institut Fourier, 32 no. 4 (1982), p. 1-11.
Christopher Hooley, On the greatest prime factor of a quadratic polynomial, Acta Mathematica 117 (1967), pp. 281-299.
Jori Merikoski, Largest prime factor of n^2+1, arXiv:1908.08816 [math.NT], 2019.
R. Ozols, Problem 11831, The American Mathematical Monthly, Vol. 122, No. 4 (April 2015), page 390
A. Schinzel, On two theorems of Gelfond and some of their applications, Acta Arith. 13 (1967), 177--236.
Carl Størmer, Quelques théorèmes sur l'équation de Pell x^2 - Dy^2 = +-1 et leurs applications (in French), Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl. I (2), 48 pp.

Includes primes from A002496.
Cf. A002144 (Pythagorean primes: primes of form 4n+1).
Cf. A256011.
Cf. A076605 (largest prime factor of n^2 - 1).

List([1..60],n->Reversed(Factors(n^2+1))[1]); # Muniru A Asiru, Oct 27 2018

[Maximum(PrimeDivisors(n^2+1)): n in [1..60]]; // Vincenzo Librandi, Jun 17 2015

seq(max(numtheory:-factorset(n^2+1)),n=1..100) ; # Robert Israel, Jun 11 2015

Table[FactorInteger[n^2+1,FactorComplete->True][[ -1,1]],{n,5!}] ..and/or.. Table[Last[Table[ #[[1]]]&/@FactorInteger[n^2+1]],{n,5!}] ..and/or.. PrimeFactors[n_]:=Flatten[Table[ #[[1]],{1}]&/@FactorInteger[n]]; Table[PrimeFactors[n^2+1][[ -1]],{n,5!}] (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)
a[ n_] := If[ n < 1, 0, FactorInteger[n n + 1][[All, 1]] // Last]; (* Michael Somos, May 11 2015 *)
Table[FactorInteger[n^2 + 1][[-1, 1]], {n, 80}] (* Vincenzo Librandi, Jun 17 2015 *)

largeasqp1(m) = { for(a=1,m, y=a^2 + 1; f = factor(y); v = component(f,1); v1 = v[length(v)]; print1(v1",") ) } \\ Cino Hilliard, Jun 12 2004

{a(n) = if( n<1, 0, Vecrev(factor(n*n + 1)[,1])[1])}; /* Michael Somos, May 11 2015 */

a(n) = A006530(1+n^2). - R. J. Mathar, Jan 28 2017

A175607 Largest number k such that the greatest prime factor of k^2 - 1 is prime(n).

Original entry on oeis.org

3, 17, 161, 8749, 19601, 246401, 672281, 23718421, 10285001, 354365441, 3222617399, 9447152318, 127855050751, 842277599279, 2218993446251, 2907159732049, 41257182408961, 63774701665793, 25640240468751, 238178082107393, 4573663454608289, 19182937474703818751, 34903240221563713, 332110803172167361, 99913980938200001

Offset: 1

Charles R Greathouse IV, Jul 23 2010

For any prime p, there are finitely many k such that k^2-1 has p as its largest prime factor.
For every prime p, is there some k where the greatest prime factor of k^2-1 is p? Answer from Artur Jasinski, Oct 22 2010: Yes.
As mentioned by Luca and Najman, this problem is closely related to the one in A002071.
The terms give an upper bound with a method for the simultaneous computation of logarithms of small primes, see the fxtbook link. - Joerg Arndt, Jul 03 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..25
Joerg Arndt, Matters Computational (The Fxtbook), section 32.4, pp.632-633.
Florian Luca and Filip Najman, "On the largest prime factor of x^2-1", Mathematics of Computation 80:273 (2011), pp. 429-435. (Paper has errata that was posted on the MOC website.)
Filip Najman, Home Page (gives all 16223 numbers k such that k^2-1 has no prime factor greater than 97)

Cf. A214093 (largest primes p such that the greatest prime factor of p^2-1 is prime(n)).
Cf. A076605 (largest prime divisor of n^2-1).
Cf. A285283 (equivalent for k^2+1). - Tomohiro Yamada, Apr 22 2017
Cf. A006530, A005563. - M. F. Hasler, Jun 13 2018

/* up to term for p=97 */
/* S[] is the list computed by Filip Najman (16223 elements) */
S=[2,3,4, ... ,332110803172167361, 19182937474703818751];
lpf(n)={ vecmax(factor(n)[, 1]) } /* largest prime factor */
{ forprime (p=2, 97,
  t = 0;
  for (n=1,#S, if ( lpf(S[n]^2-1)==p, t=n ) );
  print1(S[t],", ");
);}
/* Joerg Arndt, Jul 03 2012 */

More terms (using Filip Najman's list) by Joerg Arndt, Jul 03 2012

A074399 a(n) is the largest prime divisor of n(n+1).

Original entry on oeis.org

2, 3, 3, 5, 5, 7, 7, 3, 5, 11, 11, 13, 13, 7, 5, 17, 17, 19, 19, 7, 11, 23, 23, 5, 13, 13, 7, 29, 29, 31, 31, 11, 17, 17, 7, 37, 37, 19, 13, 41, 41, 43, 43, 11, 23, 47, 47, 7, 7, 17, 17, 53, 53, 11, 11, 19, 29, 59, 59, 61, 61, 31, 7, 13, 13, 67, 67, 23, 23, 71, 71, 73, 73, 37, 19

Offset: 1

N. J. A. Sloane, Nov 29 2002

Størmer shows that a(n) tends to infinity with n. Pólya generalized this result to other polynomials.
Kotov shows that a(n) >> log log n. - Charles R Greathouse IV, Mar 26 2012
Keates and Schinzel give effective constants for the above; in particular the latter shows that lim inf a(n)/log log n >= 2/7. - Charles R Greathouse IV, Nov 12 2012
Erdős conjectures ("on very flimsy probabilistic grounds") that for every e > 0, a(n) < (log n)^(2+e) infinitely often, while a(n) < (log n)^(2-e) only finitely often. - Charles R Greathouse IV, Mar 11 2015

S. V. Kotov, The greatest prime factor of a polynomial (in Russian), Mat. Zametki 13 (1973), pp. 515-522.
K. Mahler, Über den größten Primteiler spezieller Polynome zweiten Grades, Archiv for mathematik og naturvidenskab 41:6 (1934), pp. 3-26.
Georg Pólya, Zur arithmetischen Untersuchung der Polynome, Math. Zeitschrift 1 (1918), pp. 143-148.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
P. Erdős, Problems and results on number theoretic properties of consecutive integers and related questions, Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975), Congress. Numer. XVI , pp. 25-44, Utilitas Math., Winnipeg, Man., 1976.
M. Keates, On the greatest prime factor of a polynomial (1968), pp. 301-303.
Hector Pasten, The largest prime factor of n^2+1 and improvements on subexponential ABC, arXiv:2312.03566 [math.NT] (2024)
A. Schinzel, On two theorems of Gelfond and some of their applications, Acta Arithmetica 13:2 (1967-1968), pp. 177-236.
Carl Størmer, Quelques théorèmes sur l'équation de Pell x^2 - Dy^2 = +-1 et leurs applications (in French), Skrifter udgivne af Videnskabsselskabet i Christiania: Mathematisk-naturvidenskabelig Klasse (1897).

With A037464, the bisections of A076605.
Essentially the same as A069902.
Positions of primes <= p: A085152 (p=5), A085153 (p=7), A252494 (p=11), A252493 (p=13), A252492 (p=17).
Last position of each prime: A002072.

Table[ Last[ Table[ # [[1]]] & /@ FactorInteger[n^2 - 1]], {n, 3, 160, 2}]
Table[FactorInteger[n(n+1)][[-1,1]],{n,80}] (* Harvey P. Dale, Sep 28 2021 *)

gpf(n)=my(f=factor(n)[,1]); f[#f]
a(n)=if(n<3, n+1, max(gpf(n),gpf(n+1))) \\ Charles R Greathouse IV, Sep 14 2015

a(n) = Max (A006530(2n), A006530(2n+2)).

Pasten proves that a(n) >> (log log n)^2/(log log log n), see Corollary 1.5. - Charles R Greathouse IV, Oct 14 2024

Extended by Robert G. Wilson v, Dec 02 2002

A181452 Numbers k such that 17 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

16, 33, 35, 50, 67, 69, 101, 103, 118, 120, 169, 188, 239, 271, 307, 339, 441, 511, 545, 577, 749, 883, 1121, 1189, 1376, 1429, 1665, 1871, 2024, 2177, 2311, 2449, 2549, 3401, 4115, 4861, 4999, 5201, 9827, 11663, 24751, 28799, 57121, 62425, 74359, 388961, 672281

Offset: 1

Artur Jasinski, Oct 21 2010

Numbers k such that A076605(k) = 17.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(7) = 672281; primepi(17) = 7.

Artur Jasinski, Table of n, a(n) for n = 1..47

Cf. A076605, A175607, A181447-A181451, A181453-A181470, A181568.

[ n: n in [2..350000] | m eq 17 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 19 2011

p:=(97*89*83*79*73*71)^5 *(67*61*59*53*47*43*41)^6 *(37*31*29)^7 *(23*19*17)^8 *13^9 *11^10 *7^13 *5^15 *3^23 *2^36; [ n: n in [2..700000] | p mod (n^2-1) eq 0 and (D[#D] eq 17 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 24 2011

jj = 2^36*3^23*5^15*7^13*11^10*13^9*17^8*19^8*23^8*29^7*31^7*37^7*41^6 *43^6*47^6*53^6*59^6*61^6*67^6*71^5*73^5*79^5*83^5*89^5*97^5; rr = {}; n = 2; While[n < 700000, If[GCD[jj, n^2 - 1] == n^2 - 1, k = FactorInteger[n^2 - 1]; kk = Last[k][[1]]; If[kk == 17, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[680000], FactorInteger[#^2-1][[-1, 1]]==17&]

is(n)=n=n^2-1; forprime(p=2, 13, n/=p^valuation(n, p)); n>1 && 17^valuation(n, 17)==n \\ Charles R Greathouse IV, Jul 01 2013

A181453 Numbers k such that 19 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

18, 20, 37, 39, 56, 77, 113, 134, 151, 153, 170, 191, 246, 265, 305, 324, 341, 362, 379, 417, 419, 571, 626, 647, 664, 685, 721, 799, 911, 951, 989, 1025, 1616, 1937, 2431, 2661, 2889, 3041, 3079, 3212, 3457, 3970, 4751, 4863, 5851, 6271, 6499, 8399, 11551, 11857

Offset: 1

Artur Jasinski, Oct 21 2010

Numbers k such that A076605(k) = 19.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(8) = 23718421; primepi(19) = 8.

Artur Jasinski, Table of n, a(n) for n = 1..72

Cf. A076605, A175607, A181447-A181452, A181454-A181470, A181568.

[ n: n in [2..300000] | m eq 19 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 18 2011

p:=(97*89*83*79*73*71)^5 *(67*61*59*53*47*43*41)^6 *(37*31*29)^7 *(23*19*17)^8 *13^9 *11^10 *7^13 *5^15 *3^23 *2^36; [ n: n in [2..24000000] | p mod (n^2-1) eq 0 and (D[#D] eq 19 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 24 2011

jj=2^36*3^23*5^15*7^13*11^10*13^9*17^8*19^8*23^8*29^7*31^7*37^7*41^6 *43^6*47^6*53^6*59^6*61^6*67^6*71^5*73^5*79^5*83^5*89^5*97^5; rr ={};n = 2; While[n < 24000000, If[GCD[jj, n^2 - 1] == n^2 - 1, k = FactorInteger[n^2 - 1]; kk = Last[k][[1]]; If[kk == 19, AppendTo[rr, n]]]; n++ ]; rr
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==19&]

is(n)=n=n^2-1; forprime(p=2, 17, n/=p^valuation(n, p)); n>1 && 19^valuation(n, 19)==n \\ Charles R Greathouse IV, Jul 01 2013

A181455 Numbers k such that 29 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

28, 57, 59, 86, 115, 144, 146, 175, 231, 233, 289, 349, 376, 407, 436, 463, 494, 521, 579, 639, 666, 755, 811, 987, 1101, 1103, 1217, 1275, 1451, 1565, 1567, 1681, 2029, 2089, 2551, 2872, 2899, 3191, 3249, 3365, 4001, 4003, 4351, 4409, 4523, 4929, 5279

Offset: 1

Artur Jasinski, Oct 21 2010

Numbers k such that A076605(k) = 29.
Sequence is finite. For proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(10) = 354365441; primepi(29) = 10.

Artur Jasinski, Table of n, a(n) for n = 1..126

Cf. A076605, A175607, A181447-A181470, A181568.

[ n: n in [2..6000] | m eq 29 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 17 2011

Select[Range[5600],FactorInteger[#^2-1][[-1,1]]==29&]  (* Harvey P. Dale, Feb 16 2011 *)

is(n)=n=n^2-1; forprime(p=2, 23, n/=p^valuation(n, p)); n>1 && 29^valuation(n, 29)==n \\ Charles R Greathouse IV, Jul 01 2013

A181457 Numbers k such that 37 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

36, 38, 73, 75, 149, 186, 221, 223, 260, 295, 369, 371, 406, 443, 482, 519, 593, 628, 776, 813, 815, 961, 1000, 1072, 1259, 1331, 1333, 1405, 1407, 1444, 1481, 1701, 1814, 1849, 1886, 1923, 1999, 2071, 2367, 2591, 2663, 2737, 2887, 2959, 3329, 3331, 3403

Offset: 1

Artur Jasinski, Oct 21 2010

Numbers k such that A076605(k) = 37.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(12) = 9447152318; primepi(37) = 12.

Artur Jasinski, Table of n, a(n) for n = 1..208

Cf. A076605, A175607, A181447-A181456, A181458-A181470, A181568.

[ n: n in [2..300000] | m eq 37 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 19 2011

p:=(97*89*83*79*73*71)^5 *(67*61*59*53*47*43*41)^6 *(37*31*29)^7 *(23*19*17)^8 *13^9 *11^10 *7^13 *5^15 *3^23 *2^36; [ n: n in [2..50000000] | p mod (n^2-1) eq 0 and (D[#D] eq 37 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 20 2011

jj = 2^36*3^23*5^15*7^13*11^10*13^9*17^8*19^8*23^8*29^7*31^7*37^7*41^6 *43^6*47^6*53^6*59^6*61^6*67^6*71^5*73^5*79^5*83^5*89^5*97^5; rr = {}; n = 2; While[n < 3222617400, If[GCD[jj, n^2 - 1] == n^2 - 1, k = FactorInteger[n^2 - 1]; kk = Last[k][[1]]; If[kk == 37, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==37&]

is(n)=n=n^2-1; forprime(p=2, 31, n/=p^valuation(n, p)); n>1 && 37^valuation(n, 37)==n \\ Charles R Greathouse IV, Jul 01 2013

A181458 Numbers k such that 41 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

40, 81, 83, 122, 124, 163, 204, 206, 247, 286, 288, 329, 409, 491, 493, 573, 575, 737, 739, 778, 901, 944, 985, 1024, 1065, 1106, 1149, 1231, 1393, 1518, 1559, 1639, 1682, 2049, 2051, 2092, 2295, 2377, 2379, 2623, 2705, 2789, 3035, 3158, 3199, 3361, 3363

Offset: 1

Artur Jasinski, Oct 21 2010

Numbers k such that A076605(k) = 41.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(13) = 127855050751; primepi(41) = 13.

Artur Jasinski, Table of n, a(n) for n = 1..262

Cf. A076605, A175607, A181447-A181457, A181459-A181470, A181568.

[ n: n in [2..300000] | m eq 41 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 19 2011

p:=(97*89*83*79*73*71)^5 *(67*61*59*53*47*43*41)^6 *(37*31*29)^7 *(23*19*17)^8 *13^9 *11^10 *7^13 *5^15 *3^23 *2^36; [ n: n in [2..50000000] | p mod (n^2-1) eq 0 and (D[#D] eq 41 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 20 2011

jj = 2^36*3^23*5^15*7^13*11^10*13^9*17^8*19^8*23^8*29^7*31^7*37^7*41^6 *43^6*47^6*53^6*59^6*61^6*67^6*71^5*73^5*79^5*83^5*89^5*97^5; rr = {}; n = 2; While[n < 3222617400, If[GCD[jj, n^2 - 1] == n^2 - 1, k = FactorInteger[n^2 - 1]; kk = Last[k][[1]]; If[kk == 41, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==41&]

is(n)=n=n^2-1; forprime(p=2, 37, n/=p^valuation(n, p)); n>1 && 41^valuation(n, 41)==n \\ Charles R Greathouse IV, Jul 01 2013

A181459 Numbers k such that 43 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

42, 44, 85, 87, 171, 173, 216, 257, 259, 300, 343, 386, 431, 474, 517, 560, 601, 687, 689, 730, 818, 859, 1074, 1117, 1119, 1289, 1291, 1332, 1420, 1549, 1633, 1721, 1805, 1891, 1977, 1979, 2108, 2321, 2495, 2665, 2667, 2751, 2753, 2794, 2925, 3095, 3484

Offset: 1

Artur Jasinski, Oct 21 2010

Numbers k such that A076605(k) = 43.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(14) = 842277599279; primepi(43) = 14.

Artur Jasinski, Table of n, a(n) for n = 1..343

Cf. A076605, A175607, A181447-A181458, A181460-A181470, A181568.

[ n: n in [2..300000] | m eq 43 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 19 2011

p:=(97*89*83*79*73*71)^5 *(67*61*59*53*47*43*41)^6 *(37*31*29)^7 *(23*19*17)^8 *13^9 *11^10 *7^13 *5^15 *3^23 *2^36; [ n: n in [2..50000000] | p mod (n^2-1) eq 0 and (D[#D] eq 43 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 20 2011

jj = 2^36*3^23*5^15*7^13*11^10*13^9*17^8*19^8*23^8*29^7*31^7*37^7*41^6 *43^6*47^6*53^6*59^6*61^6*67^6*71^5*73^5*79^5*83^5*89^5*97^5; rr = {}; n = 2; While[n < 3222617400, If[GCD[jj, n^2 - 1] == n^2 - 1, k = FactorInteger[n^2 - 1]; kk = Last[k][[1]]; If[kk == 43, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==43&]

is(n)=n=n^2-1; forprime(p=2, 41, n/=p^valuation(n, p)); n>1 && 43^valuation(n, 43)==n \\ Charles R Greathouse IV, Jul 01 2013

cp's OEIS Frontend

A037464 Bisection of A076605.

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A014442 Largest prime factor of n^2 + 1.

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A175607 Largest number k such that the greatest prime factor of k^2 - 1 is prime(n).

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A074399 a(n) is the largest prime divisor of n(n+1).

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A181452 Numbers k such that 17 is the largest prime factor of k^2 - 1.

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A181453 Numbers k such that 19 is the largest prime factor of k^2 - 1.

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A181455 Numbers k such that 29 is the largest prime factor of k^2 - 1.

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