A176255 -id:A176255 - cp's OEIS frontend

A176256 Numbers of the form 4k+1 with least prime divisor of the form 4m-1.

9, 21, 33, 45, 49, 57, 69, 77, 81, 93, 105, 117, 121, 129, 133, 141, 153, 161, 165, 177, 189, 201, 209, 213, 217, 225, 237, 249, 253, 261, 273, 285, 297, 301, 309, 321, 329, 333, 341, 345, 357, 361, 369, 381, 393, 405, 413, 417, 429, 437, 441, 453, 465, 469

Offset: 1

Vladimir Shevelev, Apr 13 2010

nonn

By definition, all terms are composite numbers.

Cannot be the hypotenuse of a primitive Pythagorean triangle. - Robert G. Wilson v, Mar 16 2014

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

Cf. A176255, A002148, A002145, A016813, A004767.

Complement of A020882 in 1 == Mod 4.

fQ[n_] := Mod[ n, 4] == 1 && Mod[ FactorInteger[n][[1, 1]], 4] == 3; Select[Range@470, fQ] (* Robert G. Wilson v, Apr 08 2014 *)

isok(n) = ((n % 4) == 1) && (f = factor(n)) && ((f[1, 1] % 4) == 3); \\ Michel Marcus, Mar 16 2014

More terms from Michel Marcus, Mar 16 2014

A176257 Numbers of the form 4k-1 with greatest prime divisor of the form 4m+1.

Original entry on oeis.org

15, 39, 51, 75, 87, 91, 111, 119, 123, 135, 143, 159, 183, 187, 195, 203, 219, 255, 259, 267, 287, 291, 303, 319, 327, 339, 351, 371, 375, 407, 411, 427, 435, 447, 451, 455, 459, 471, 507, 511, 519, 543, 551, 555, 579, 583, 591, 595, 615, 623, 663, 667, 671

Offset: 1

Vladimir Shevelev, Apr 13 2010

nonn

By definition, all terms are composite numbers.

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

Cf. A176256, A176255, A002148, A002145, A016813, A004767.

Select[4*Range[200]-1,Divisible[FactorInteger[#][[-1,1]]-1,4]&] (* Harvey P. Dale, May 17 2013 *)

isok(n) = ((n % 4) == 3) && ((vecmax(factor(n)[,1]) % 4) == 1); \\ Michel Marcus, Feb 07 2016

Corrected and extended by Harvey P. Dale, May 17 2013

A176258 Numbers of the form 4k+1 with greatest prime divisor of the form 4m-1.

Original entry on oeis.org

9, 21, 33, 49, 57, 69, 77, 81, 93, 105, 121, 129, 133, 141, 161, 165, 177, 189, 201, 209, 213, 217, 237, 245, 249, 253, 285, 297, 301, 309, 321, 329, 341, 345, 361, 381, 385, 393, 413, 417, 437, 441, 453, 465, 469, 473, 489, 497, 501, 513, 517, 525, 529, 537, 553, 573

Offset: 1

Vladimir Shevelev, Apr 13 2010

nonn

All terms of A107978 are in the sequence.

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

Cf. A176257, A176256, A176255, A107978, A002148, A002145, A016813, A004767.

Select[4 Range@ 150 + 1, Mod[#, 4] == 3 &[FactorInteger[#][[-1, 1]]] &] (* Michael De Vlieger, Feb 07 2016 *)

isok(n) = (n != 1) && ((n % 4) == 1) && ((vecmax(factor(n)[,1]) % 4) == 3); \\ Michel Marcus, Feb 07 2016

Corrected and extended by Michel Marcus, Feb 07 2016

A176262 Numbers of the form 3k+1 with greatest prime divisor of the form 3m-1.

Original entry on oeis.org

4, 10, 16, 22, 25, 34, 40, 46, 55, 58, 64, 82, 85, 88, 94, 100, 106, 115, 118, 121, 136, 142, 145, 154, 160, 166, 178, 184, 187, 202, 205, 214, 220, 226, 232, 235, 238, 250, 253, 256, 262, 265, 274, 289, 295, 298, 319, 322, 328, 334, 340, 346, 352, 355, 358, 376

Offset: 1

Vladimir Shevelev, Apr 13 2010

nonn

All numbers of the form 2p, where p==2(mod 3) is prime, are in the sequence.

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

Cf. A176258, A176257, A176256, A176255, A002148, A002145, A016813, A004767.

Select[3 Range@ 120 + 1, Mod[#, 3] == 2 &[FactorInteger[#][[-1, 1]]] &] (* Michael De Vlieger, Feb 07 2016 *)

isok(n) = ((n % 3) == 1) && (n != 1) && ((vecmax(factor(n)[,1]) % 3) == 2); \\ Michel Marcus, Feb 07 2016

Corrected and extended by Michel Marcus, Feb 07 2016

A176274 Numbers of the form 3k-1 with greatest prime divisor of the form 3m+1.

Original entry on oeis.org

14, 26, 35, 38, 56, 62, 65, 74, 86, 95, 98, 104, 122, 134, 140, 143, 146, 152, 155, 158, 182, 185, 194, 206, 209, 215, 218, 224, 245, 248, 254, 260, 266, 278, 296, 302, 305, 314, 323, 326, 335, 338, 341, 344, 350, 362, 365, 380, 386, 392, 395, 398, 407, 416, 422, 434, 446

Offset: 1

Vladimir Shevelev, Apr 14 2010

nonn

All numbers of the form 2p, where p==1(mod 3) is prime, are in the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A176262, A176258, A176257, A176256, A176255, A002148, A002145, A016813, A004767.

Select[Range[500],Divisible[#+1,3]&&Divisible[FactorInteger[#] [[-1,1]]-1, 3]&] (* Harvey P. Dale, Jul 29 2019 *)

isok(n) = ((n % 3) == 2) && ((vecmax(factor(n)[,1]) % 3) == 1); \\ Michel Marcus, Feb 08 2016

More terms from Michel Marcus, Feb 08 2016

A176275 Numbers of the form 6k+1 with the least prime divisor of the form 6m-1.

Original entry on oeis.org

25, 55, 85, 115, 121, 145, 175, 187, 205, 235, 253, 265, 289, 295, 319, 325, 355, 385, 391, 415, 445, 451, 475, 493, 505, 517, 529, 535, 565, 583, 595, 625, 649, 655, 667, 685, 697, 715, 745, 775, 781, 799, 805, 835, 841, 865, 895, 901, 913, 925, 943, 955, 979, 985, 1003, 1015, 1045, 1075

Offset: 1

Vladimir Shevelev, Apr 14 2010

nonn

All terms of A108166 are in the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A176274, A176262, A176256, A176257, A176258, A176255, A108166, A016969, A007528.

Select[6*Range[200] + 1, IntegerQ[(FactorInteger[#][[1, 1]] + 1)/6] &] (* Harvey P. Dale, Sep 19 2018 *)

isok(n) = ((n % 6) == 1) && (n != 1) && ((vecmin(factor(n)[,1]) % 6) == 5); \\ Michel Marcus, Feb 08 2016

Corrected by R. J. Mathar, May 03 2013

A218459 a(n) is the smallest positive integer d such that prime(n) = x^2 + dy^2 has a solution (x,y) in integers.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 1, 2, 7, 1, 3, 1, 1, 2, 11, 1, 2, 1, 2, 7, 1, 3, 2, 1, 1, 1, 3, 2, 1, 1, 3, 2, 1, 2, 1, 3, 1, 2, 23, 1, 2, 1, 7, 1, 1, 3, 2, 3, 2, 1, 1, 7, 1, 2, 1, 7, 1, 3, 1, 1, 2, 1, 2, 11, 1, 1, 2, 1, 2, 1, 1, 7, 3, 1, 2, 22, 1, 1, 1, 1, 2, 1, 7, 1, 3, 2, 1, 1, 1, 3, 2, 19, 3, 2, 2

Offset: 1

Alonso del Arte, Oct 29 2012

a(n) = smallest positive integer d such that prime(n) is reducible in the ring Z[sqrt(-d)].
If prime(n) == 1 or 2 mod 4, then a(n) = 1. If prime(n) == 3 mod 8, then a(n) = 2. If prime(n) == 7 mod 24 then a(n) = 3.
If prime(n) == 23 mod 24, a(n) >= 7. In particular, the above conditions are if and only if. - Charles R Greathouse IV, Oct 31 2012
a(n) = 7 if and only if prime(n) is 11, 15, or 23 mod 28. - Charles R Greathouse IV, Nov 09 2012
It appears 75% of values are 1 or 2, with the vast majority of the rest prime, though many are duplicates. Conjecture: Odd composite values belong to A176255. - Bill McEachen, Sep 03 2023

			a(1) = 1 because the first prime is 2, which is 1^2 + 1^2.
a(2) = 2 because the second prime is 3, which is 1^2 + 2*1^2, but not of the form x^2 + y^2 for any integers x, y.
a(3) = 1 because the third prime is 5, which is 2^2 + 1*1^2.
a(4) = 3 because the third prime is 7, which is 2^2 + 3*1^2, but not of the form x^2 + y^2 or x^2 + 2y^2 for any integers x, y.

Ethan D. Bolker, Elementary Number Theory: An Algebraic Approach. Mineola, New York: Dover Publications (1969, reprinted 2007): p. 68, Theorem 24.5; p. 74, Theorem 25.4.
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, Section 9, "Ring class fields and p = x^2 + n y^2." - From N. J. A. Sloane, Dec 26 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Alonso del Arte, Diagram illustrating first six terms on the complex plane

Cf. A088192, A176255.

r[n_, d_] := Reduce[ Prime[n] == x^2 + d*y^2, {x, y}, Integers]; a[n_] := For[d = 1, True, d++, If[r[n, d] =!= False, Return[d] ] ]; Table[a[n], {n, 1, 95}] (* Jean-François Alcover, Apr 04 2013 *)

ndv(d, p)=(#bnfisintnorm(bnfinit(y^2+d), p))==0
forprime(p=2, 500, for(d=1, p, if(!ndv(d, p), print1(d, ", "); break))) \\ Georgi Guninski, Oct 27 2012

check(d,p)={
   if(kronecker(-d,p)<0 || #bnfisintnorm(bnfinit('x^2+d),p)==0, return(0));
   for(y=1,sqrtint(p\d),if(issquare(p-d*y^2),return(1)));
   0
};
do(p)={
   if(p%24<23,return(if(p%4<3,1,if(p%8==3,2,3))));
   if(kronecker(p,7)>0, return(7));
   if(check(11,p), return(11));
   for(d=19,p,
    if(issquarefree(d) && check(d,p), return(d))
   )
};
apply(do, primes(100)) \\ Charles R Greathouse IV, Oct 31 2012

A218459(n)={my(p=prime(n),d);while(d++,for(y=1,sqrtint((p-1)\d), issquare(p-d*y^2)&&return(d)))} \\ M. F. Hasler, May 05 2013

a(n) >= A088192(n). - Charles R Greathouse IV, Oct 31 2012

a(76) corrected by Charles R Greathouse IV, Nov 13 2012

Edited by N. J. A. Sloane, Dec 07 2012, Dec 26 2012

A176278 Numbers of the form 6k-1 with the least prime divisor of the form 6m+1.

Original entry on oeis.org

77, 119, 161, 203, 221, 287, 299, 329, 371, 377, 413, 437, 497, 533, 539, 551, 581, 611, 623, 689, 707, 749, 767, 779, 791, 833, 893, 917, 923, 959, 1001, 1007, 1043, 1079, 1121, 1127, 1157, 1169, 1211, 1253, 1271, 1313, 1337, 1349, 1379, 1391, 1421, 1457

Offset: 1

Vladimir Shevelev, Apr 14 2010

nonn

By definition, all terms are composite numbers.

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

Cf. A176275, A176274, A176262, A176258, A176257, A176256, A176255, A108166, A016969, A007528.

Select[Range[11,2581,6],1==Mod[FactorInteger[ # ][[1,1]],6]&] (* Zak Seidov, Apr 14 2010 *)

isok(n) = ((n % 6) == 5) && ((vecmin(factor(n)[,1]) % 6) == 1); \\ Michel Marcus, Feb 08 2016

Corrected (erroneous 341 deleted) and extended by Zak Seidov, Apr 14 2010

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A176256 Numbers of the form 4k+1 with least prime divisor of the form 4m-1.

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A176257 Numbers of the form 4k-1 with greatest prime divisor of the form 4m+1.

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A176258 Numbers of the form 4k+1 with greatest prime divisor of the form 4m-1.

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A176262 Numbers of the form 3k+1 with greatest prime divisor of the form 3m-1.

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A176274 Numbers of the form 3k-1 with greatest prime divisor of the form 3m+1.

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A176275 Numbers of the form 6k+1 with the least prime divisor of the form 6m-1.

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A218459 a(n) is the smallest positive integer d such that prime(n) = x^2 + dy^2 has a solution (x,y) in integers.

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