A002148 -id:A002148 - cp's OEIS frontend

A176255 Numbers of the form 4k-1 with least prime divisor of the form 4m+1.

35, 55, 95, 115, 155, 175, 215, 235, 247, 275, 295, 299, 323, 335, 355, 391, 395, 403, 415, 455, 475, 515, 527, 535, 559, 575, 595, 611, 635, 655, 695, 715, 731, 755, 767, 775, 799, 815, 835, 871, 875, 895, 899, 923, 935, 955, 995, 1003, 1015, 1027, 1055

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. A002148, A002145, A016813, A004767.

[n: n in [1..1500] |  (n mod 4 eq 3) and (Min(PrimeFactors(n)) mod 4) eq 1]; // Vincenzo Librandi, Feb 07 2016

A020639 := proc(n) if n = 1 then 1; else min(op(numtheory[factorset](n))) ; end if; end proc:
isA176255 := proc(n) (n mod 4 = 3) and ( A020639(n) mod 4 = 1) ; end proc:
for n from 3 to 1200 by 4 do if isA176255(n) then printf("%d,",n); end if; end do:
# R. J. Mathar, Oct 30 2010

Select[4 Range@ 265 - 1, Mod[#, 4] == 1 &[FactorInteger[#][[1, 1]]] &] (* Michael De Vlieger, Feb 07 2016 *)

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

Terms > 559 from R. J. Mathar, Oct 30 2010

A002146 Smallest prime == 7 (mod 8) where Q(sqrt(-p)) has class number 2n+1.

Original entry on oeis.org

7, 23, 47, 71, 199, 167, 191, 239, 383, 311, 431, 647, 479, 983, 887, 719, 839, 1031, 1487, 1439, 1151, 1847, 1319, 3023, 1511, 1559, 2711, 4463, 2591, 2399, 3863, 2351, 3527, 3719

Offset: 0

N. J. A. Sloane, Mira Bernstein

nonn

Conjecture: a(n) < A002148(n) for all n >= 1. - Jianing Song, Jul 20 2022

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David Broadhurst and T. D. Noe, Table of n, a(n) for n = 0..28603
D. Shanks, Review of R. B. Lakein and S. Kuroda, Tables of class numbers h(-p) for fields Q(sqrt(-p)), p <= 465071, Math. Comp., 24 (1970), 491-492.

Cf. A002147, A002148, A060651, A002143 (class numbers).

a(n) = forprime(p=2, oo, if ((p % 8) == 7, if (qfbclassno(-p) == 2*n+1, return(p)))); \\ Michel Marcus, Jul 20 2022

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

Original entry on oeis.org

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

A060649 Smallest number k==3 (mod 4) such that Q(sqrt(-k)) has class number n, or 0 if no such k exists.

Original entry on oeis.org

3, 15, 23, 39, 47, 87, 71, 95, 199, 119, 167, 231, 191, 215, 239, 399, 383, 335, 311, 455, 431, 591, 647, 695, 479, 551, 983, 831, 887, 671, 719, 791, 839, 1079, 1031, 959, 1487, 1199, 1439, 1271, 1151, 1959, 1847, 1391, 1319, 2615, 3023, 1751, 1511, 1799

Offset: 1

Robert G. Wilson v, Apr 17 2001

nonn

From Jianing Song, May 08 2021: (Start)
Conjecture 1: a(n) > 0 for all n;
Conjecture 2: a(n) = o(n^2). (End)
Conjecture: this is also the smallest absolute value of negative fundamental discriminant d for class number n. This is to say, for even n, if a(n) > 0 and A344072(n/2) > 0, then A344072(n/2) > a(n). - Jianing Song, Oct 03 2022

Jianing Song, Table of n, a(n) for n = 1..1162
Eric Weisstein's World of Mathematics, Class Number.

Cf. A002148, A081319, A344072, A038552.

Mathematica
```
(* First do <
				
```

a(n) = my(d=3); while(!isfundamental(-d) || qfbclassno(-d)!=n, d+=4); d \\ Jianing Song, May 08 2021

Edited by Dean Hickerson, Mar 17 2003

Escape clause added by Jianing Song, May 08 2021

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

A355876 Smallest prime p == 1 (mod 8) such that Q(sqrt(p)) has class number 2n+1.

Original entry on oeis.org

17, 257, 401, 577, 1129, 1297, 13033, 11321, 11257, 38569, 7057, 23593, 27689, 8761, 56857, 284561, 63361, 25601, 24337, 55441, 458929, 14401, 32401, 78401, 70969, 69697, 376897, 106537, 41617, 160001, 193601, 57601, 197137, 367721, 414433, 1506473, 444089, 331777, 156817

Offset: 0

Jianing Song, Jul 20 2022

nonn

It seems that a(n) < A355877(n) for most n. a(n) > A355877(n) for n = 0, 1, 6, 9, 15, 20, 35, ...

			p = 257 is the smallest prime congruent to 1 modulo 8 such that Q(sqrt(p)) has class number 3, so a(1) = 257.

Jianing Song, Table of n, a(n) for n = 0..57
Wikipedia, Class numbers of quadratic fields
Index entries for sequences related to quadratic fields

Cf. A355878.

Similar sequences: A002148 (p == 3 (mod 8)), A355877 (p == 5 (mod 8)), A002146 (p == 7 (mod 8)).

a(n) = forprime(p=2, oo, if(p%8==1 && qfbclassno(p)==2*n+1, return(p)))

A002149 Largest prime p==3 (mod 8) such that Q(sqrt(-p)) has class number 2n+1.

Original entry on oeis.org

163, 907, 2683, 5923, 10627, 15667, 20563, 34483, 37123, 38707, 61483, 90787, 93307, 103387, 166147, 133387, 222643, 210907, 158923, 253507, 296587

Offset: 0

N. J. A. Sloane

nonn

Most of these values are only conjectured to be correct.
Apr 15 2008: David Broadhurst says: I computed class numbers for prime discriminants with |D| < 10^9, but stopped when the first case with |D| > 5*10^8 was observed. That factor of 2 seems to me to be a reasonable margin of error, when you look at the pattern of what is included.
Arno, Robinson, & Wheeler prove a(0)-a(11). - Charles R Greathouse IV, Apr 25 2013

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David Broadhurst, Table of n, a(n) for n = 0..739 (conjectural; see comment)
Steven Arno, M. L. Robinson, and Ferrell S. Wheeler, Imaginary quadratic fields with small odd class number, Acta Arith. 83 (1998), pp. 295-330.
D. Shanks, Review of R. B. Lakein and S. Kuroda, Tables of class numbers h(-p) for fields Q(sqrt(-p)), p <= 465071, Math. Comp., 24 (1970), 491-492.

Cf. A002148, A003173, A006203.

Edited by Dean Hickerson, Mar 17 2003

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A176255 Numbers of the form 4k-1 with least prime divisor of the form 4m+1.

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A002146 Smallest prime == 7 (mod 8) where Q(sqrt(-p)) has class number 2n+1.

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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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A060649 Smallest number k==3 (mod 4) such that Q(sqrt(-k)) has class number n, or 0 if no such k exists.

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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.