A154778 -id:A154778 - cp's OEIS frontend

A155717 Numbers of the form N = a^2 + 7b^2 for some positive integers a,b.

8, 11, 16, 23, 29, 32, 37, 43, 44, 53, 56, 64, 67, 71, 72, 77, 79, 88, 92, 99, 107, 109, 112, 113, 116, 121, 127, 128, 137, 144, 148, 149, 151, 161, 163, 172, 176, 179, 184, 191, 193, 197, 200, 203, 207, 211, 212, 224, 232, 233, 239, 253, 256, 259, 261, 263, 268

Offset: 1

M. F. Hasler, Jan 25 2009

Subsequence of A020670 (which allows for a and b to be zero).

If N=a^2+7*b^2 is a term then 7*N=(7*b)^2+7*a^2 is also a term. Conversely,if 7*N is a term then N is a term. Example: N=56; N/7=8 is a term, N*7=7^2+7*7^2 is a term. Sequences A154777, A092572 and A154778 have the same property with 7 replaced by prime numbers 2,3 and 5 respectively. - Jerzy R Borysowicz, May 22 2020

Cf. A000404, A154777, A092572, A097268, A154778, A155707-A155716, A155560-A155578.

Select[Range[300], Reduce[a>0 && b>0 && # == a^2 + 7b^2, {a, b}, Integers] =!= False&] (* Jean-François Alcover, Nov 17 2016 *)

isA155717(n,/* optional 2nd arg allows us to get other sequences */c=7) = { for(b=1,sqrtint((n-1)\c), issquare(n-c*b^2) & return(1))}
for( n=1,300, isA155717(n) & print1(n","))

def aupto(limit):
    cands = range(1, int(limit**.5)+2)
    nums = [a**2 + 7*b**2 for a in cands for b in cands]
    return sorted(set(k for k in nums if k <= limit))
print(aupto(268)) # Michael S. Branicky, Aug 11 2021

A155715 Least number expressible as a^2 + k b^2 with positive integers a,b, for each k=1,...,n.

Original entry on oeis.org

2, 17, 73, 73, 241, 241, 1009, 1009, 1009, 1009, 7561, 7561, 21961, 32356, 32356, 32356, 44641, 44641, 349924, 349924, 349924, 349924, 1399696, 1399696, 1399696, 3027249, 3027249, 3027249, 4349601, 4349601, 18567396, 18567396, 18567396

Offset: 1

M. F. Hasler, Jan 27 2009

nonn

Sequence A028372 considers primes with this property, but allowing for nonzero a,b (which obviously is irrelevant for n>2). Up to n=13, the terms of the present sequence are prime without imposing it explicitely and thus coincide with A028372 except for n=2.

a(n) > 10^9 for n >= 47. [From Donovan Johnson, Sep 29 2009]

			a(1) = 2 = 1^2 + 1^2 is the least number of the sequence A000404 (sum of positive squares). a(2) = 17 = 1^2 + 4^2 = 3^2 + 2*2^2 is the least number in sequence A000404 to be in sequence A154777 (a^2+2b^2)as well. a(3) = 73 = 3^2 + 8^2 = 1^2 + 2*6^2 = 5^2 + 3*4^2 is the least number in the intersection of sequences A000404, A154777 and A092572 (a^2+3b^2).

Donovan Johnson, Table of n, a(n) for n=1..46

Cf. A028372, A000404, A154777, A092572, A097268, A154778, A155707-A155716, A155560-A155578.

k=1; for( n=1,10^9, forstep( c=k,1,-1, for( b=1,sqrtint((n-1)\c), issquare(n-c*b^2) & next(2));next(2)); print1(n",");k++;n--)

a(23)-a(46) and b-file from Donovan Johnson, Sep 29 2009

A155708 Numbers expressible as a^2 + k*b^2 with nonzero integers a,b, for k=2, k=3 and k=5.

Original entry on oeis.org

36, 129, 144, 201, 241, 324, 409, 441, 489, 516, 576, 601, 769, 804, 849, 900, 921, 964, 1009, 1129, 1161, 1201, 1249, 1296, 1321, 1489, 1521, 1569, 1609, 1636, 1641, 1764, 1801, 1809, 1849, 1929, 1956, 2064, 2089, 2161, 2169, 2281, 2304, 2361, 2404, 2521

Offset: 1

M. F. Hasler, Feb 10 2009

nonn

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

Cf. A155715, A028372, A000404, A154777, A092572, A097268, A154778, A155707-A155716, A155560-A155578.

N:= 10000: # to get all terms <= N
S[2]:= {}: S[3]:= {}: S[5]:= {}:
for a from 1 to floor(sqrt(N)) do
  for k in [2,3,5] do
    S[k]:= S[k] union {seq(a^2 + k*b^2, b = 1 .. floor(sqrt((N-a^2)/k)))}
  od
od:
R:= S[2] intersect S[3] intersect S[5]:
sort(convert(R,list)); # Robert Israel, Jul 11 2018

isA155708(n, /* optional 2nd arg allows us to get other sequences */c=[5, 3, 2]) = { for(i=1, #c, for(b=1, sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) & next(2)); return); 1}
for(n=1,9999, isA155708(n) & print1(n","))

A155712 Intersection of A092572 and A155716: N = a^2 + 3b^2 = c^2 + 6d^2 for some positive integers a,b,c,d.

Original entry on oeis.org

7, 28, 31, 49, 63, 73, 79, 97, 100, 103, 112, 124, 127, 151, 175, 193, 196, 199, 217, 223, 241, 252, 271, 279, 292, 313, 316, 337, 343, 367, 388, 400, 409, 412, 433, 439, 441, 448, 457, 463, 484, 487, 496, 508, 511, 553, 567, 577, 601, 604, 607, 631, 657, 673

Offset: 1

M. F. Hasler, Jan 25 2009

From Robert Israel, Jan 19 2025: (Start)
If k is a term, then so is j^2 * k for all positive integers j.
The primes in this sequence appear to be A033199.
(End)

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

Cf. A000404, A033199, A092572, A097268, A154777, A154778, A155716, A155717, A155560-A155578.

N:= 1000: # for terms <= N
A:= {seq(seq(a^2 + 3*b^2, b=1 .. floor(sqrt((N-a^2)/3))),a=1..floor(sqrt(N)))}
   intersect {seq(seq(c^2 + 6*d^2, d = 1 .. floor(sqrt((N-c^2)/6))),c=1..floor(sqrt(N)))}:
sort(convert(A,list)); # Robert Israel, Jan 19 2025

isA155712(n,/* optional 2nd arg allows to get other sequences */c=[6,3]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) && next(2)); return);1}
for( n=1,999, isA155712(n) && print1(n",")) \\ Update to modern PARI syntax (& -> &&) by M. F. Hasler, Jan 18 2025

A155714 Least number expressible as a^2 + p b^2 with positive integers a,b, for each prime p <= prime(n) = A000040(n).

Original entry on oeis.org

3, 12, 36, 144, 144, 4356, 4356, 4356, 7056, 17424, 176400, 2547216, 2547216, 6290064, 6780816, 6780816, 6780816, 6780816, 93315600, 93315600, 271986064, 271986064, 271986064, 271986064, 271986064, 308213136, 308213136, 308213136

Offset: 1

M. F. Hasler, Feb 10 2009

nonn

a(n) > 10^9 for n >= 33. [From Donovan Johnson, Sep 29 2009]

Donovan Johnson, Table of n, a(n) for n=1..32

Cf. A155715, A028372, A000404, A154777, A092572, A097268, A154778, A155707-A155716, A155560-A155578.

A155714(k,n=1) = { local(p); until( !n++, p=prime(k); until( !p=precprime(p-1), for( b=1, sqrtint((n-1)\p), issquare(n-p*b^2) & next(2)); next(2)); break);n}
t=1; for(k=1,30, print1(t=A155714(k,t),","))

a(12)-a(32) and b-file from Donovan Johnson, Sep 29 2009

A155574 Intersection of A154777 and A092572: N = a^2 + 2b^2 = c^2 + 3d^2 for some positive integers a,b,c,d.

Original entry on oeis.org

12, 19, 36, 43, 48, 57, 67, 73, 76, 97, 108, 129, 139, 144, 147, 163, 171, 172, 192, 193, 201, 211, 219, 228, 241, 268, 283, 291, 292, 300, 304, 307, 313, 324, 331, 337, 361, 379, 387, 388, 409, 417, 432, 433, 441, 457, 475, 484, 489, 499, 507, 513, 516, 523

Offset: 1

M. F. Hasler, Jan 25 2009

Subsequence of A155564 (where a,b,c,d may be zero).

Cf. A000404, A154777, A092572, A097268, A154778, A155716, A155717, A155560-A155578.

isA155574(n,/* optional 2nd arg allows us to get other sequences */c=[3,2]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,999, isA155574(n) & print1(n","))

A155576 Intersection of A000404 and A155716: N = a^2 + b^2 = c^2 + 6d^2 for some positive integers a,b,c,d.

Original entry on oeis.org

10, 25, 40, 58, 73, 90, 97, 100, 106, 145, 160, 193, 202, 225, 232, 241, 250, 265, 292, 298, 313, 337, 346, 360, 388, 394, 400, 409, 424, 433, 457, 490, 505, 522, 538, 577, 580, 586, 601, 625, 634, 640, 657, 673, 730, 745, 769, 772, 778, 808, 810, 841, 865

Offset: 1

M. F. Hasler, Jan 25 2009

Subsequence of A155566 (where a,b,c,d may be zero).

Cf. A000404, A154777, A092572, A097268, A154778, A155716, ...

isA155576(n,/* optional 2nd arg allows us to get other sequences */c=[6,1]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,999, isA155576(n) & print1(n","))

A155709 Intersection of A154777 and A155716: N = a^2 + 2b^2 = c^2 + 6d^2 for some positive integers a,b,c,d.

Original entry on oeis.org

22, 33, 73, 88, 97, 118, 121, 132, 150, 166, 177, 193, 198, 214, 225, 241, 249, 262, 292, 294, 297, 313, 321, 337, 352, 358, 388, 393, 409, 433, 438, 441, 454, 457, 472, 484, 502, 528, 537, 550, 577, 582, 600, 601, 649, 657, 664, 673, 681, 694, 708, 726, 753

Offset: 1

M. F. Hasler, Jan 25 2009

Subsequence of A155569 (where a,b,c,d may be zero).

Cf. A000404, A154777, A092572, A097268, A154778, A155716, ...

isA155709(n,/* optional 2nd arg allows us to get other sequences */c=[6,2]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,999, isA155709(n) & print1(n","))

A218697 Semiprimes that can be written in the form x^2 + 5*y^2 with x, y > 0.

Original entry on oeis.org

6, 9, 14, 21, 46, 49, 69, 86, 94, 129, 134, 141, 145, 161, 166, 201, 205, 206, 214, 249, 254, 301, 305, 309, 321, 326, 329, 334, 381, 445, 446, 454, 469, 489, 501, 505, 526, 529, 545, 566, 581, 614, 669, 681, 694, 721, 734, 745, 749, 766, 789, 841, 849, 886, 889

Offset: 1

Arkadiusz Wesolowski, Nov 04 2012

nonn

If two primes which end in 3 or 7 and surpass by 3 a multiple of 4 are multiplied, then their product will be composed of a square and the quintuple of another square. (Fermat (1654))

			94 = 7^2 + 5*3^2, therefore 94 is a term.

Dedekind R., Theory of Algebraic Integers, Cambridge Univ. Press, 1996 (translation of the 1877 French original), pp. 12-13.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

Cf. A001358, A154778.

n = 889; limx = Sqrt[n]; limy = Sqrt[n/5]; Select[Union@Flatten@Table[x^2 + 5*y^2, {x, limx}, {y, limy}], # <= n && PrimeOmega[#] == 2 &]
Select[Select[Range[889], PrimeOmega[#] == 2 &], Length@FindInstance[y > 0 && x^2 + 5*y^2 == #, {x, y}, Integers] > 0 &] (* Arkadiusz Wesolowski, Jan 13 2013 *)

A154778 INTERSECT A001358.

A155573 Intersection of A000404, A154777 and A092572: N = a^2 + b^2 = c^2 + 2d^2 = e^2 + 3f^2 for some positive integers a,b,c,d,e,f.

Original entry on oeis.org

73, 97, 193, 241, 292, 313, 337, 388, 409, 433, 457, 577, 601, 657, 673, 769, 772, 873, 900, 937, 964, 1009, 1033, 1129, 1153, 1156, 1168, 1201, 1249, 1252, 1297, 1321, 1348, 1489, 1521, 1552, 1609, 1636, 1657, 1732, 1737, 1753, 1777, 1801, 1825, 1828

Offset: 1

M. F. Hasler, Jan 25 2009

Cf. A000404, A154777, A092572, A097268, A154778, A155716, A155717, A155560-A155578.

isA155573(n,/* optional 2nd arg allows us to get other sequences */c=[3,2,1]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,1999, isA155573(n) & print1(n","))

cp's OEIS Frontend

A155717 Numbers of the form N = a^2 + 7b^2 for some positive integers a,b.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Python

A155715 Least number expressible as a^2 + k b^2 with positive integers a,b, for each k=1,...,n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A155708 Numbers expressible as a^2 + k*b^2 with nonzero integers a,b, for k=2, k=3 and k=5.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

A155712 Intersection of A092572 and A155716: N = a^2 + 3b^2 = c^2 + 6d^2 for some positive integers a,b,c,d.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

A155714 Least number expressible as a^2 + p b^2 with positive integers a,b, for each prime p <= prime(n) = A000040(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A155574 Intersection of A154777 and A092572: N = a^2 + 2b^2 = c^2 + 3d^2 for some positive integers a,b,c,d.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A155576 Intersection of A000404 and A155716: N = a^2 + b^2 = c^2 + 6d^2 for some positive integers a,b,c,d.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A155709 Intersection of A154777 and A155716: N = a^2 + 2b^2 = c^2 + 6d^2 for some positive integers a,b,c,d.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A218697 Semiprimes that can be written in the form x^2 + 5*y^2 with x, y > 0.

Views

Author

Keywords

Comments