A155716 -id:A155716 - cp's OEIS frontend

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

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

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","))

A380295 Numbers that can be written as a^2 + 3b^2 for some a, b in A155716 and also as c^2 + 6d^2 for some c, d in A092572.

Original entry on oeis.org

1552, 1975, 4753, 5047, 5425, 7825, 8167, 9175, 10096, 11025, 11536, 12007, 16528, 16807, 16993, 18823, 19600, 23863, 24832, 25633, 25767, 26983, 27223, 29200, 30919, 31600, 31927, 32791, 33175, 35329, 35623, 41953, 43063, 43687, 51943, 54775, 57303, 59575, 60016, 61783, 63175, 71575, 72103

Offset: 1

Robert Israel, Jan 19 2025

nonn

If k is a term, then so is j^4 * k for all positive integers j.

			a(5) = 5425 is a term because 5425 = 73^2 + 6 * 4^2 = 25^2 + 3 * 40^2 with 73 = 5^2 + 3 * 4^2, 4 = 1^2 + 3 * 1^2, 25 = 1^2 + 6 * 2^2 and 40 = 4^2 + 6 * 2^2.

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

Cf. A092572, A155712, A155716.

N:= 500: # for terms <= N^2
A:= {seq(seq(a^2 + 3*b^2,a=1..floor(sqrt(N-3*b^2))),b=1..floor(sqrt(N/3)))}:
B:= {seq(seq(a^2 + 6*b^2,a=1..floor(sqrt(N-6*b^2))),b=1..floor(sqrt(N/6)))}:
C:= select(`<=`,{seq(seq(a^2+6*b^2,a=A),b=A)},N^2):
E:= select(`<=`,{seq(seq(a^2+3*b^2,a=B),b=B)},N^2):
sort(convert(C intersect E,list));

A155713 Intersection of A154778 and A155716: N = a^2 + 5b^2 = c^2 + 6d^2 for some positive integers a,b,c,d.

Original entry on oeis.org

49, 70, 105, 145, 150, 166, 196, 214, 225, 241, 249, 280, 294, 321, 406, 409, 420, 441, 454, 505, 580, 600, 601, 609, 630, 664, 681, 694, 721, 726, 745, 769, 784, 841, 856, 870, 886, 889, 900, 934, 945, 964, 996, 1009, 1030, 1041, 1089, 1120, 1126, 1129

Offset: 1

M. F. Hasler, Jan 25 2009

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

isA155713(n,/* optional 2nd arg allows us to get other sequences */c=[6,5]) = { 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, isA155713(n) & print1(n","))

A155578 Intersection of A000404 and A155717: N = a^2 + b^2 = c^2 + 7*d^2 for some positive integers a,b,c,d.

Original entry on oeis.org

8, 29, 32, 37, 53, 72, 109, 113, 116, 128, 137, 148, 149, 193, 197, 200, 212, 232, 233, 261, 277, 281, 288, 296, 317, 333, 337, 373, 389, 392, 400, 401, 421, 424, 436, 449, 452, 457, 464, 477, 512, 541, 548, 557, 569, 592, 596, 613, 617, 641, 648, 653, 673

Offset: 1

M. F. Hasler, Jan 25 2009

Subsequence of A155568 (where a,b,c,d may be zero).

Cf. A000404, A154777, A092572, A097268, A154778, A155716, A155717.

isA155578(n,/* optional 2nd arg allows us to get other sequences */c=[7,1]) = { 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, isA155578(n) & print1(n","))

from math import isqrt
def aupto(limit):
    cands = range(1, isqrt(limit)+1)
    left =  set(a**2 +   b**2 for a in cands for b in cands)
    right = set(c**2 + 7*d**2 for c in cands for d in cands)
    return sorted(k for k in left & right if k <= limit)
print(aupto(673)) # Michael S. Branicky, Aug 29 2021

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

Original entry on oeis.org

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","))

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

cp's OEIS Frontend

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

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

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

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A380295 Numbers that can be written as a^2 + 3*b^2 for some a, b in A155716 and also as c^2 + 6*d^2 for some c, d in A092572.

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A155713 Intersection of A154778 and A155716: N = a^2 + 5b^2 = c^2 + 6d^2 for some positive integers a,b,c,d.

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A155578 Intersection of A000404 and A155717: N = a^2 + b^2 = c^2 + 7*d^2 for some positive integers a,b,c,d.

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

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

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

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A380295 Numbers that can be written as a^2 + 3b^2 for some a, b in A155716 and also as c^2 + 6d^2 for some c, d in A092572.