A092572 -id:A092572 - cp's OEIS frontend

A092575 Number of representations of n of the form x^2 + 3y^2 with (x,y)=1.

0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0

Offset: 1

Eric W. Weisstein, Feb 28 2004

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
E. Akhtarkavan, M. F. M. Salleh and O. Sidek, Multiple Descriptions Video Coding Using Coinciding Lattice Vector Quantizer for H.264/AVC and Motion JPEG2000, World Applied Sciences Journal 21 (2): 157-169, 2013. - From _N. J. A. Sloane_, Feb 11 2013
Eric Weisstein's World of Mathematics, Eulers 6n Plus 1 Theorem

Cf. A002476, A092572, A092573, A092574.

N:= 200: # to get a(1)..a(N)
V:= Vector(N):
for y from 1 to floor(sqrt(N/3-1)) do
  for x from 1 to floor(sqrt(N-3*y^2)) do
    if igcd(x,y) = 1 then V[x^2 + 3*y^2]:= V[x^2+3*y^2]+1
    fi
od od:
convert(V,list); # Robert Israel, Apr 03 2017

r[n_] := Reduce[ x > 0 && y > 0 && GCD[x, y] == 1 && n == x^2 + 3 y^2, {x, y}, Integers]; a[n_] := Which[ r[n] === False, 0, r[n][[0]] === And, 1, True, Length[r[n]]]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, Oct 31 2012 *)

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

A092574 Positive integers that can be represented in the form x^2 + 3y^2 with (x,y) = 1 and x and y positive.

Original entry on oeis.org

4, 7, 12, 13, 19, 21, 28, 31, 37, 39, 43, 49, 52, 57, 61, 67, 73, 76, 79, 84, 91, 93, 97, 103, 109, 111, 124, 127, 129, 133, 139, 147, 148, 151, 156, 157, 163, 169, 172, 181, 183, 193, 196, 199, 201, 211, 217, 219, 223, 228, 229, 237, 241, 244, 247, 259, 268

Offset: 1

Eric W. Weisstein, Feb 28 2004

nonn

Superset of primes of the form 6n+1 (A002476).

For all proper solutions with nonnegative x and y see A244819. - Wolfdieter Lang, Mar 02 2021

E. Akhtarkavan, M. F. M. Salleh and O. Sidek, Multiple Descriptions Video Coding Using Coinciding Lattice Vector Quantizer for H.264/AVC and Motion JPEG2000, World Applied Sciences Journal 21 (2): 157-169, 2013. - From _N. J. A. Sloane_, Feb 11 2013
Eric Weisstein's World of Mathematics, Eulers 6n Plus 1 Theorem

Cf. A002476, A092572, A092573, A092575, A244819.

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

A301534 Number of ways to write the n-th prime congruent to 7 modulo 12 as x^2 + 3y^2 + 152^z with x,y,z nonnegative integers.

Original entry on oeis.org

0, 2, 3, 4, 5, 5, 2, 6, 6, 4, 7, 4, 9, 6, 6, 6, 7, 9, 5, 10, 3, 9, 7, 9, 8, 11, 9, 8, 10, 5, 8, 9, 4, 10, 7, 7, 7, 8, 7, 13, 8, 6, 6, 14, 7, 15, 3, 11, 8, 10, 8, 7, 7, 9, 6, 9, 7, 7, 10, 12, 6, 9, 4, 7, 10, 12, 12, 7, 13, 9, 12, 6, 7, 10, 5, 8, 7, 12, 12, 10

Offset: 1

Zhi-Wei Sun, Apr 16 2018

nonn

Conjecture: a(n) > 0 for all n > 1. In other words, any prime p > 7 with p == 7 (mod 12) can be written as x^2 + 3*y^2 + 15*2^z with x,y,z nonnegative integers.

We have verified the conjecture for all primes p == 7 (mod 12) with 7 < p < 8*10^9.

			a(1) = 0 since 7 cannot be written as x^2 + 3*y^2 + 15*2^z with x,y,z nonnegative integers.
a(2) = 2 since the second prime congruent to 7 modulo 12 is 19 and 19 = 1^2 + 3*1^2 + 15*2^0 = 2^2 + 3*0^2 + 15*2^0.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A000079, A000290, A092572, A299924, A299537, A299794, A300219, A300362, A300396, A300510, A301376, A301391, A301452, A301471, A301472, A302920.

p[n_]:=p[n]=Prime[n];
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[MemberQ[{2},Mod[Part[Part[f[n],i],1],3]]&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=n==0||(n>0&&g[n]);
n=0;Do[If[Mod[p[m],12]!=7,Goto[aa]];n=n+1;r=0;Do[If[QQ[p[m]-15*2^k],Do[If[SQ[p[m]-15*2^k-3x^2],r=r+1],{x,0,Sqrt[(p[m]-15*2^k)/3]}]],{k,0,Log[2,p[m]/15]}];Print[n," ",r];Label[aa],{m,1,315}]

A155575 Intersection of A000404 and A154778: N = a^2 + b^2 = c^2 + 5d^2 for some positive integers a,b,c,d.

Original entry on oeis.org

29, 41, 45, 61, 89, 101, 109, 116, 145, 149, 164, 180, 181, 205, 225, 229, 241, 244, 245, 261, 269, 281, 305, 349, 356, 369, 389, 401, 404, 405, 409, 421, 436, 445, 449, 461, 464, 505, 509, 521, 541, 545, 549, 569, 580, 596, 601, 641, 656, 661, 701, 709, 720

Offset: 1

M. F. Hasler, Jan 25 2009

Subsequence of A155565 (where a,b,c,d may be zero).

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

isA155575(n,/* optional 2nd arg allows us to get other sequences */c=[5,1]) = { 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, isA155575(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","))

A155577 Intersection of A154777 and A154778: N = a^2 + 2b^2 = c^2 + 5d^2 for some positive integers a,b,c,d.

Original entry on oeis.org

6, 9, 24, 36, 41, 54, 81, 86, 89, 96, 129, 134, 144, 150, 164, 166, 201, 214, 216, 225, 241, 246, 249, 281, 294, 321, 324, 326, 344, 356, 369, 384, 401, 409, 441, 449, 454, 486, 489, 516, 521, 534, 536, 566, 569, 576, 600, 601, 614, 641, 656, 664, 681, 694

Offset: 1

M. F. Hasler, Jan 25 2009

Subsequence of A155567 (where a,b,c,d may be zero).

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

isA155577(n,/* optional 2nd arg allows us to get other sequences */c=[5,2]) = { 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, isA155577(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","))

cp's OEIS Frontend

A092575 Number of representations of n of the form x^2 + 3y^2 with (x,y)=1.

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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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A092574 Positive integers that can be represented in the form x^2 + 3y^2 with (x,y) = 1 and x and y positive.

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

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A301534 Number of ways to write the n-th prime congruent to 7 modulo 12 as x^2 + 3*y^2 + 15*2^z with x,y,z nonnegative integers.

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Mathematica

A155575 Intersection of A000404 and A154778: N = a^2 + b^2 = c^2 + 5d^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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A155577 Intersection of A154777 and A154778: N = a^2 + 2b^2 = c^2 + 5d^2 for some positive integers a,b,c,d.

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A301534 Number of ways to write the n-th prime congruent to 7 modulo 12 as x^2 + 3y^2 + 152^z with x,y,z nonnegative integers.