A154777 -id:A154777 - cp's OEIS frontend

A155707 Numbers expressible as a^2 + k b^2 with nonzero integers a,b, for k=2, k=3, k=5 and k=7.

144, 576, 1009, 1129, 1201, 1296, 1801, 1849, 2304, 2521, 2689, 2881, 3049, 3361, 3529, 3600, 3889, 4036, 4201, 4356, 4489, 4516, 4561, 4729, 4804, 5184, 5209, 5569, 5881, 5929, 6841, 7009, 7056, 7204, 7396, 7561, 7681, 8089, 8521, 8689, 8761, 8929

Offset: 1

M. F. Hasler, Feb 10 2009

nonn

Subsequence of A155708.

Robert Israel, Table of n, a(n) for n = 1..2500

Subsequence of A155708.

Cf. A028372, A000404, A154777, A092572, A097268, A155714, A155560.

filter:= proc(x) local k,S;
   if numtheory:-quadres(x,3*5*7)<> 1 then return false fi;
   for k in [2,3,5,7] do
     S:= [isolve(x = a^2 + k*b^2)];
     if andmap(t -> subs(t,a*b) = 0, S) then return false fi;
   od;
   true
end proc;
select(filter, [$1..10000]); # Robert Israel, May 14 2025

isA155707(n,/* optional 2nd arg allows us to get other sequences */c=[7, 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, isA155707(n) & print1(n","))

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

A201613 Primes of the form p^2 + 2q^2 with p and q odd primes.

Original entry on oeis.org

43, 59, 67, 107, 139, 251, 307, 347, 379, 547, 587, 859, 1699, 1867, 1931, 3371, 3499, 3739, 4507, 5059, 5347, 6907, 6971, 7451, 10091, 10627, 10667, 11467, 12491, 18787, 20411, 21227, 22907, 29947, 32059, 32779, 37547, 38651, 39619, 49307, 49747, 53147, 55787

Offset: 1

Zak Seidov, Dec 03 2011

nonn

One of primes p, q must be 3, hence we have two sets of primes: 9+2*p^2 and p^2+18 with p > 3.

Note that if we allow 2 for p or q then there is another "set" of primes of the form p^2+8 (q=2) with odd prime p -- this set contains only the prime 17=3^2+8.

			43=5^2+2*3^2, 59=3^2+2*5^2, 67=7^2+2*3^2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A260553 and of A154777.

list(lim)=my(v=List(),t); forprime(p=5,sqrtint(lim\1-18), if(isprime(t=p^2+18), listput(v,t))); forprime(q=5,sqrtint((lim-9)\2), if(isprime(t=2*q^2+9), listput(v,t))); Set(v) \\ Charles R Greathouse IV, Aug 26 2015

Corrected by Charles R Greathouse IV, Aug 26 2015

A338432 Triangle read by rows: T(n, k) = (n - k + 1)^2 + 2*k^2, for n >= 1, and k = 1, 2, ..., n.

Original entry on oeis.org

3, 6, 9, 11, 12, 19, 18, 17, 22, 33, 27, 24, 27, 36, 51, 38, 33, 34, 41, 54, 73, 51, 44, 43, 48, 59, 76, 99, 66, 57, 54, 57, 66, 81, 102, 129, 83, 72, 67, 68, 75, 88, 107, 132, 163, 102, 89, 82, 81, 86, 97, 114, 137, 166, 201

Offset: 1

Wolfdieter Lang, Dec 09 2020

This triangle is obtained from the array A(m, k) = m^2 + 2*k^2, for k and m >= 1, read by upwards antidiagonals. This array A is of interest for representing numbers as a sum of three non-vanishing squares with two squares coinciding.
For the numbers represented this way, see A154777. To find the actual values for m and k (taken positive), given a representable number from A154777, one can also use the number triangle T(n, k) = A(n-k+1, k).
To find the number of representations of value N (from A154777), it is sufficient to consider the rows n >= 1 not exceeding n_{max} = Floor(N, Min), where the sequence Min gives the minima of the numbers in each row: Min = {min(n)}_{n>=1} with min(n) = min(T(n, 1), T(n, 2), ..., T(n, n)) and Floor(N, Min) is the greatest member of Min not exceeding N.
Conjecture: min(n) = T(n, ceiling(n/3)), n >= 1. This is the sequence (n+1)^2 - ceiling(n/3)*(2*(n+1) - 3*ceiling(n/3)) = A071619(n+1) = ceiling((2/3)*(n+1)^2) = (n+1)^2 - floor((1/3)*(n+1)^2) = 3, 6, 11, 17, 24, 33, 43, .... (Proof of these identities by considering the three n (mod 3) cases.)
For the multiplicities of the representable values A154777(n), see A339047.
The author met this representation problem in connection with special triples of integer curvatures in the Descartes-Steiner five circle problem.

			The triangle T(n, k) begins:
n \ k  1   2   3   4   5   6   7   8   9  10  11  12 ...
1:     3
2:     6   9
3:    11  12  19
4:    18  17  22  33
5:    27  24  27  36  51
6:    38  33  34  41  54  73
7:    51  44  43  48  59  76  99
8:    66  57  54  57  66  81 102 129
9:    83  72  67  68  75  88 107 132 163
10:  102  89  82  81  86  97 114 137 166 201
11:  123 108  99  96  99 108 123 144 171 204 243
12:  146 129 118 113 114 121 134 153 178 209 246 289
...
----------------------------------------------------
T(5, 1) = 5^2 + 2*1^2 = 27 = T(5, 3) = 3^2 + 2*3^2. A338433(11) = 2 for A154777(11) = 27.
T(4, 4) = 1^2 + 2*4^2 = 33 = T(6, 2) = 5^2 + 2*2^2. A338433(12) = 2 for A154777(12) = 33.
T(5, 5) = 1^2 + 2*5^2 = 51 = T(7, 1) = 7^2 + 2*1^2. A338433(20) = 2 for A154777(20) = 51.
T(7, 7) = 1^1 - 2*7^2 = 99 = T(11, 3) = 9^2 + 2*3^2 = 99 = T(11, 5) = 7^2 + 2*5^2. A338433(39) = 3 for A154777(39) = 99.
The first multiplicity 4 appears for 297.

Cf. A154777, A339047.
Cf. Columns k = 1..3: A059100, A189833, A241848.
Cf. Diagonals m = 1..4: A058331, A255843, A339048, A255847.

T(n, k) = A(n - k + 1, k), with the array A(m, k) = m^2 + 2*k^2, for n >= 1 and k = 1, 2, ..., n, and 0 otherwise.
G.f. of T and A column k (offset 0): G(k, x) = (1 + x + 2*(1 - x)^2*k^2)/(1-x)^3, for k >= 1.
G.f. of T diagonal m (A row m) (offset 0): D(m, x) = ((2*(1+x) + (1-x)^2*m^2)/(1-x)^3), for m >= 1.
G.f. of row polynomials in x (that is, g.f. of the triangle): G(z,x) = (3 - 3*z + (2 - 6*x + x^2)*z^2 + (2 + x)*x*z^3)*x*z / ((1 - z)*(1 - x*z))^3.

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

A200977 Smallest number expressible in the form a^2 + 2b^2, with positive integers a and b, in exactly n ways.

Original entry on oeis.org

3, 27, 99, 297, 891, 1683, 8019, 5049, 18513, 15147, 649539, 31977, 314721, 136323, 166617, 95931, 10673289, 351747, 64304361, 287793, 1499553, 8450649, 127680201, 863379, 20160657, 99379467, 5979699, 5180274, 1235641473, 3165723, 50984802, 3933171

Offset: 1

Zak Seidov, Nov 29 2011

nonn

Incidentally all terms are multiples of 3.

			3 = 1+2*1^2.
27 = 3+2*3^2 = 5^2+2*1^2.
99 = 1+2*7^2 = 7^2+2*5^2 = 9^2+2*3^2.

Cf. A154777 (numbers of the form a^2 + 2b^2 with positive integers a, b).

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","))

cp's OEIS Frontend

A155707 Numbers expressible as a^2 + k b^2 with nonzero integers a,b, for k=2, k=3, k=5 and k=7.

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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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A201613 Primes of the form p^2 + 2q^2 with p and q odd primes.

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A338432 Triangle read by rows: T(n, k) = (n - k + 1)^2 + 2*k^2, for n >= 1, and k = 1, 2, ..., 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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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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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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A200977 Smallest number expressible in the form a^2 + 2b^2, with positive integers a and b, in exactly n ways.

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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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