A028372 -id:A028372 - 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

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

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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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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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