A155715 -id:A155715 - cp's OEIS frontend

A028372 Smallest prime that is simultaneously of forms x^2 + m*y^2 for m = 1, ..., n.

2, 2, 73, 73, 241, 241, 1009, 1009, 1009, 1009, 7561, 7561, 21961, 35281, 35281, 35281, 44641, 44641, 374089, 622561, 622561, 622561, 4379281, 4379281, 4379281, 17690689, 17690689, 17690689, 17690689, 17690689, 316234801, 1996405009, 1996405009, 1996405009

Offset: 1

John L. Drost

nonn

Sequence A155715 lists the smallest numbers of this kind with x,y > 0, but not necessarily prime. - M. F. Hasler, Jan 27 2009

a(n) > 4*10^9 for n >= 35. - Donovan Johnson, May 29 2012

p=2; for(k=1,999, forstep( c=k,1,-1, for( b=1,sqrtint(p\c), issquare(p-c*b^2) & next(2)); p=nextprime(p+1); c=k+1); print1(p",")) \\ M. F. Hasler, Jan 27 2009

a(17)-a(18) corrected and a(26)-a(31) from Donovan Johnson, Sep 29 2009

a(32)-a(34) from Donovan Johnson, May 29 2012

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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A028372 Smallest prime that is simultaneously of forms x^2 + m*y^2 for m = 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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