A075577
k^2 is a term if k^2 + (k-1)^2 and k^2 + (k+1)^2 are primes.
Original entry on oeis.org
4, 25, 625, 900, 1225, 4900, 7225, 10000, 12100, 50625, 52900, 67600, 81225, 84100, 102400, 152100, 168100, 225625, 240100, 245025, 265225, 348100, 462400, 483025, 504100, 562500, 577600, 714025, 902500, 1166400, 1210000, 1288225, 1380625, 1416100, 1428025
Offset: 1
900 = 30^2 is a term because 30^2 + 29^2 = 1741 is prime and 30^2 + 31^2 = 1861 is prime.
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Do[s=n^2+(n-1)^2; s1=n^2+(n+1)^2; If[PrimeQ[s]&&PrimeQ[s1], Print[n^2]], {n, 1, 5000}]
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from sympy import isprime
def aupto(limit):
alst, is2 = [], False
for k in range(1, int(limit**.5) + 2):
is1, is2 = is2, isprime(k**2 + (k+1)**2)
if is1 and is2: alst.append(k**2)
return alst
print(aupto(1500000)) # Michael S. Branicky, Apr 25 2021
A261803
a(n) is the smallest number satisfying a(n)^2+1 = p(n)*q(n), p(n) < q(n) both prime, such that q(n+1)/p(n+1) < q(n)/p(n) with the initial condition q(1)/p(1) < 3/2.
Original entry on oeis.org
50, 334, 516, 670, 844, 1164, 1250, 1800, 2450, 9800, 14450, 20000, 24200, 101250, 105800, 135200, 162450, 168200, 204800, 304200, 336200, 451250, 480200, 490050, 530450, 696200, 924800, 966050, 1008200, 1125000, 1155200, 1428050, 1805000, 2332800, 2420000
Offset: 1
a(1) = 50 because 50^2+1 = 41*61 => 61/41 = 1.4878... < 1.5
a(2) = 334 because 334^2+1 = 281*397 => 397/281 = 1.4128... < 1.4878...
a(3) = 516 because 516^2+1 = 449*593 => 593/449 = 1.3207... < 1.4128...
a(4) = 670 because 670^2+1 = 593*757 => 757/593 = 1.2765... < 1.3207...
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with(numtheory):nn:=100:d:=1.5:
for n from 1 to nn do:
x:=factorset(n^2+1):n0:=bigomega(n^2+1):
if n0=2
then
q:=evalf(x[2]/x[1]):
if q
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(* Assumption: n>7 ==> a(n)=0 mod 50 *) a[n_] := a[n] = For[k = Which[n==1, 0, n <= 7, a[n-1]+1, True, a[n-1] + 50], True, k = Which[n <= 7, k+1, k == a[n-1]+1, k+49, True, k+50], f = FactorInteger[k^2+1]; If[Length[f] == 2, If[f[[All, 2]] == {1, 1}, {p1, q1} = f[[All, 1]]; If[q1/p1 < If[n == 1, 3/2, q[n-1]/p[n-1]], p[n] = p1; q[n] = q1; Return[k]]]]]; Table[Print["a(", n, ") = ", a[n], " p = ", p[n], " q = ", q[n], " q/p = ", N[q[n]/p[n], 10], " q-p = ", q[n]-p[n]]; a[n], {n, 1, 100}] (* Jean-François Alcover, Sep 28 2015 *)
A348594
Numbers m such that m^2 + 1 = p*q with p, q primes and m = (p + q)/2 - 1.
Original entry on oeis.org
8, 50, 1250, 1800, 2450, 9800, 14450, 20000, 24200, 101250, 105800, 135200, 162450, 168200, 204800, 304200, 336200, 451250, 480200, 490050, 530450, 696200, 924800, 966050, 1008200, 1125000, 1155200, 1428050, 1805000, 2332800, 2420000, 2576450, 2761250, 2832200
Offset: 1
50 = 2*5^2 is in the sequence because 50^2 + 1 = 41*61 with 50 = (41 + 61)/2 - 1.
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with(numtheory):nn:=250:printf(`%d, `,8):
for k from 0 to nn do:
n:=50*k^2:d:=factorset(n^2+1):
if bigomega(n^2+1)=2 and (d[1]+d[2])/2 - 1 = n
then
printf(`%d, `,n):
else
fi:
od:
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q[n_] := Module[{f = FactorInteger[n^2 + 1]}, f[[;; , 2]] == {1, 1} && f[[1, 1]] + f[[2, 1]] == 2*n + 2]; Select[Range[3*10^5], q] (* Amiram Eldar, Jan 26 2022 *)
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isok(m) = my(x); if (bigomega(x=m^2+1)==2, my(f=factor(x)); (f[1,1]+f[2,1] == 2*(m+1))); \\ Michel Marcus, Jan 26 2022
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