A236382
Values of y corresponding to the smallest value of x such that x^2 + y^2 = P, where P is the product of the first n primes of the form 4k + 1, and 0 < x < y.
Original entry on oeis.org
2, 8, 33, 179, 1087, 6972, 50741, 396437, 3387169, 31954512, 314715456, 3162851149, 33021136387, 351019495376, 4108577867504, 50151584231673, 628397549315889, 8265279309778838, 111197960474124352, 1544811437679174169, 21682461771278746182, 328115185564476599824
Offset: 1
a(3) = 33 because the solutions to x^2 + y^2 = 5*13*17 are (x,y) = (23,24), (9,32), (4,33), (12,31) and the value of y corresponding to the smallest value of x is 33.
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a(n) = my(t=1, v); forprimestep(p=5, oo, 4, t*=p; if(!n--, break)); v=qfbsolve(Qfb(1, 0, 1), t, 3); vecmax(vector(#v, i, vecmax(abs(v[i])))); \\ Jinyuan Wang, Apr 30 2025
A341678
Irregular triangle read by rows: row n consists of all numbers x such that x^2 + y^2 = A006278(n), with 0 < x < y.
Original entry on oeis.org
1, 1, 4, 4, 9, 12, 23, 2, 19, 46, 67, 74, 86, 109, 122, 64, 103, 167, 191, 236, 281, 292, 359, 449, 512, 568, 601, 607, 664, 673, 743, 59, 132, 531, 581, 627, 876, 1008, 1284, 1588, 1659, 1723, 2092, 2136, 2317, 2373, 2736, 2757, 2803, 3072, 3164, 3333, 3469, 3704, 3821, 4028, 4077, 4136, 4371, 4596, 4668, 4712, 4851
Offset: 1
Triangle starts:
1,
1, 4,
4, 9, 12, 23,
2, 19, 46, 67, 74, 86, 109, 122,
64, 103, 167, 191, 236, 281, 292, 359, 449, 512, 568, 601, 607, 664, 673, 743,
...
In the second row, calculations are as follows. 5*13 is the product of the first two primes congruent to 1 (mod 4), and 65 = 1^2 + 8^2 = 4^2 + 7^2, so the second row is 1, 4.
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row(n) = {my(t=1, q=3, v=vector(2^n/2)); for(k=1, n, until(q%4==1, q=nextprime(q+1)); t*=q); q=0; for(k=1, #v, until(issquare(t-q^2), q++); v[k]=q); v; } \\ Jinyuan Wang, Mar 03 2021
A236421
Largest value of x such that x^2 + y^2 = P, where P is the product of the first n primes of the form 4k + 1, and 0 < x < y.
Original entry on oeis.org
1, 4, 23, 122, 743, 4851, 35579, 279644, 2390953, 22510892, 222505867, 2236298893, 23344760669, 248205367341, 2905133255201, 35462464074278, 444339941124303, 5844308003978966, 78628682982091847, 1092345114491353169, 15331801966256295943
Offset: 1
a(3) = 23 because the solutions to x^2 + y^2 = 5*13*17 are (x,y) = (23,24), (9,32), (4,33), (12,31) of which the largest value of x is 23.
A236422
Values of y corresponding to the largest value of x such that x^2 + y^2 = P, where P is the product of the first n primes of the form 4k + 1, and 0 < x < y.
Original entry on oeis.org
2, 7, 24, 131, 796, 5008, 36202, 281003, 2399224, 22679301, 222569004, 2236648234, 23354177528, 248211165052, 2905273285888, 35462586540039, 444348395841976, 5844562089950893, 78628980833594936, 1092348171981581852, 15331829536310136066
Offset: 1
a(3) = 24 because the solutions to x^2 + y^2 = 5*13*17 are (x,y) = (23,24), (9,32), (4,33), (12,31) and the value of y corresponding to the largest value of x is 24.
A349708
a(n) is the smallest positive number k such that (product of the first n odd primes) + k^2 is a square.
Original entry on oeis.org
1, 1, 4, 1, 19, 53, 58, 97, 181, 4244, 2122, 31126, 16451, 297392, 2444006, 622249, 2909047, 216182072, 62801719, 769709491, 32522441312, 37859955467, 129549407177, 286721160343, 101419856449, 107709289064864, 72441253480727, 56099073382147, 5249126879235893
Offset: 1
a(4)=1 because the product of the first 4 odd primes, 3*5*7*11 = 1155, is 34^2 - 1. a(5)=19 because 15015=3*5*7*11*13=124^2-19^2, and no positive integer less than 19 will work in this situation.
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a(n) = my(k=1, p=prod(k=2, n+1, prime(k))); while (!issquare(k^2+p), k++); k; \\ Michel Marcus, Jan 10 2022
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from math import isqrt
from sympy import primorial, divisors
def A349708(n):
m = primorial(n+1)//2
a = isqrt(m)
d = max(filter(lambda d: d <= a, divisors(m,generator=True)))
return (m//d-d)//2 # Chai Wah Wu, Mar 29 2022
a(15)-a(26) and corrections to a(9) and a(11) from
Jinyuan Wang, Jan 07 2022
A350813
a(n) is the least positive number k such that the product of the first n primes that are congruent to 1 (mod 4) is equal to y^2 - k^2 for some integer y.
Original entry on oeis.org
2, 4, 24, 38, 16, 588, 5782, 5528, 80872, 319296, 3217476, 32301914, 20085008, 166518276, 2049477188, 17443412442, 27905362944, 233647747282, 886295348972, 134684992249108, 98002282636962, 392994156083892, 5283713761100536, 76642755213473624, 923250078609721236
Offset: 1
For n=3, m = 5*13*17. The "middle" most nearly equal divisor and codivisor of m are y-k=17 and y+k=65, whence a(n) = (65 - 17)/2 = 24.
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from math import prod, isqrt
from itertools import islice
from sympy import sieve, divisors
def A350813(n):
m = prod(islice(filter(lambda p: p % 4 == 1, sieve),n))
a = isqrt(m)
d = max(filter(lambda d: d <= a, divisors(m,generator=True)))
return (m//d-d)//2 # Chai Wah Wu, Mar 29 2022
Terms corrected by and more terms from
Jinyuan Wang, Mar 17 2022
A348635
a(n) is the smallest positive number k coprime to (2n+1)!! such that (2n+1)!! + k^2 is a square.
Original entry on oeis.org
1, 1, 4, 4, 29, 17, 436, 356, 569, 1847, 27704, 72944, 1283333, 726079, 23833532, 45232276, 302068799, 616565857, 26369361188, 23157514888, 70991664061, 505527042479, 1150735735948, 13238389944712, 58668785675111, 209280259070287, 7809609503808088, 530566746979816
Offset: 1
a(5)=29 since 106^2 - 29^2 = 10395 = 3*5*7*9*11 and 29 is relatively prime to 10395 and is as small as possible.
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df(n) = (2*n)! / n! / 2^n; \\ A001147
a(n) = my(d=df(n+1), k=1); while (!((gcd(d,k)==1) && issquare(d+k^2)), k++); k; \\ Michel Marcus, Jan 06 2022
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df(n) = (2*n)! / n! / 2^n; \\ A001147
a(n) = my(d=df(n+1), m=sqrtint(d), k); while (!(issquare(m^2-d, &k) && gcd(d,k)==1), m++); k; \\ Michel Marcus, Jan 06 2022
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