A145299
Smallest k such that k^2+1 is divisible by A002144(n)^6.
Original entry on oeis.org
1068, 1999509, 390112, 253879357, 756360062, 2363588163, 5041394261, 9435321777, 41865466758, 102666405913, 197177418061, 316411915250, 171829799914, 625667121807, 182312430890, 1095001339019, 6390289199260
Offset: 1
a(1) = 1068 since A002144(1) = 5, 1068^2+1 = 1140625 = 5^6*73 and for no k < 1068 does 5^6 divide k^2+1. a(11) = 197177418061 since A002144(11) = 97, 197177418061^2+1 = 38878934193202368999722 = 2*97^6*23337479509 and for no k < 197177418061 does 97^6 divide k^2+1.
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{ e=6; forprime(p=2, 1000, if(p%4==1, k=lift(sqrt(-1+O(p^e))); if(k>p^e/2,k=p^e-k); print1(k, ", "))) }
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from itertools import islice
from sympy import nextprime, sqrt_mod_iter
def A145299_gen(): # generator of terms
p = 1
while (p:=nextprime(p)):
if p&3==1:
yield min(sqrt_mod_iter(-1,p**6))
A145299_list = list(islice(A145299_gen(),20)) # Chai Wah Wu, May 04 2024
More terms and efficient PARI program from. -
Max Alekseyev, Oct 28 2008
A145296
Smallest k such that k^2 + 1 is divisible by A002144(n)^3.
Original entry on oeis.org
57, 239, 1985, 10133, 9466, 11389, 27590, 51412, 153765, 344464, 107551, 296344, 172078, 432436, 931837, 753090, 676541, 2321221, 2027724, 3394758, 1706203, 4841182, 1438398, 2947125, 398366, 5657795, 4942017, 9400802, 11906503
Offset: 1
a(3) = 1985 since A002144(3) = 17, 1985^2 + 1 = 3940226 = 2*17^3*401 and for no k < 1985 does 17^3 divide k^2+1.
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{m=12000000; pmax=300; z=70; v=vector(z); for(n=1, m, fac=factor(n^2+1); for(j=1, #fac[, 1], if(fac[j, 2]>=3&&fac[j, 1]<=pmax, q=primepi(fac[j, 1]); if(q<=z&&v[q]==0, v[q]=n)))); t=1; j=0; while(t&&j
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{e=3; forprime(p=2, 300, if(p%4==1, q=p^e; m=q; while(!ispower(m-1,2,&n), m=m+q); print1(n, ",")))} \\ Klaus Brockhaus, Oct 09 2008
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from itertools import islice
from sympy import nextprime, sqrt_mod_iter
def A145296_gen(): # generator of terms
p = 1
while (p:=nextprime(p)):
if p&3==1:
yield min(sqrt_mod_iter(-1,p**3))
A145296_list = list(islice(A145296_gen(),20)) # Chai Wah Wu, May 04 2024
A145297
Smallest k such that k^2+1 is divisible by A002144(n)^4.
Original entry on oeis.org
182, 239, 27493, 34522, 800982, 1251967, 623098, 6304056, 6459524, 20099637, 22709274, 35764191, 40317977, 54397650, 166206108, 187800003, 165728858, 152475014, 282599844, 312923750, 154613663, 485200742, 912190662, 548850444
Offset: 1
a(1) = 182 since A002144(1) = 5, 182^2+1 = 33125 = 5^4*53 and for no k < 182 does 5^4 divide k^2+1.
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{e=4; forprime(p=2, 250, if(p%4==1, q=p^e; m=q; while(!ispower(m-1,2,&n), m=m+q); print1(n, ",")))}
-
from itertools import islice
from sympy import nextprime, sqrt_mod_iter
def A145297_gen(): # generator of terms
p = 1
while (p:=nextprime(p)):
if p&3==1:
yield min(sqrt_mod_iter(-1,p**4))
A145297_list = list(islice(A145297_gen(),20)) # Chai Wah Wu, May 04 2024
A145298
Smallest k such that k^2+1 is divisible by A002144(n)^5.
Original entry on oeis.org
1068, 143044, 390112, 7745569, 6423465, 46464143, 23048345, 144762466, 404034898, 2153335831, 331407850, 1108900220, 2581164875, 760839155, 10734466938, 6595297216, 773302059, 61063137802, 31915893786, 112699451831
Offset: 1
a(4) = 7745569 since A002144(4) = 29, 7745569^2+1 = 59993839133762 = 2*29^5*97*15077 and for no k < 7745569 does 29^5 divide k^2+1.
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{e=5; forprime(p=2, 200, if(p%4==1, q=p^e; m=q; while(!ispower(m-1,2,&n), m=m+q); print1(n, ",")))}
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from itertools import islice
from sympy import nextprime, sqrt_mod_iter
def A145298_gen(): # generator of terms
p = 1
while (p:=nextprime(p)):
if p&3==1:
yield min(sqrt_mod_iter(-1,p**5))
A145298_list = list(islice(A145298_gen(),20)) # Chai Wah Wu, May 04 2024
A145871
Smallest k such that k^2+1 is divisible by A002144(n)^7.
Original entry on oeis.org
32318, 6826318, 96940388, 7986582530, 24900904028, 92615568742, 416081467190, 988322434636, 3219884218827, 4867146503697, 26457926739667, 47023298541694, 26661771973542, 90980209992989, 257680081342861, 283410689912607
Offset: 1
a(2) = 6826318 since A002144(2) = 13, 6826318^2+1 = 46598617437125 = 5^3*13^7*13*457 and for no k < 6826318 does 13^7 divide k^2+1. a(4) = 7986582530 since A002144(4) = 29, 7986582530^2+1 = 63785500508501200901 = 29^7*197*409*45893 and for no k < 7986582530 does 29^7 divide k^2+1.
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{e=7; forprime(p=2, 40, if(p%4==1, q=p^e; m=q; while(!issquare(m-1, &n), m=m+q); print1(n, ",")))}
A145872
Smallest k such that k^2+1 is divisible by A002144(n)^8.
Original entry on oeis.org
110443, 6826318, 3379649772, 61012922706, 1019349744435, 287369842623, 11331029931180, 71294762793847, 239822883201307, 923990886302412, 2369608176604944, 3156215819652023, 521749964271465, 2026364722410364
Offset: 1
a(1) = 110443 since A002144(1) = 5, 110443^2+1 = 12197656250 = 2*5^8*13*1201 and for no k < 110443 does 5^8 divide k^2+1. a(3) = 3379649772 since A002144(3) = 17, 3379649772^2+1 = 11422032581379651985 = 5*13*17^8*97*259697 and for no k < 3379649772 does 17^8 divide k^2+1.
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{e=8; forprime(p=2, 40, if(p%4==1, q=p^e; m=q; while(!issquare(m-1, &n), m=m+q); print1(n, ",")))}
A145873
Smallest k such that k^2+1 is divisible by A002144(n)^9.
Original entry on oeis.org
280182, 822557039, 24306922095, 4563230639355, 15069267560119, 112076323050317, 50928660480181, 3138611770750343, 9110883894036198, 50251663587824641, 76004727767164666, 310872228812491206, 521749964271465
Offset: 1
a(1) = 280182 since A002144(1) = 5, 280182^2+1 = 78501953125 = 5^9*40193 and for no k < 280182 does 5^9 divide k^2+1. a(3) = 24306922095 since A002144(3) = 17, 24306922095^2+1 = 590826461732399189026 = 2*17^9*29*673*127637 and for no k < 24306922095 does 17^9 divide k^2+1.
A174492
a(n) = the smallest k such that k^2+1 = p*A002144(n)^2, p prime of A002144 .
Original entry on oeis.org
18, 70, 540, 800, 1486, 2984, 500, 6760, 776, 4060, 5604, 4030, 5744, 1710, 1744, 46146, 186174, 162886, 62064, 32150, 37416, 16610, 26884, 15006, 130026, 58724
Offset: 1
a(1) = 18 because 18^2 + 1 = 13*A002144(1) ^2 = 13*5^2 ;
a(2) = 70 because 70^2 + 1 = 29*A002144(2) ^2 = 29*13^2 ;
a(3) = 540 because 540^2 + 1 = 1009*A002144(3) ^2 = 1009*17^2 .
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with(numtheory):nn:=400:T:=array(1..nn):k:=1:for x from 1 to nn do: p:=4*x+1:if
type(p,prime)=true then T[k]:=p:k:=k+1:else fi:od:for n from 1 to k do: ind:=0:for
m from 1 to 500000 while(ind=0) do:y:=m^2+1:x:= factorset(y) : n1:=nops(x):n2
:=bigomega(y):if n1=2 and n2 = 3 and x[1]=T[n] and ind=0 then ind:=1:printf(`%d,
`,m):else fi:od:od:
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