A216330
Squares equal to the difference between two successive primes of the form n^2+1.
Original entry on oeis.org
64, 144, 100, 1024, 4900, 10816, 11664, 12544, 18496, 102400, 41616, 46656, 331776, 298116, 44100, 451584, 270400, 141376, 372100, 678976, 504100, 1849600, 524176, 2890000, 3504384, 602176, 685584, 8702500, 1768900, 2160900, 868624, 532900, 624100, 12960000
Offset: 1
64 is in the sequence because 6^2 + 1 = 37, 10^2+1 = 101 and 101 - 37 = 64 is square.
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q:=2:for n from 2 to 100 do:p:=n^2+1:if type(p,prime)=true then x:=p-q:q:=p: z:=sqrt(x):if z=floor(z) then printf(`%d, `,z):else fi:od:
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Select[#[[2]]-#[[1]]&/@Partition[Select[Range[2000000]^2+1,PrimeQ],2,1], IntegerQ[ Sqrt[#]]&] (* Harvey P. Dale, Nov 08 2017 *)
A339007
Least k such that p = k^2 + 1 and q = (k+2n)^2 + 1 are prime numbers with q - p square.
Original entry on oeis.org
24, 6, 312984, 16896, 120, 734994, 10640, 10, 1946016, 150, 171864, 180, 31200, 17136, 120, 84, 8976, 54, 137256, 300, 231504, 66, 184, 360126, 24, 5824, 2496, 224, 261696, 90, 4359344, 66, 50160, 68816, 280, 864, 1524696, 570, 219336, 11520, 8487984, 126, 22704
Offset: 1
a(1) = 24 because 24^2 + 1 = 577, (24 + 2)^2 + 1 = 677 and 677 - 577 = 10^2 is a square. The other values m such that p = m^2 + 1 and q = (m+2)^2 + 1 are primes with q - p square are 11024, 133224, 156024, 342224, 416024,...
a(2) = 6 because 6^2 + 1 = 37, (6 + 4)^2 + 1 = 101 and 101 - 37 = 8^2 is a square. The other values m such that p = m^2 + 1 and q = (m+4)^2 + 1 are primes with q - p square are 16, 126, 1350, 1456, 1566, 2310, 5200,...
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for n from 1 to 50 do:
ii:=0:
for k from 2 by 2 to 10^9 while(ii=0) do:
p:=k^2+1:q:=(k+2*n)^2 +1:
if isprime(p) and isprime(q) and sqrt(q-p)=floor(sqrt(q-p))
then
ii:=1:printf(`%d %d \n`,n,k):
else
fi:
od:
od:
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a(n) = my(k=1); while (!(isprime(p=k^2+1) && isprime(q=(k+2*n)^2 + 1) && issquare(q-p)), k++); k; \\ Michel Marcus, Nov 18 2020
A339008
Least k such that p = k^2 + 1 and q = (k+2n)^2 + 1 are two consecutive prime numbers of the same form with q - p square.
Original entry on oeis.org
24, 6, 312984, 16896, 240, 734994, 10640, 10360, 1946016, 2550, 13189264, 72996, 416520, 2184336, 1584360, 202484, 232696, 1700150, 2394456, 375360, 8736504, 9237866, 53629744, 360126, 87000, 574339974, 82404216, 23237760, 1249877496, 826650, 127119344, 1527720
Offset: 1
a(1) = 24 because 24^2 + 1 = 577, (24 + 2)^2 + 1 = 677. The numbers 577 and 677 are two consecutive primes of the form m^2+1, and 677 - 577 = 10^2 is a square. The other values m such that p = m^2 + 1 and q = (m+2)^2 + 1 are consecutive primes with q - p square are 11024, 133224, 156024, 342224, 416024, ...
a(2) = 6 because 6^2 + 1 = 37, (6 + 4)^2 + 1 = 101. The numbers 37 and 101 are two consecutive primes of the form m^2+1, and 101 - 37 = 8^2 is a square. The other values m such that p = m^2 + 1 and q = (m+4)^2 + 1 are consecutive primes with q - p square are 16, 126, 1350, 1456, 1566, 2310, 5200, ...
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for n from 1 to 25 do:
ii:=0:n1:=0:q:=2:
for k from 2 by 2 to 10^9 while(ii=0) do:
p:=k^2+1:
if isprime(p)
then
x:=p-q:q:=p:z:=sqrt(x):
if z=floor(z) and k-n1=2*n
then
ii:=1:printf(`%d %d \n`,n,n1):
else
n1:=k:
fi:
fi:
od:
od:
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consecutive(p, q) = {forprime(r = nextprime(p+1), precprime(q-1), if (isprime(r) && issquare(r-1), return(0));); return(1);}
a(n) = my(k=1); while (!(isprime(p=k^2+1) && isprime(q=(k+2*n)^2 + 1) && issquare(q-p) && consecutive(p, q)), k++); k; \\ Michel Marcus, Nov 30 2020
A247592
Numbers n such that A002496(n) mod A002496(n-1) is a perfect square.
Original entry on oeis.org
2, 8, 10, 25, 42, 147, 160, 169, 238, 260, 491, 544, 869, 890, 923, 1140, 1337, 1386, 1465, 1643, 1927, 3371, 4614, 5038, 5086, 5225, 5832, 5909, 5995, 7118, 7157, 8540, 9859, 12543, 13505, 13795, 13841, 14211, 15347, 17079, 17263, 18643, 20211, 21184, 21245
Offset: 1
a(3)=10 because A002496(10) mod A002496(9)= 677 mod 577 = 10^2.
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with(numtheory):nn:=360000:T:=array(1..nn):kk:=0:
for n from 1 to nn do:
if type(n^2+1,prime)=true then
kk:=kk+1:T[kk]:=n^2+1:
else
fi:
od:
for m from 1 to kk-1 do:
r:=irem(T[m+1],T[m]):z:=sqrt(r):
if z=floor(z)
then printf(`%d, `, m+1):
else
fi:
od:
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lst={};lst1={};nn=400000;Do[If[PrimeQ[n^2+1],AppendTo[lst,n^2+1]],{n,1,nn}];nn1:=Length[lst];
Do[If[IntegerQ[Sqrt[Mod[lst[[m]],lst[[m-1]]]]],AppendTo[lst1,m]],{m,2,nn1}];lst1
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from gmpy2 import t_mod, is_square, is_prime
A247592_list, A002496_list, m, c = [], [2], 2, 2
for n in range(1, 10**7):
m += 2*n+1
if is_prime(m):
if is_square(t_mod(m, A002496_list[-1])):
A247592_list.append(c)
A002496_list.append(m)
c += 1 # Chai Wah Wu, Sep 20 2014
A351141
Pairs of primes (p,q) = (A002496(m), A002496(m+1)) such that q-p is a power r of the product of its prime factors for some m.
Original entry on oeis.org
37, 101, 577, 677, 15877, 16901, 57601, 62501, 33988901, 34035557, 113209601, 113507717, 121528577, 121572677, 345960001, 346332101, 635040001, 635544101, 7821633601, 7823402501, 17748634177, 17749167077, 24343488577, 24344112677, 97958984257, 97962740101
Offset: 1
The pair (257, 401) = (16^2+1, 20^2+1) is not in the sequence because 401 - 257 = 144 = 2^4*3^2.
The pair (577, 677) = (24^2+1, 26^2+1) is in the sequence because 577 - 677 = 100 = 2^2*5^2.
The pair (33988901, 34035557) = (5830^2+1, 5834^2+1) is in the sequence because 33988901 - 34035557 = 46656 = 2^6*3^6.
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with(numtheory):
T:=array(1..26):nn:=350000:q:=5:j:=1:
for n from 4 by 2 to nn do:
p:=n^2+1:
if type(p, prime)=true
then
x:=p-q:r:=q:q:=p:
u:=factorset(x):n0:=nops(u):ii:=0:d:=product(u[i],i=1..n0):
for k from 2 to 20 while(ii=0) do:
if d^k=x
then ii=1:T[j]:=r:T[j+1]:=q:j:=j+2:
else
fi:
od:
fi:
od:
print(T):
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lista(nn) = my(lastp=2); forprime(p=nextprime(lastp+1), nn, if (issquare(p-1), if (ispowerful(p-lastp), my(f=factor(p-lastp)[,2]); if (vecmin(f) == vecmax(f), print1(lastp, ", ", p, ", "));); lastp = p;);); \\ Michel Marcus, Feb 03 2022
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