A203619 -id:A203619 - cp's OEIS frontend

A204765 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+239)^2 = y^2.

0, 217, 220, 717, 1900, 1917, 4780, 11661, 11760, 28441, 68544, 69121, 166344, 400081, 403444, 970101, 2332420, 2352021, 5654740, 13594917, 13709160, 32958817, 79237560, 79903417, 192098640, 461830921, 465711820, 1119633501, 2691748444, 2714367981

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 09 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A185394, A198294, A203619, A206426

LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,217,220,717,1900,1917,4780}, 70]

G.f.: x^2*(119*x^5+x^4+119*x^3-497*x^2-3*x-217)/((x-1)*(x^6-6*x^3+1)). [Colin Barker, Aug 05 2012]

A205644 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+287)^2 = y^2.

Original entry on oeis.org

0, 25, 36, 205, 252, 273, 328, 705, 748, 861, 988, 1045, 1968, 2233, 2352, 2665, 4836, 5085, 5740, 6477, 6808, 12177, 13720, 14413, 16236, 28885, 30336, 34153, 38448, 40377, 71668, 80661, 84700, 95325, 169048, 177505, 199752, 224785, 236028, 418405, 470820

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 09 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,6,-6,0,0,0,0,0,0,0,-1,1).

Cf. A185394, A198294, A203619, A204765, A206426.

LinearRecurrence[ {1,0,0,0,0,0,0,0,6,-6,0,0,0,0,0,0,0,-1,1}, {0,25,36,205,252,273,328,705,748,861,988,1045,1968,2233,2352,2665,4836,5085,5740}, 70]

G.f.: x^2*(23*x^17 +9*x^16 +91*x^15 +17*x^14 +7*x^13 +17*x^12 +91*x^11 +9*x^10 +23*x^9 -113*x^8 -43*x^7 -377*x^6 -55*x^5 -21*x^4 -47*x^3 -169*x^2 -11*x -25)/((x -1)*(x^18 -6*x^9 +1)). - Colin Barker, Aug 05 2012

A205672 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+329)^2 = y^2.

Original entry on oeis.org

0, 87, 112, 184, 235, 376, 451, 595, 660, 987, 1440, 1575, 1971, 2256, 3055, 3484, 4312, 4687, 6580, 9211, 9996, 12300, 13959, 18612, 21111, 25935, 28120, 39151, 54484, 59059, 72487, 82156, 109275, 123840, 151956, 164691, 228984, 318351, 345016, 423280

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 09 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,6,-6,0,0,0,0,0,0,0,-1,1).

Cf. A185394, A198294, A203619, A204765, A205644, A206426.

LinearRecurrence[ {1,0,0,0,0,0,0,0,6,-6,0,0,0,0,0,0,0,-1,1}, {0,87,112,184,235,376,451,595,660,987,1440,1575,1971,2256,3055,3484,4312,4687,6580}, 120]

G.f.: x^2*(69*x^17 +15*x^16 +36*x^15 +21*x^14 +47*x^13 +21*x^12 +36*x^11 +15*x^10 +69*x^9 -327*x^8 -65*x^7 -144*x^6 -75*x^5 -141*x^4 -51*x^3 -72*x^2 -25*x -87)/((x-1)*(x^18 -6*x^9 +1)). - Colin Barker, Aug 05 2012

A323052 Numbers that are the sum of m = 5 successive primes and also the product of m = 5 (other) successive primes.

Original entry on oeis.org

2775683761181, 10945513774549181, 31285407706348267, 43861128120750079, 100441814079170659, 159395121707397143, 402260157804827743, 1340537842364790347, 4738876023641493659, 16292356006439865799, 48185685922249598383, 64649628495078140851, 74655966842055716569

Offset: 1

Zak Seidov, Jan 03 2019

nonn

			For a(1) = 2775683761181, the first addend (summand) is prime(21350924509) = 555136752211, and the first factor is prime(62) = 293.

Daniel Suteu, Table of n, a(n) for n = 1..10000

Cf. A203619 (case m = 3).

a(11) corrected and a(12)-a(13) added by Daniel Suteu, Jan 05 2019

A352065 a(n) is the least prime p that starts a run of 2n+1 consecutive primes whose product is a sum of the same number of (others or same) consecutive primes.

Original entry on oeis.org

2, 29, 293, 229, 3119, 67, 18121, 59629, 10247, 15391, 5903, 24007, 11783, 39359, 21013, 104917, 38273, 61129, 23663, 2423

Offset: 0

Jean-Marc Rebert, Mar 05 2022

			a(0) = 2, because 2 = 2, and there is no smaller prime.
a(1) = 29, because 29 * 31 * 37 = 33263 = 11083 + 11087 + 11093, and there is no smaller prime that starts a run of 3 consecutive primes whose product is a sum of 3 consecutive primes.
a(2) = 293, because 293 * 307 * 311 * 313 * 317 = 2775683761181 = 555136752211 + 555136752221 + 555136752227 + 555136752251 + 555136752271, and there is no smaller prime that starts a run of 5 consecutive primes whose product is a sum of 5 consecutive primes.
Let y be the product of the 2n+1 consecutive primes starting with a(n) and let q be the first prime in the sum of 2n+1 consecutive primes. For n = 0..3 we have:
.
  n  2n+1  a(n)                  y  #dgts(y)                 q  #dgts(q)
  -  ----  ----  -----------------  --------  ----------------  --------
  0     1     2                  2         1                 2         1
  1     3    29              33263         5             11083         5
  2     5   293      2775683761181        13      555136752211        12
  3     7   229  52139749485151463        17  7448535640735789        16
.
For more examples, see the "doubleDecomposition" link.

Jean-Marc Rebert, doubleDecomposition
Carlos Rivera, Puzzle 1077. These numbers that are..., The Prime Puzzles and Problems Connection.

Cf. A203619, A323052.

from math import prod
from sympy import prime, nextprime, prevprime
def A352065(n):
    plist = [prime(k) for k in range(1,2*n+2)]
    pd = prod(plist)
    while True:
        mlist = [nextprime(pd//(2*n+1)-1)]
        for _ in range(n):
            mlist = [prevprime(mlist[0])]+mlist+[nextprime(mlist[-1])]
        if sum(mlist) <= pd:
            while (s := sum(mlist)) <= pd:
                if s == pd:
                    return plist[0]
                mlist = mlist[1:]+[nextprime(mlist[-1])]
        else:
            while (s := sum(mlist)) >= pd:
                if s == pd:
                    return plist[0]
                mlist = [prevprime(mlist[0])]+mlist[:-1]
        pd //= plist[0]
        plist = plist[1:] + [nextprime(plist[-1])]
        pd *= plist[-1] # Chai Wah Wu, Apr 21 2022

a(15)-a(19) from Chai Wah Wu, Apr 21 2022

A206854 Smallest integer m such that m is a product of 2n-1 consecutive primes and a sum of 2n-1 consecutive primes.

Original entry on oeis.org

2, 33263, 2775683761181, 52139749485151463, 31359251876786281892441299570699, 2385018819218440287149, 23509572623777698757692123744388316389653416929069870587, 436178570920976645136650311902311012102337977560516289614008518576769313, 166345108784858794943225366868487068031523855419640057875257310044811

Offset: 1

Zak Seidov, Feb 13 2012

nonn

n=1: m = 2 (trivial case: product and sum of single prime, 2);
n=2: m = 33263 = product{29, 31, 37} = sum{11083, 11087, 11093};
n=3: m = 2775683761181 = product({293, 307, 311, 313, 317}) = sum({555136752211, 555136752221, 555136752227, 555136752251, 555136752271});
n=4: m = 52139749485151463=product({229, 233, 239, 241, 251, 257, 263})= sum({7448535640735789, 7448535640735843, 7448535640735867, 7448535640735877, 7448535640735991, 7448535640736009, 7448535640736087});
n=5: m = 31359251876786281892441299570699 = product({3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191}) = sum({3484361319642920210271255507593, 3484361319642920210271255507619, 3484361319642920210271255507719, 3484361319642920210271255507767, 3484361319642920210271255507923, 3484361319642920210271255507937, 3484361319642920210271255507941, 3484361319642920210271255508067, 3484361319642920210271255508133});
n=6: m = 2385018819218440287149 = product({67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109}) = sum({216819892656221844131, 216819892656221844133, 216819892656221844139, 216819892656221844169, 216819892656221844307, 216819892656221844331, 216819892656221844347, 216819892656221844373, 216819892656221844397, 216819892656221844401, 216819892656221844421}).

Cf. A203619.

scp:= proc(x,n) local P,i,s;
  P:= Vector(n);
  P[1]:= nextprime(ceil(x/n));
  for i from 2 to n do P[i]:= nextprime(P[i-1]) od;
  s:= convert(P,`+`);
  while s > x do
    s:= s - P[n];
    P[2..n]:= P[1..n-1];
    if P[2] = 2 then return false fi;
    P[1]:= prevprime(P[2]);
    s:= s + P[1];
  od;
  evalb(s=x)
end proc:
f:= proc(n) local i,P,r;
     P:= ;
     r:= convert(P,`*`);
     while not scp(r,2*n-1) do
       r:= r/P[1];
       P[1..2*n-2]:= P[2..2*n-1];
       P[2*n-1]:= nextprime(P[2*n-2]);
       r:= r*P[2*n-1];
     od;
end proc:
f(1):= 2:
map(f, [$1..8]); # Robert Israel, Mar 13 2023

a(7)-a(9) from Robert Israel, Mar 13 2023

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A204765 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+239)^2 = y^2.

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A205644 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+287)^2 = y^2.

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A205672 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+329)^2 = y^2.

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A323052 Numbers that are the sum of m = 5 successive primes and also the product of m = 5 (other) successive primes.

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A352065 a(n) is the least prime p that starts a run of 2n+1 consecutive primes whose product is a sum of the same number of (others or same) consecutive primes.

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A206854 Smallest integer m such that m is a product of 2n-1 consecutive primes and a sum of 2n-1 consecutive primes.

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