A347594 -id:A347594 - cp's OEIS frontend

A347754 a(n) = sqrt(A347594(n-1)^2 + n^2 + A347594(n)).

2, 3, 4, 8, 14, 28, 21, 33, 65, 50, 97, 73, 14, 30, 32, 22, 18, 32, 31, 32, 53, 68, 50, 43, 55, 100, 112, 154, 135, 226, 449, 832, 640, 194, 382, 302, 509, 665, 1213, 905, 213, 43, 57, 113, 49, 99, 126, 217, 269, 269, 173, 116, 153, 161, 212, 309, 540, 1057, 863, 1690, 3157, 2593, 1343, 1401, 1506, 1797, 2829, 1170, 87

Offset: 1

Seiichi Manyama, Sep 12 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A347594.

b[0]=1;b[m_]:=b[m]=(k=1;While[!IntegerQ@Sqrt[b[m-1]^2+m^2+k],k++];k);
a[n_]:=a[n]=Sqrt[b[n-1]^2+n^2+b[n]];Array[a,100] (* Giorgos Kalogeropoulos, Sep 12 2021 *)

lista(nn) = {my(prec = 1, list=List(), x); for (n=1, nn, my(k = 1); while (!issquare(x = prec^2+n^2+k), k++); listput(list, sqrtint(x)); prec = k;); Vec(list);} \\ Michel Marcus, Sep 13 2021

from math import isqrt
A347754_list, a = [], 1
for n in range(1,10**3):
    m = a**2+n**2
    k = isqrt(m)+1
    a = k**2-m
    A347754_list.append(k) # Chai Wah Wu, Sep 13 2021

def A347754(n)
  s = 1
  ary = []
  (1..n).each{|i|
    j = i * i + s * s
    k = Math.sqrt(j).floor + 1
    ary << k
    s = k * k - j
  }
  ary
end
p A347754(100)

a(n) = floor(sqrt(A347594(n-1)^2 + n^2)) + 1.

A347595 a(0) = 1; for n>0, a(n) is the smallest positive integer that has not previously occurred such that a(n-1)^2 + n^2 + a(n) is a square.

Original entry on oeis.org

1, 2, 8, 27, 39, 54, 73, 98, 133, 186, 273, 426, 709, 1250, 2305, 4386, 8517, 16746, 33169, 65978, 131557, 262674, 524865, 1049202, 2097829, 4195034, 8389393, 16778058, 33555333, 67109826, 134218753, 268436546, 536872069, 1073743050, 2147484945, 4294968666, 8589936037, 17179870706

Offset: 0

Scott R. Shannon, Sep 08 2021

nonn

This sequence uses the same rules as A347594 except here all numbers must be unique. Up to 10^5 terms all terms are larger than the previous term; it is unknown if this holds for all terms as n->infinity.

			a(1) = 2 as a(0)^2 + 1^2 = 1 + 1 = 2, and 2 + 2 = 4 = 2^2 is the next smallest square.
a(2) = 8 as a(1)^2 + 2^2 = 4 + 4 = 8, and 8 + 8 = 16 = 4^2. Note that although 8 + 1 = 9 = 3^2, 1 cannot be chosen as a(0) = 1.
a(3) = 27 as a(2)^2 + 3^2 = 64 + 9 = 73 and 73 + 27 = 100 = 10^2.  Note that although 73 + 8 = 81 = 9^2, 8 cannot be chosen as a(2) = 8.
a(4) = 39 as a(3)^2 + 4^2 = 729 + 16 = 745, and 745 + 39 = 784 = 28^2 is the next smallest square.

Cf. A347594, A000290, A103605, A009000, A005408, A347754.

A349119 a(0) = 1; for n>0, a(n) is the smallest positive integer that has not previously occurred such that |n - a(n-1)| + a(n) is a square.

Original entry on oeis.org

1, 4, 2, 3, 8, 6, 9, 7, 15, 10, 16, 11, 24, 5, 27, 13, 22, 20, 14, 31, 25, 12, 26, 33, 40, 21, 44, 19, 55, 23, 18, 36, 32, 35, 48, 51, 34, 46, 17, 42, 47, 30, 37, 43, 63, 82, 28, 45, 61, 52, 62, 38, 50, 78, 57, 79, 41, 65, 29, 70, 39, 59, 97, 66, 98, 67, 80, 68, 49, 101, 69, 119, 53, 124, 71, 60

Offset: 0

Scott R. Shannon, Nov 08 2021

			a(1) = 4 as |1 - a(0)| = |1 - 1| = 0, and 0 + 4 = 4 = 2^2 is the next smallest square. Note a(1) cannot be 1 as a(0) = 1.
a(4) = 8 as |4 - a(3)| = |4 - 3| = 1, and 1 + 8 = 9 = 3^2 is the next smallest square. Note a(4) cannot be 3 as a(3) = 3.
a(8) = 15 as |8 - a(7)| = |8 - 7| = 1, and 1 + 15 = 16 = 4^2 is the next smallest square. Note a(8) cannot be 3 or 8 as these have previously occurred.

Scott R. Shannon, Image of the first 1000000 terms.

Cf. A349182, A000290, A347595, A347594, A009000, A347754.

A349182 a(0) = 1; for n>0, a(n) is the smallest positive integer such that |n - a(n-1)| + a(n) is a square.

Original entry on oeis.org

1, 1, 3, 1, 1, 5, 3, 5, 1, 1, 7, 5, 2, 5, 7, 1, 1, 9, 7, 4, 9, 4, 7, 9, 1, 1, 11, 9, 6, 2, 8, 2, 6, 9, 11, 1, 1, 13, 11, 8, 4, 12, 6, 12, 4, 8, 11, 13, 1, 1, 15, 13, 10, 6, 1, 10, 3, 10, 1, 6, 10, 13, 15, 1, 1, 17, 15, 12, 8, 3, 14, 7, 16, 7, 14, 3, 8, 12, 15, 17, 1, 1, 19, 17, 14, 10, 5, 18, 11

Offset: 0

Scott R. Shannon, Nov 09 2021

nonn

			a(1) = 1 as |1 - a(0)| = |1 - 1| = 0, and 0 + 1 = 1 = 1^2 is the next smallest square.
a(2) = 3 as |2 - a(1)| = |2 - 1| = 1, and 1 + 3 = 4 = 2^2 is the next smallest square.
a(5) = 5 as |5 - a(4)| = |5 - 1| = 4, and 4 + 5 = 9 = 3^2 is the next smallest square.

Scott R. Shannon, Image for the first 50000 terms.

Cf. A349119, A000290, A347595, A347594, A009000, A347754.

nxt[{n_,a_}]:={n+1,Module[{k=1},While[!IntegerQ[Sqrt[(Abs[n+1-a])+k]],k++];k]}; NestList[ nxt,{0,1},90][[;;,2]] (* Harvey P. Dale, Aug 20 2024 *)

A358444 a(1) = 1, a(2) = 2; for n > 2, a(n) = smallest positive number which has not appeared that has a common factor with a(n-2)^2 + a(n-1)^2.

Original entry on oeis.org

1, 2, 5, 29, 4, 857, 10, 734549, 539562233501, 6, 12433, 15, 8, 17, 353, 12, 124753, 13, 14, 20, 16, 18, 22, 24, 25, 1201, 26, 41, 2357, 28, 5556233, 37, 30, 2269, 39, 32, 35, 52, 3929, 40, 15438641, 82, 45, 65, 34, 5381, 78, 50, 36, 38, 42, 44, 46, 48, 51, 3, 9, 21, 27, 33, 54, 55, 91

Offset: 1

Scott R. Shannon, Nov 16 2022

nonn

The majority of terms are concentrated along or just above the line a(n) = n, resulting in 51 fixed points in the first 5000 terms. However, some terms are much larger because the sum of the squares of the previous two terms is a prime number.

Conjecture: the sequence is a permutation of the positive integers.

			a(5) = 4 as a(3)^2 + a(4)^2 = 25 + 841 = 866, and 4 is the smallest unused number that shares a factor with 866.
a(9) = 539562233501 as a(7)^2 + a(8)^2 = 100 + 539562233401 = 539562233501, which is a prime number.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 10000 terms where a(n) is within 10% of n. The green line is a(n) = n.
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..24857. a(24858) is a multiple of a prime factor of 2345424289569907866042152579118178340801^2 + 24922^2.
Michael De Vlieger, Table of n, a(n) for n = 1..65536.

Cf. A337136, A347594, A098550, A336957.

nn = 120; c[] = False; q[] = 1; Do[Set[{a[i], c[i], q[i]}, {i, True, 2}], {i, 2}]; i = a[1]^2; j = a[2]^2; Do[k = i + j; s = FactorInteger[k][[All, 1]]; Do[(m = q[#]; While[c[# m], m++]; q[#] = m; If[# m < k, k = # m]) &[s[[n]]], {n, Length[s]}]; Set[{a[n], c[k], i, j}, {k, True, j, k^2}], {n, 3, nn}]; Array[a, nn] (* Michael De Vlieger, Nov 17 2022 *)

cp's OEIS Frontend

A347754 a(n) = sqrt(A347594(n-1)^2 + n^2 + A347594(n)).

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A347595 a(0) = 1; for n>0, a(n) is the smallest positive integer that has not previously occurred such that a(n-1)^2 + n^2 + a(n) is a square.

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A349119 a(0) = 1; for n>0, a(n) is the smallest positive integer that has not previously occurred such that |n - a(n-1)| + a(n) is a square.

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A349182 a(0) = 1; for n>0, a(n) is the smallest positive integer such that |n - a(n-1)| + a(n) is a square.

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A358444 a(1) = 1, a(2) = 2; for n > 2, a(n) = smallest positive number which has not appeared that has a common factor with a(n-2)^2 + a(n-1)^2.

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