Burak Muslu - cp's OEIS frontend

A348569 a(n) = (prime(n) + prime(n+2))^2 + prime(n+1)^2.

58, 125, 305, 521, 953, 1313, 1961, 2833, 3757, 5317, 6553, 8081, 9593, 11425, 14045, 16477, 19597, 21913, 24641, 27829, 30577, 35113, 40321, 45509, 50201, 53873, 56393, 60281, 68465, 75665, 86857, 91669, 101117, 106301

Offset: 1

Burak Muslu, Oct 23 2021

The square of the shortest distance between the start and end points of the path followed by a person moving 90 degrees left and right (or left and right) at the end of each path on a path of three consecutive prime numbers.

			For n = 1 the a(1) = (prime(1) + prime(3))^2 + prime(2)^2 = (2 + 5)^2 + 3^2 = 49 + 9 = 58.

Cf. A000040, A090076, A133529.

Table[(Prime[n] + Prime[n + 2])^2 + Prime[n + 1]^2,{n,34}]

a(n) = prime(n)^2 + prime(n+1)^2 + prime(n+2)^2 + 2*prime(n)*prime(n+2).

A347702 Prime numbers that give a remainder of 1 when divided by the sum of their digits.

Original entry on oeis.org

11, 13, 17, 41, 43, 97, 101, 131, 157, 181, 233, 239, 271, 311, 353, 401, 421, 491, 521, 541, 599, 617, 631, 647, 673, 743, 811, 859, 953, 1021, 1031, 1051, 1093, 1171, 1201, 1249, 1259, 1301, 1303, 1327, 1373, 1531, 1601, 1621, 1801, 1871, 2029, 2111, 2129, 2161

Offset: 1

Burak Muslu, Sep 10 2021

			97 is a term since its sum of digits is 9+7 = 16, and 97 mod 16 = 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A007605, A136251.
Subsequence of A209871.
A259866 \ {31}, and the primes associated with A056804 \ {1, 2} and A056797 are subsequences.

select(t -> isprime(t) and t mod convert(convert(t,base,10),`+`) = 1, [seq(i,i=3..10000,2)]); # Robert Israel, Mar 05 2024

Select[Range[2000], PrimeQ[#] && Mod[#, Plus @@ IntegerDigits[#]] == 1 &] (* Amiram Eldar, Sep 10 2021 *)

isok(p) = isprime(p) && ((p % sumdigits(p)) == 1); \\ Michel Marcus, Sep 10 2021

from sympy import primerange
def ok(p): return p%sum(map(int, str(p))) == 1
print(list(filter(ok, primerange(1, 2130)))) # Michael S. Branicky, Sep 10 2021

A347530 Primes of the form (p^2 + 9)/2 where p is prime.

Original entry on oeis.org

17, 29, 89, 149, 269, 929, 1109, 1409, 3449, 5309, 6389, 8069, 12329, 14969, 33029, 34589, 42929, 47129, 48989, 60209, 67349, 78809, 98129, 109049, 118589, 136769, 158489, 175829, 213209, 264269, 317609, 338669, 363809, 367229, 389849, 438989, 454109, 467549

Offset: 1

Burak Muslu, Sep 05 2021

nonn

Each p is an odd number, so p^2 == 1 (mod 8), thus (p^2 + 9)/2 == 1 (mod 4).

			17 is in the sequence as 17 = (p^2 + 9)/2 where p = 5 is prime.
29 is in the sequence as 29 = (p^2 + 9)/2 where p = 7 is prime.

Cf. A000040, A045637, A062324.

Subsequence of A076727 and of A103739.

Select[(Select[Range[3, 1000], PrimeQ]^2 + 9)/2, PrimeQ] (* Amiram Eldar, Sep 05 2021 *)

A343008 a(n) = F(n+5) * F(n+2) - 12 * (-1)^n where F(n) = A000045(n) are the Fibonacci numbers.

Original entry on oeis.org

28, 27, 117, 260, 727, 1857, 4908, 12803, 33565, 87828, 229983, 602057, 1576252, 4126635, 10803717, 28284452, 74049703, 193864593, 507544140, 1328767763, 3478759213, 9107509812, 23843770287, 62423800985, 163427632732, 427859097147, 1120149658773

Offset: 1

Burak Muslu, Apr 02 2021

nonn

First differences of A341928.
Second differences of A341208.
Third differences of A338225.
Fourth differences of A226205.
Fourth differences between the areas of consecutive rectangles with side lengths F(n+3) and F(n).
Twice the fourth differences between the areas of consecutive deltoids with cross lengths F(n+3) and F(n).
Twice the fourth differences between the areas of consecutive triangles with the height and base length are F(n+3) and F(n).

			For n = 2, a(2) = F(2+5) * F(2+2) - 12 * (-1)^2 = 13 * 3 - 12 = 27.

B. Muslu, Sayılar ve Bağlantılar, Luna, 2021, p. 52.

Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A000045, A226205, A338225, A341208, A341928.

a[n_]:=Fibonacci[n+5]*Fibonacci[n+2]-12(-1)^n
Array[a,30] (* Giorgos Kalogeropoulos, Apr 02 2021 *)

a(n) = F(n+5) * F(n+2) - 12 * (-1)^n.

G.f.: x*(28 - 29*x + 7*x^2)/(1 - 2*x - 2*x^2 + x^3).

A341928 a(n) = F(n+4) * F(n+2) + 7 * (-1)^n where F(n) = A000045(n) are the Fibonacci numbers.

Original entry on oeis.org

3, 31, 58, 175, 435, 1162, 3019, 7927, 20730, 54295, 142123, 372106, 974163, 2550415, 6677050, 17480767, 45765219, 119814922, 313679515, 821223655, 2149991418, 5628750631, 14736260443, 38580030730, 101003831715, 264431464447, 692290561594, 1812440220367

Offset: 1

Burak Muslu, Feb 23 2021

First differences of A341208.
Second differences of A338225.
Third differences of A226205 n > 2.
Third differences between the areas of consecutive rectangles with side lengths F(n+3) and F(n).
Twice the third differences between the areas of consecutive deltoids with cross lengths F(n+3) and F(n).
Twice the third differences between the areas of consecutive triangles with the height and base length are F(n+3) and F(n).

			For n = 2, a(2) = F(2+4) * F(2+2) + 7 * (-1)^2 = 8 * 3 + 7 = 31.

Burak Muslu, Sayılar ve Bağlantılar, Luna, 2021, p. 51 (in Turkish).

Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A000045, A341208, A338225, A226205.

Table[Fibonacci[n + 4] * Fibonacci[n + 2] + 7 * (-1)^n, {n, 1, 28}] (* Amiram Eldar, Feb 23 2021 *)

a(n) = fibonacci(n+4)*fibonacci(n+2) + 7*(-1)^n; \\ Michel Marcus, Feb 23 2021

a(n) = F(n+4) * F(n+2) + 7 * (-1)^n.

G.f.: x*(3 + 25*x - 10*x^2)/(1 - 2*x - 2*x^2 + x^3).

A338225 a(n) = F(n+3) * F(n+1) + (-1)^n where F(n) = A000045(n) are the Fibonacci numbers.

Original entry on oeis.org

2, 11, 23, 66, 167, 443, 1154, 3027, 7919, 20738, 54287, 142131, 372098, 974171, 2550407, 6677058, 17480759, 45765227, 119814914, 313679523, 821223647, 2149991426, 5628750623, 14736260451, 38580030722, 101003831723, 264431464439, 692290561602, 1812440220359, 4745030099483

Offset: 1

Burak Muslu, Jan 30 2021

Twice the difference between the areas of consecutive deltoids with cross lengths F(n+3) and F(n).
Also it is the difference between the areas of consecutive rectangles with side lengths F(n+3) and F(n).
Also first differences of A226205 for n > 2.

			For n = 2, a(2) = F(2+3) * F(2+1) + (-1)^2 = 5 * 2 + 1 = 11.

Burak Muslu, Sayılar ve Bağlantılar, Luna, 2021, p. 50.

Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A000045, A007598, A226205, A341208.

a[n_] := Fibonacci[n + 3] * Fibonacci[n + 1] + (-1)^n; Array[a, 30] (* Amiram Eldar, Jan 30 2021 *)

a(n) = fibonacci(n+3)*fibonacci(n+1) + (-1)^n; \\ Michel Marcus, Mar 25 2021

a(n) = F(n+3) * F(n+1) + (-1)^n, for n > 0.
G.f.: x*(2 + 7*x - 3*x^2)/(1 - 2*x - 2*x^2 + x^3). - Stefano Spezia, Jan 30 2021
a(n) = F(n+2)^2 + 2*(-1)^n = A007598(n+2) + 2*(-1)^n. - Amiram Eldar, Jan 11 2022

A341208 a(n) = F(n+4) * F(n+1) - 4 * (-1)^n where F(n) = A000045(n) are the Fibonacci numbers.

Original entry on oeis.org

9, 12, 43, 101, 276, 711, 1873, 4892, 12819, 33549, 87844, 229967, 602073, 1576236, 4126651, 10803701, 28284468, 74049687, 193864609, 507544124, 1328767779, 3478759197, 9107509828, 23843770271, 62423801001, 163427632716, 427859097163, 1120149658757

Offset: 1

Burak Muslu, Feb 06 2021

First differences of A338225.

Also it is second differences between the areas of consecutive rectangles with side lengths F(n+3) and F(n).

			For n = 2, a(2) = F(2+4) * F(2+1) - 4 * (-1)^2 = 8 * 2 - 4 = 12.

Burak Muslu, Sayılar ve Bağlantılar, Luna, 2021, p. 51.

Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A000045, A338225

a(n) = fibonacci(n+4)*fibonacci(n+1) - 4*(-1)^n; \\ Michel Marcus, Feb 06 2021

a(n) = F(n+4) * F(n+1) - 4 * (-1)^n for n > 0.

G.f.: x*(9 - 6*x + x^2)/(1 - 2*x - 2*x^2 + x^3).

A337188 a(n) = determinant([a(n-1), a(n-2); a(n-4), a(n-3)]) for n >= 5, a(n) = n otherwise.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 13, 37, 194, 2263, 81209, 15670815, 35447299799, 2878604306322646, 45110072663945746399499, 1599030269628449375351280360624211, 4602975420092714513333476912306224941820648781605

Offset: 1

Burak Muslu, Jan 29 2021

nonn

B. Muslu, Sayılar ve Bağlantılar, Luna, 2021, pp. 18-22.

a:= proc(n) option remember; `if`(n<5, n,
      a(n-1)*a(n-3)-a(n-2)*a(n-4))
    end:
seq(a(n), n=1..18);  # Alois P. Heinz, Jan 29 2021

a[n_] := a[n] = If[n < 5, n, Det @ Map[a, n - {{1, 2}, {4, 3}}, {2}]]; Array[a, 20] (* Amiram Eldar, Jan 29 2021 *)
nxt[{a_,b_,c_,d_}]:={b,c,d,Det[{{d,c},{a,b}}]}; NestList[nxt,{1,2,3,4},20][[All,1]] (* Harvey P. Dale, Oct 23 2022 *)

a(n) = if (n<=4, n, a(n-1)*a(n-3) - a(n-2)*a(n-4)); \\ Michel Marcus, Jan 29 2021

a(n) = a(n-1)*a(n-3) - a(n-2)*a(n-4) for n >= 5, a(n) = n for n <= 4.

cp's OEIS Frontend

User: Burak Muslu

A348569 a(n) = (prime(n) + prime(n+2))^2 + prime(n+1)^2.

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A347702 Prime numbers that give a remainder of 1 when divided by the sum of their digits.

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A347530 Primes of the form (p^2 + 9)/2 where p is prime.

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A343008 a(n) = F(n+5) * F(n+2) - 12 * (-1)^n where F(n) = A000045(n) are the Fibonacci numbers.

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A341928 a(n) = F(n+4) * F(n+2) + 7 * (-1)^n where F(n) = A000045(n) are the Fibonacci numbers.

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A338225 a(n) = F(n+3) * F(n+1) + (-1)^n where F(n) = A000045(n) are the Fibonacci numbers.

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A341208 a(n) = F(n+4) * F(n+1) - 4 * (-1)^n where F(n) = A000045(n) are the Fibonacci numbers.

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