A005074 -id:A005074 - cp's OEIS frontend

A090306 a(n) = 17*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 17.

2, 17, 291, 4964, 84679, 1444507, 24641298, 420346573, 7170533039, 122319408236, 2086600473051, 35594527450103, 607193567124802, 10357885168571737, 176691241432844331, 3014108989526925364, 51416544063390575519

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 25 2004

Lim_{n-> infinity} a(n)/a(n+1) = 0.058621... = 2/(17+sqrt(293)) = (sqrt(293)-17)/2.
Lim_{n-> infinity} a(n+1)/a(n) = 17.058621... = (17+sqrt(293))/2 = 2/(sqrt(293)-17).
For more information about this type of recurrence follow the Khovanova link and see A054413, A086902 and A178765. - Johannes W. Meijer, Jun 12 2010

			a(4) = 17*a(3) + a(2) = 17*4964 + 291=((17+sqrt(293))/2)^4 + ((17-sqrt(293))/2)^4 = 84678.999988190 + 0.000011809 = 84679.

G. C. Greubel, Table of n, a(n) for n = 0..500
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for linear recurrences with constant coefficients, signature (17,1).

Cf. A005074.

Lucas polynomials Lucas(n,m): A000032 (m=1), A002203 (m=2), A006497 (m=3), A014448 (m=4), A087130 (m=5), A085447 (m=6), A086902 (m=7), A086594 (m=8), A087798 (m=9), A086927 (m=10), A001946 (m=11), A086928 (m=12), A088316 (m=13), A090300 (m=14), A090301 (m=15), A090305 (m=16), this sequence (m=17), A090307 (m=18), A090308 (m=19), A090309 (m=20), A090310 (m=21), A090313 (m=22), A090314 (m=23), A090316 (m=24), A330767 (m=25).

m:=17;; a:=[2,m];; for n in [3..20] do a[n]:=m*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Dec 30 2019

m:=17; I:=[2,m]; [n le 2 select I[n] else m*Self(n-1) +Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 30 2019

seq(simplify(2*(-I)^n*ChebyshevT(n, 17*I/2)), n = 0..20); # G. C. Greubel, Dec 30 2019

LinearRecurrence[{17,1},{2,17},30] (* Harvey P. Dale, Jan 24 2018 *)
LucasL[Range[20]-1, 17] (* G. C. Greubel, Dec 30 2019 *)

vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 17*I/2) ) \\ G. C. Greubel, Dec 30 2019

[2*(-I)^n*chebyshev_T(n, 17*I/2) for n in (0..20)] # G. C. Greubel, Dec 30 2019

a(n) = 17*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 17.
a(n) = ((17+sqrt(293))/2)^n + ((17-sqrt(293))/2)^n.
(a(n))^2 = a(2n) - 2 if n=1, 3, 5, ...
(a(n))^2 = a(2n) + 2 if n=2, 4, 6, ...
G.f.: (2-17*x)/(1-17*x-x^2). - Philippe Deléham, Nov 02 2008
From Johannes W. Meijer, Jun 12 2010: (Start)
a(2n+1) = 17*A098249(n).
a(3n+1) = A041550(5n), a(3n+2) = A041550(5n+3), a(3n+3) = 2*A041550(5n+4).
Lim_{k-> infinity} a(n+k)/a(k) = (A090306(n) + A178765(n)*sqrt(293))/2.
Lim_{n-> infinity} A090306(n)/A178765(n) = sqrt(293). (End)
a(n) = Lucas(n, 17) = 2*(-i)^n * ChebyshevT(n, 17*i/2). - G. C. Greubel, Dec 30 2019
E.g.f.: 2*exp(17*x/2)*cosh(sqrt(293)*x/2). - Stefano Spezia, Dec 31 2019

More terms from Ray Chandler, Feb 14 2004

A005075 Sum of squares of primes = 2 mod 3 dividing n.

Original entry on oeis.org

0, 4, 0, 4, 25, 4, 0, 4, 0, 29, 121, 4, 0, 4, 25, 4, 289, 4, 0, 29, 0, 125, 529, 4, 25, 4, 0, 4, 841, 29, 0, 4, 121, 293, 25, 4, 0, 4, 0, 29, 1681, 4, 0, 125, 25, 533, 2209, 4, 0, 29, 289, 4, 2809, 4, 146, 4, 0, 845, 3481, 29, 0, 4, 0, 4, 25, 125, 0, 293, 529, 29, 5041, 4, 0, 4, 25, 4, 121, 4, 0, 29, 0, 1685, 6889, 4, 314

Offset: 1

N. J. A. Sloane

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000290, A005063, A005071, A005074, A005076, A005077, A008472, A079978.

Array[DivisorSum[#, #^2 &, And[PrimeQ@ #, Mod[#, 3] == 2] &] &, 85] (* Michael De Vlieger, Jul 11 2017 *)
f[p_, e_] := If[Mod[p, 3] == 2, p^2, 0]; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 21 2022 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%3) == 2, p^2)); \\ Michel Marcus, Jul 11 2017

(define (A005075 n) (if (= 1 n) 0 (+ (A000290 (if (= 2 (modulo (A020639 n) 3)) (A020639 n) 0)) (A005075 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

Additive with a(p^e) = p^2 if p = 2 (mod 3), 0 otherwise.

a(n) = A005063(n) - A005071(n) - 9*A079978(n). - Antti Karttunen, Jul 10 2017

More terms from Antti Karttunen, Jul 10 2017

A005076 Sum of cubes of primes = 2 mod 3 dividing n.

Original entry on oeis.org

0, 8, 0, 8, 125, 8, 0, 8, 0, 133, 1331, 8, 0, 8, 125, 8, 4913, 8, 0, 133, 0, 1339, 12167, 8, 125, 8, 0, 8, 24389, 133, 0, 8, 1331, 4921, 125, 8, 0, 8, 0, 133, 68921, 8, 0, 1339, 125, 12175, 103823, 8, 0, 133, 4913, 8, 148877, 8, 1456, 8, 0, 24397, 205379, 133, 0, 8, 0, 8, 125, 1339, 0, 4921, 12167, 133, 357911

Offset: 1

N. J. A. Sloane

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000578, A005064, A005072, A005074, A005075, A005077, A008472, A079978.

Array[DivisorSum[#, #^3 &, And[PrimeQ@ #, Mod[#, 3] == 2] &] &, 71] (* Michael De Vlieger, Jul 11 2017 *)
f[p_, e_] := If[Mod[p, 3] == 2, p^3, 0]; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 21 2022 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%3) == 2, p^3)); \\ Michel Marcus, Jul 11 2017

(define (A005076 n) (if (= 1 n) 0 (+ (A000578 (if (= 2 (modulo (A020639 n) 3)) (A020639 n) 0)) (A005076 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

Additive with a(p^e) = p^3 if p = 2 (mod 3), 0 otherwise.

a(n) = A005064(n) - A005072(n) - 27*A079978(n). - Antti Karttunen, Jul 10 2017

More terms from Antti Karttunen, Jul 10 2017

A005077 Sum of 4th powers of primes = 2 mod 3 dividing n.

Original entry on oeis.org

0, 16, 0, 16, 625, 16, 0, 16, 0, 641, 14641, 16, 0, 16, 625, 16, 83521, 16, 0, 641, 0, 14657, 279841, 16, 625, 16, 0, 16, 707281, 641, 0, 16, 14641, 83537, 625, 16, 0, 16, 0, 641, 2825761, 16, 0, 14657, 625, 279857, 4879681, 16, 0, 641, 83521, 16, 7890481, 16, 15266, 16, 0, 707297, 12117361, 641, 0, 16, 0, 16, 625, 14657, 0, 83537

Offset: 1

N. J. A. Sloane

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000583, A005065, A005073, A005074, A005075, A005076, A008472, A079978.

Array[DivisorSum[#, #^4 &, And[PrimeQ@ #, Mod[#, 3] == 2] &] &, 68] (* Michael De Vlieger, Jul 11 2017 *)
f[p_, e_] := If[Mod[p, 3] == 2, p^4, 0]; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 21 2022 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%3) == 2, p^4)); \\ Michel Marcus, Jul 11 2017

(define (A005077 n) (if (= 1 n) 0 (+ (A000583 (if (= 2 (modulo (A020639 n) 3)) (A020639 n) 0)) (A005077 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

Additive with a(p^e) = p^4 if p = 2 (mod 3), 0 otherwise.

a(n) = A005065(n) - A005073(n) - 81*A079978(n). - Antti Karttunen, Jul 10 2017

More terms from Antti Karttunen, Jul 10 2017

A005090 Number of primes == 2 mod 3 dividing n.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 1, 1, 1, 1, 0, 2, 0, 2, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 2, 1, 2, 1, 1, 0, 2, 1, 1, 1, 1, 2, 1, 0, 2, 1, 2, 0, 1, 0, 1, 1, 2, 0, 2, 1, 2, 1, 1, 0, 1, 1, 1, 1, 1, 0, 2, 0, 2, 1, 1, 2, 1, 1, 2, 1, 2, 0, 2, 0, 2, 1, 1, 0, 1, 1, 2, 1, 2, 0, 1, 1, 2, 1, 1, 0, 3, 0, 1, 1, 1, 2, 2, 0, 2, 1, 2

Offset: 1

N. J. A. Sloane

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001221, A005074, A005088, A079978.

Array[DivisorSum[#, 1 &, And[PrimeQ@ #, Mod[#, 3] == 2] &] &, 120] (* Michael De Vlieger, Jul 11 2017 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, (f[k,1] % 3) == 2); \\ Michel Marcus, Jul 11 2017

from sympy import primefactors
def a(n): return sum(1 for p in primefactors(n) if p%3==2)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 11 2017

(define (A005090 n) (if (= 1 n) 0 (+ (A004526 (modulo (A020639 n) 3)) (A005090 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

Additive with a(p^e) = 1 if p = 2 (mod 3), 0 otherwise.
From Antti Karttunen, Jul 10 2017: (Start)
a(1) = 0; for n > 1, floor((A020639(n) mod 3)/2) + a(A028234(n)).
a(n) = A001221(n) - A005088(n) - A079978(n).
(End)

More terms from Antti Karttunen, Jul 10 2017

A344908 Sum of the distinct odd-indexed prime divisors, p_{2k-1}, of n.

Original entry on oeis.org

0, 2, 0, 2, 5, 2, 0, 2, 0, 7, 11, 2, 0, 2, 5, 2, 17, 2, 0, 7, 0, 13, 23, 2, 5, 2, 0, 2, 0, 7, 31, 2, 11, 19, 5, 2, 0, 2, 0, 7, 41, 2, 0, 13, 5, 25, 47, 2, 0, 7, 17, 2, 0, 2, 16, 2, 0, 2, 59, 7, 0, 33, 0, 2, 5, 13, 67, 19, 23, 7, 0, 2, 73, 2, 5, 2, 11, 2, 0, 7, 0, 43, 83, 2, 22

Offset: 1

Wesley Ivan Hurt, Jun 02 2021

nonn

a(m) = 0 for m in A066207. - Michel Marcus, Jun 12 2021

Inverse Möbius transform of n * c(n) * (pi(n) mod 2), where c(n) is the prime characteristic (A010051). - Wesley Ivan Hurt, Jun 23 2024

			a(6) = Sum_{p|6} p * (pi(p) mod 2) = 2*(pi(2) mod 2) + 3*(pi(3) mod 2) = 2*1 + 3*0 = 2.

Martin Ehrenstein, Table of n, a(n) for n = 1..20000

Cf. A000720 (pi), A008472 (sopf), A005074, A010051, A066207, A324966.

Cf. A344931 (sum of distinct even-indexed prime divisors).

Table[Sum[k*Mod[PrimePi[k], 2] (PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]

a(n) = my(f=factor(n)); sum(k=1, #f~, if (primepi(f[k,1]) % 2, f[k,1])); \\ Michel Marcus, Jun 12 2021

a(n) = Sum_{p|n} p * (pi(p) mod 2).
G.f.: Sum_{k>=1} prime(2*k-1) * x^prime(2*k-1) / (1 - x^prime(2*k-1)). - Ilya Gutkovskiy, Oct 24 2023
a(n) = Sum_{d|n} d * c(d) * (pi(d) mod 2), where c = A010051. - Wesley Ivan Hurt, Jun 23 2024

A344931 Sum of the distinct even-indexed prime divisors, p_{2k}, of n.

Original entry on oeis.org

0, 0, 3, 0, 0, 3, 7, 0, 3, 0, 0, 3, 13, 7, 3, 0, 0, 3, 19, 0, 10, 0, 0, 3, 0, 13, 3, 7, 29, 3, 0, 0, 3, 0, 7, 3, 37, 19, 16, 0, 0, 10, 43, 0, 3, 0, 0, 3, 7, 0, 3, 13, 53, 3, 0, 7, 22, 29, 0, 3, 61, 0, 10, 0, 13, 3, 0, 0, 3, 7, 71, 3, 0, 37, 3, 19, 7, 16, 79, 0, 3, 0, 0, 10, 0, 43, 32

Offset: 1

Wesley Ivan Hurt, Jun 02 2021

nonn

Inverse Möbius transform of n * c(n) * ((pi(n)+1) mod 2), where c(n) is the prime characteristic (A010051). - Wesley Ivan Hurt, Jun 23 2024

			a(12) = Sum_{p|12} p * ((pi(p)+1) mod 2) = 2*0 + 3*1 = 3.

Martin Ehrenstein, Table of n, a(n) for n = 1..20000

Cf. A000720 (pi), A008472 (sopf), A005074, A324966.

Cf. A344908 (sum of distinct odd-indexed prime divisors).

Table[Sum[k*Mod[PrimePi[k] + 1, 2] (PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]

a(n) = my(f=factor(n)); sum(k=1, #f~, if (!(primepi(f[k,1]) % 2), f[k,1])); \\ Michel Marcus, Jun 12 2021

a(n) = Sum_{p|n} p * ((pi(p)+1) mod 2).
G.f.: Sum_{k>=1} prime(2*k) * x^prime(2*k) / (1 - x^prime(2*k)). - Ilya Gutkovskiy, Oct 24 2023
a(n) = Sum_{d|n} d * c(d) * ((pi(d)+1) mod 2), where c = A010051. - Wesley Ivan Hurt, Jun 23 2024

cp's OEIS Frontend

A090306 a(n) = 17*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 17.

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A005075 Sum of squares of primes = 2 mod 3 dividing n.

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A005076 Sum of cubes of primes = 2 mod 3 dividing n.

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A005077 Sum of 4th powers of primes = 2 mod 3 dividing n.

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A005090 Number of primes == 2 mod 3 dividing n.

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A344908 Sum of the distinct odd-indexed prime divisors, p_{2k-1}, of n.

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Mathematica

PARI

Formula

A344931 Sum of the distinct even-indexed prime divisors, p_{2k}, of n.