A085985 -id:A085985 - cp's OEIS frontend

A085818 For n > 1: a(n) = p if n = p^e with p prime and e > 1, otherwise a(n) = (n-m)-th prime, where m = number of nonprime prime powers <= n; a(1)=1.

1, 2, 3, 2, 5, 7, 11, 2, 3, 13, 17, 19, 23, 29, 31, 2, 37, 41, 43, 47, 53, 59, 61, 67, 5, 71, 3, 73, 79, 83, 89, 2, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 7, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241

Offset: 1

Reinhard Zumkeller, Jul 04 2003

nonn

a(n) = A025473(n) if n = p^e with p prime and e > 1, otherwise a(n) = A008578(n-A085501(n));
n divides A085819(n) = Product_{k<=n} a(k), as by construction: a(1)=1; if n divides A085819(n-1) then a(n) = smallest prime not occurring earlier; if n does not divide A085819(n-1) then a(n) = greatest prime factor of n (A006530);
A000040 occurs infinitely many times as a subsequence.
a(A085971(n))=A000040(n) and for all k > 1: a(A000040(n)^k)=A000040(n); A085985(n)=A049084(a(n)). - Reinhard Zumkeller, Jul 06 2003

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A000040, A008578, A085971.

Cf. A006530, A025473, A085501, A085819, A085985, A049084.

f(n) = 1 + sum(k=2, n, isprimepower(k) && !isprime(k));  \\ A085501
a(n) = {if (n==1, return (1)); my(p); if (isprimepower(n, &p) && !isprime(n), p, prime(n-f(n)));} \\ Michel Marcus, Jan 28 2021

from sympy import primefactors, prime, primepi, integer_nthroot
def A085818(n): return 1 if n==1 else (f[0] if len(f:=primefactors(n))==1 and f[0]Chai Wah Wu, Aug 20 2024

A090729 a(n) = 21a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 21.

Original entry on oeis.org

2, 21, 439, 9198, 192719, 4037901, 84603202, 1772629341, 37140612959, 778180242798, 16304644485799, 341619353958981, 7157701788652802, 149970118207749861, 3142214780574094279, 65836540273848229998

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 18 2004

A Chebyshev T-sequence with Diophantine property.

a(n) gives the general (nonnegative integer) solution of the Pell equation a^2 - 437*b^2 =+4 with companion sequence b(n)=A092499(n), n>=0.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Indranil Ghosh, Table of n, a(n) for n = 0..755
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (21,-1).

Cf. A085985.
a(n)=sqrt(4 + 437*A092499(n)^2), n>=1, (Pell equation d=437, +4).
Cf. A077428, A078355 (Pell +4 equations).

a[0] = 2; a[1] = 21; a[n_] := 21a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *)

[lucas_number2(n,21,1) for n in range(0,20)] # Zerinvary Lajos, Jun 27 2008

a(n) = S(n, 21) - S(n-2, 21) = 2*T(n, 21/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 21)=A092499(n+1). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.
a(n) = ap^n + am^n, with ap := (21+sqrt(437))/2 and am := (21-sqrt(437))/2.
G.f.: (2-21*x)/(1-21*x+x^2).

Chebyshev and Pell comments from Wolfdieter Lang, Sep 10 2004

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A085818 For n > 1: a(n) = p if n = p^e with p prime and e > 1, otherwise a(n) = (n-m)-th prime, where m = number of nonprime prime powers <= n; a(1)=1.

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A090729 a(n) = 21a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 21.

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