A091529 -id:A091529 - cp's OEIS frontend

A254636 Numbers that cannot be represented as x*y + x + y, where x>=y>1.

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 16, 18, 21, 22, 25, 28, 30, 33, 36, 37, 40, 42, 45, 46, 52, 57, 58, 60, 61, 66, 70, 72, 73, 78, 81, 82, 85, 88, 93, 96, 100, 102, 105, 106, 108, 112, 117, 121, 126, 130, 133, 136, 138, 141, 145, 148, 150, 156, 157, 162, 165, 166, 172

Offset: 1

Alex Ratushnyak, Feb 03 2015

nonn

0, 7 and numbers n such that n+1 is either prime or twice a prime. - Robert Israel, Aug 05 2015

Cf. A091529 (appears to be essentially the same, except first few terms).

Cf. A253975.

sort([0,7, op(select(t -> isprime(t+1), [$1..10^4])), op(select(t -> isprime((t+1)/2),[2*i+1$i=1..5*10^3]))]); # Robert Israel, Aug 05 2015

r[n_] := Reduce[x >= y > 1 && n == x y + x + y, {x, y}, Integers];
Reap[For[n = 0, n <= 200, n++, If[r[n] === False, Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 28 2019 *)

from sympy import primepi
def A254636(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n-1+x-(x>=7)-primepi(x+1)-primepi(x+1>>1))
    return bisection(f,n-1,n-1) # Chai Wah Wu, Oct 14 2024

A091528 a(n) = (Sum_{k=1..n} H(k)k!(n-k)!) mod (n+1), where H(k) is the k-th harmonic number.

Original entry on oeis.org

1, 1, 0, 3, 4, 2, 0, 6, 6, 5, 0, 3, 8, 0, 0, 13, 0, 3, 0, 0, 12, 17, 0, 0, 14, 0, 0, 1, 0, 6, 0, 0, 18, 0, 0, 1, 20, 0, 0, 23, 0, 25, 0, 0, 24, 44, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 30, 8, 0, 36, 32, 0, 0, 0, 0, 10, 0, 0, 0, 2, 0, 56, 38, 0, 0, 0, 0, 19, 0, 0, 42, 48, 0, 0, 44, 0, 0, 6, 0, 0, 0, 0, 48, 0, 0

Offset: 1

Leroy Quet, Jan 08 2004

nonn

Cf. A091529, A091530.

Table[ Mod[ Sum[ HarmonicNumber[k]k!(n - k)!, {k, 1, n}], n + 1], {n, 1, 95}] (* or *) (* Robert G. Wilson v, Jan 14 2004 *)
h[n_] := If[ EvenQ[n], (1/2)HarmonicNumber[n/2], HarmonicNumber[n] - (1/2)HarmonicNumber[ Floor[n/2]]]; Table[ Mod[ n!h[n], n + 1], {n, 1, 95}]
(* or *) h[n_] := Sum[1/(2k - If[ EvenQ[n], 0, 1]), {k, 1, Floor[(n + 1)/2]}]; Table[ Mod[ n!h[n], n + 1], {n, 1, 95}]

It appears that a(n) is congruent to n!*h(n) (mod {n+1}) where h(n) = (1/2)*H(n/2) for even n and h(n) = H(n) - (1/2)*H(floor(n/2)) for odd n.

Edited and extended by Robert G. Wilson v, Jan 14 2004

cp's OEIS Frontend

A254636 Numbers that cannot be represented as x*y + x + y, where x>=y>1.

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A091528 a(n) = (Sum_{k=1..n} H(k)k!(n-k)!) mod (n+1), where H(k) is the k-th harmonic number.

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cp's OEIS Frontend

A254636 Numbers that cannot be represented as x*y + x + y, where x>=y>1.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

Python

A091528 a(n) = (Sum_{k=1..n} H(k)*k!*(n-k)!) mod (n+1), where H(k) is the k-th harmonic number.

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Mathematica

Formula

Extensions

A091528 a(n) = (Sum_{k=1..n} H(k)k!(n-k)!) mod (n+1), where H(k) is the k-th harmonic number.