A090864 Complement of generalized pentagonal numbers (A001318).

3, 4, 6, 8, 9, 10, 11, 13, 14, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84

Offset: 1

Author

Jon Perry, Feb 12 2004

Keywords

• nonn
• easy

Comments

Also n for which A006906(n) is even, or equivalently n for which A000009(n) is even (since A006906 and A000009 have the same parity).

The number of partitions of a(n) into distinct parts with an even number of parts equals the number of such partitions with an odd number of parts: A067661(a(n)) = A067659(a(n)). See, e.g., the Freitag-Busam reference, p. 410 given in A036499. - Wolfdieter Lang, Jan 19 2016

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000009, A001318, A067659, A067661, A080995, A006906, A010815, A118303.

Cf. A085141, A111651.

Programs

Mathematica

Complement[Range[200], Select[Accumulate[Range[0,200]]/3, IntegerQ]] (* G. C. Greubel, Jun 06 2017 *)

PARI

a(n) = my(q,r); [q,r]=divrem(sqrtint(24*n),3); n + q + (r >= bitnegimply(1,q)); \\ Kevin Ryde, Sep 15 2024

Python

from math import isqrt
def A090864(n):
    def f(x): return n+(m:=isqrt(24*x+1)+1)//6+(m-2)//6
    kmin, kmax = 0,1
    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 # Chai Wah Wu, Aug 29 2024

Formula

A080995(a(n)) = 0; A000009(a(n)) = A118303(n). - Reinhard Zumkeller, Apr 22 2006

A010815(a(n)) = A067661(a(n)) - A067659(a(n)) = 0, n >= 1. See a comment above. - Wolfdieter Lang, Jan 19 2016

a(n) = n+1 + A085141(n-1) + A111651(n). - Kevin Ryde, Sep 15 2024

Extensions

More terms from Reinhard Zumkeller, Apr 22 2006

Edited by Ray Chandler, Dec 14 2011

Edited by Jon E. Schoenfield, Nov 25 2016