A061344 Numbers of form p^m + 1, p odd prime, m >= 1.

4, 6, 8, 10, 12, 14, 18, 20, 24, 26, 28, 30, 32, 38, 42, 44, 48, 50, 54, 60, 62, 68, 72, 74, 80, 82, 84, 90, 98, 102, 104, 108, 110, 114, 122, 126, 128, 132, 138, 140, 150, 152, 158, 164, 168, 170, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240

Offset: 1

Author

Hans Dieter Lueke (lueke(AT)ient.rwth-aachen.de), Jun 08 2001

Keywords

• nonn
• easy

Comments

Lengths of almost-binary sequences with perfect odd-periodic autocorrelation function.

As J. Arndt points out, each element of this sequence leads to a conference matrix (cf. link to Wikipedia and A000952). - M. F. Hasler, Mar 14 2008

References

• H. D. Lueke, Binary odd-periodic complementary sequences. IEEE Trans. Inform. Theory, 43, pp. 365-367, 1997.

Links

• M. F. Hasler, Table of n, a(n) for n = 1..10000
• Wikipedia, Conference matrix.

Crossrefs

Equals A061345 + 1. Cf. A000952.

Programs

PARI

A061344(n)= local(m=1,p); for(c=1,n, until( isprime(m+=2) || ispower(m,[null], && p) && isprime(p),); /*print(c," ",m+1)*/); m+1 \\ - M. F. Hasler, Mar 14 2008

Python

from sympy import primepi, integer_nthroot
def A061344(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = 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+x-sum(primepi(integer_nthroot(x,k)[0])-1 for k in range(1,x.bit_length())))
    return bisection(f,n+1,n+1)+1 # Chai Wah Wu, Feb 03 2025

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jun 12 2001

Edited by M. F. Hasler, Mar 14 2008