A246655 Prime powers: numbers of the form p^k where p is a prime and k >= 1.

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211

Offset: 1

Author

Peter Luschny and Franklin T. Adams-Watters, Sep 01 2014

Keywords

• nonn
• nice
• core
• easy

Comments

The elements are called prime powers in contrast to the powers of primes which are the numbers of the same form but with k >= 0, cf. A000961.

Every nonzero integer is the product of elements of this sequence which are relatively prime and an element of {-1, 1}. This product is up to a rearrangement of the factors unique. (This statement is the fundamental theorem of arithmetic.)

These numbers are the numbers such that the von Mangoldt function is nonzero.

These numbers are the numbers of elements in finite fields. - Franz Vrabec, Aug 11 2004

A positive integer n is a prime power if and only if nZ is a primary ideal of Z. - John Cremona, Sep 02 2014

Also, numbers n divisible by their cototients A051953(n). - Ivan Neretin, May 29 2016

Numbers n such that (theta_3(q) - theta_3(q^n)) / 2 is the g.f. of a multiplicative sequence. - Michael Somos, Oct 17 2016

Numbers that are evenly divisible by exactly one prime number. - Lee A. Newberg, May 07 2018

Ram proved that these are precisely the numbers n such that the binomial coefficients n!/(m!(n-m)!) for m = 1..n-1 have a common factor greater than 1 (which is the unique prime dividing n). See Joris, Oestreicher & Steinig for a generalization. - Charles R Greathouse IV, Apr 24 2019

Blagojević & Ziegler prove that for these n and for any convex polygon in the plane, the polygon can be partitioned into n polygons with equal area and equal perimeter. The result is conjectured (by Nandakumar & Rao, who proved the case n = 2) to hold for all n. - Charles R Greathouse IV, Apr 24 2019

Numbers n such that A367064(n) < 0. - Chai Wah Wu, Nov 06 2023

This sequence represents all positive high amplitude peaks of the inverse Riemann spectrum R(x)= Sum_{k=1..oo} -cos(log(x)*Im(z_k)) going over the imaginary part of the nontrivial zeros "Im(z_k)" in the Riemann zeta function. - Marc Morgenegg, Jul 29 2025

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Brady Haran and Günter Ziegler, Cannons and Sparrows, Numberphile video (2018)
• H. Joris, C. Oestreicher and J. Steinig, The greatest common divisor of certain sets of binomial coefficients, Journal of Number Theory 21 (1985), pp. 101-119.
• Pavle V. M. Blagojević and Günter M. Ziegler, Convex equipartitions via equivariant obstruction theory, arXiv:1202.5504 [math.AT], 2012-2013; Israel Journal of Mathematics 200:1 (June 2014), pp 49-77.
• Karl-Heinz Kuhl, Inverse Riemann Spectrum
• Hans Montanus and Ron Westdijk, Cellular Automation and Binomials, Green Blue Mathematics (2022), p. 90.
• Laurentiu Panaitopol, Some of the properties of the sequence of powers of prime numbers, Rocky Mountain Journal of Mathematics, Volume 31, Number 4, Winter 2001.
• Balak Ram, Common factors of n!/(m!(n-m)!), (m = 1, 2, ... n-1), Journal of the Indian Mathematical Club (Madras) 1 (1909), pp. 39-43.
• Eric Weisstein's World of Mathematics, Prime Power
• Eric Weisstein's World of Mathematics, Projective Plane
• Chai Wah Wu, Algorithms for complementary sequences, arXiv:2409.05844 [math.NT], 2024.
• Wikipedia, Prime power
• Index entries for "core" sequences

Crossrefs

There are four different sequences which may legitimately be called "prime powers": A000961 (p^k, k >= 0), A246655 (p^k, k >= 1), A246547 (p^k, k >= 2), A025475 (p^k, k=0 and k >= 2). When you refer to "prime powers", be sure to specify which of these you mean. Also A001597 is the sequence of nontrivial powers n^k, n >= 1, k >= 2. - N. J. A. Sloane, Mar 24 2018

Cf. A001221, A014963, A069513, A367064.

Partial sums of A275120.

Programs

Maple

select(t -> nops(numtheory:-factorset(t))=1, [$1..1000]); # Robert Israel, Sep 01 2014
A246655 := proc(n) A000961(n+1) end proc: # R. J. Mathar, Jan 09 2017
isprimepower := n -> nops(NumberTheory:-PrimeFactors(n)) = 1: # Peter Luschny, Oct 09 2022

Mathematica

Select[Range[222], PrimePowerQ]

PARI

[p| p <- [1..222], isprimepower(p)]

PARI

list(lim)=my(v=List(primes([2,lim\=1]))); for(e=2,logint(lim,2), forprime(p=2,sqrtnint(lim,e), listput(v,p^e))); Set(v) \\ Charles R Greathouse IV, Feb 03 2023

Python

from sympy import primerange
m = 10**5
A246655 = []
for p in primerange(1,m):
    pe = p
    while pe < m:
        A246655.append(pe)
        pe *= p
A246655 = sorted(A246655) # Chai Wah Wu, Sep 04 2014

Python

from sympy import primepi, integer_nthroot
def A246655(n):
    def f(x): return int(n-1+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 20 2024

Sage

[n for n in (1..222) if sloane.A001221(n) == 1]

Formula

a(n) is characterized by A001221(a(n)) = 1.

a(n) is characterized by A014963(a(n)) != 1.

Euler's A000010(a(n)) = a(n)*(1 - 1/A014963(a(n))).

All three relations above are not true for A000961(n) instead of a(n).

Sum_{k=1..n} 1/a(k) ~ log(log(a(n))) + A077761 + A136141. - François Huppé, Jul 31 2024