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A246655
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Prime powers: numbers of the form p^k where p is a prime and k >= 1.
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260
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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
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refs;
listen;
history;
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internal format)
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OFFSET
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1,1
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COMMENTS
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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
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
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LINKS
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FORMULA
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a(n) is characterized byA001221(a(n)) = 1.
a(n) is characterized byA014963(a(n))!= 1.
All three relations above are not true forA000961(n) instead of a(n).
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MAPLE
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select(t -> nops(numtheory:-factorset(t))=1, [$1..1000]); #Robert Israel,Sep 01 2014
isprimepower:= n -> nops(NumberTheory:-PrimeFactors(n)) = 1: #Peter Luschny,Oct 09 2022
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MATHEMATICA
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Select[Range[222], PrimePowerQ]
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PROG
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(Sage)
[n for n in (1..222) if sloane.A001221(n) == 1]
(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
for p in primerange(1, m):
pe = p
while pe < m:
pe *= p
(Python)
from sympy import primepi, integer_nthroot
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
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CROSSREFS
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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. AlsoA001597is the sequence of nontrivial powers n^k, n >= 1, k >= 2. -N. J. A. Sloane,Mar 24 2018
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KEYWORD
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nonn,nice,core,easy
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AUTHOR
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STATUS
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approved
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