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A000961 Prime powers p^k (p prime, k >= 0).
(Formerly M0517 N0185)
370
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, 223, 227 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Since 1 = p^0 does not have a well defined prime base p, it is sometimes not regarded as a prime power.

These numbers are (apart from 1) the numbers of elements in finite fields. - Franz Vrabec, Aug 11 2004

Numbers whose divisors form a geometrical progression. The divisors of p^k are 1, p, p^2, p^3, ...p^k. - Amarnath Murthy, Jan 09 2002

a(n) = A025473(n)^A025474(n). - David Wasserman, Feb 16 2006

a(n) = A117331(A117333(n)). - Reinhard Zumkeller, Mar 08 2006

These are also precisely the orders of those finite affine planes that are known to exist as of today. (The order of a finite affine plane is the number of points in an arbitrarily chosen line of that plane. This number is unique for all lines comprise the same number of points.) - Peter C. Heinig (algorithms(AT)gmx.de), Aug 09 2006

Except for first term, the index of the second number divisible by n in A002378, if the index equals n. - Mats Granvik, Nov 18 2007

These are precisely the numbers such that lcm(1,...,m-1)<lcm(1,...,m) (=A003418(m) for m>0; here for m=1, the l.h.s. is taken to be 0). We have a(n+1)=a(n)+1 if a(n) is a Mersenne prime or a(n)+1 is a Fermat prime; the converse is true except for n=7 (from Catalan's conjecture) and n=1, since 2^1-1 and 2^0+1 are not considered as Mersenne resp. Fermat prime. - M. F. Hasler, Jan 18 2007, Apr 18 2010

The sequence is A000015 without repetitions, or more formally, A000961=Union[A000015]. - Zak Seidov, Feb 06 2008

Except for a(1)=1, indices for which the cyclotomic polynomial Phi[k] yields a prime at x=1, cf. A020500. - M. F. Hasler, Apr 04 2008

Also, {A138929(k) ; k>1} = {2*A000961(k) ; k>1} = {4,6,8,10,14,16,18,22,26,32,34,38,46,50,54,58,62,64,74,82,86,94,98,...} are exactly the indices for which Phi[k](-1) is prime. - M. F. Hasler, Apr 04 2008

A143201(a(n)) = 1. [From Reinhard Zumkeller, Aug 12 2008]

Number of distinct primes dividing n=omega(n)<2. [From Juri-Stepan Gerasimov, Oct 30 2009]

Or, prime numbers^nonnegative numbers (without repetition). Numbers n such that sum{p-1|p is prime and divisor of n}=product{p-1|p is prime and divisor of n}. A055631(n)=A173557(n-1). [From Juri-Stepan Gerasimov, Dec 09 2009, Mar 10 2010]

Numbers n such that A028236(n) = 1. [From Klaus Brockhaus, Nov 06 2010]

A188666(k) = a(k+1) for k: 2*a(k) <= k < 2*a(k+1), k > 0; notably a(n+1) = A188666(2*a(n)). [Reinhard Zumkeller, Apr 25 2011]

A003415(a(n)) = A192015(n); A068346(a(n)) = A192016(n); a(n)=A192134(n)+A192015(n). [Reinhard Zumkeller, Jun 26 2011]

A089233(a(n)) = 0. - Reinhard Zumkeller, Sep 04 2013

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.

M. Koecher and A. Krieg, Ebene Geometrie, Springer, 1993

R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge 1986, Theorem 2.5, p. 45.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Eric Weisstein's World of Mathematics, Prime Power

Eric Weisstein's World of Mathematics, Projective Plane

Index entries for "core" sequences

FORMULA

m=a(n) for some n <=> lcm(1,...,m-1)<lcm(1,...,m), where lcm(1...0):=0 as to include a(1)=1. a(n+1)=a(n)+1 <=> a(n+1)=A019434(k) or a(n)=A000668(k) for some k (by Catalan's conjecture), except for n=1 and n=7. - M. F. Hasler, Jan 18 2007, Apr 18 2010

A001221(a(n))<2. [From Juri-Stepan Gerasimov, Oct 30 2009]

MAPLE

readlib(ifactors): for n from 1 to 250 do if nops(ifactors(n)[2])=1 then printf(`%d, `, n) fi: od:

A000961 := proc(n)

    option remember;

    local k ;

    if n = 1 then

        1;

    else

        for k from procname(n-1)+1 do

            if nops(numtheory[factorset](k)) = 1 then

                return k ;

            end if;

        end do:

    end if;

end proc: # Alois P. Heinz, Apr 08 2013

MATHEMATICA

Select[ Range[ 2, 250 ], Mod[ #, # - EulerPhi[ # ] ] == 0 & ]

Select[ Range[ 2, 250 ], Length[FactorInteger[ # ] ] == 1 & ]

max = 0; a = {}; Do[m = FactorInteger[n]; w = Sum[m[[k]][[1]]^m[[k]][[2]], {k, 1, Length[m]}]; If[w > max, AppendTo[a, n]; max = w], {n, 1, 1000}]; a(*Artur Jasinski*)

PROG

(MAGMA) [1] cat [ n : n in [2..250] | IsPrimePower(n) ]; // corrected by Arkadiusz Wesolowski, Jul 20 2012

(PARI) A000961(n, l=-1, k=0)=until(n--<1, until(l<lcm(l, k++), ); l=lcm(l, k)); k print_A000961(lim=999, l=-1)=for(k=1, lim, l==lcm(l, k)&next; l=lcm(l, k); print1(k, ", ")) - M. F. Hasler, Jan 18 2007

(PARI) isA000961(n) = (omega(n) == 1 | n == 1) [From Michael B. Porter, Sep 23 2009]

(PARI) nextA000961(n) = {local(m, r, p); m=2*n; for(e=1, ceil(log(n+0.01)/log(2)), r=(n+0.01)^(1/e); p=prime(primepi(r)+1); m=min(m, p^e)); m} [From Michael B. Porter, Nov 02 2009]

(PARI) is(n)=isprimepower(n) || n==1 \\ Charles R Greathouse IV, Nov 20 2012

(PARI) list(lim)=my(v=primes(primepi(lim)), u=List([1])); forprime(p=2, sqrtint(lim\1), for(e=2, log(lim+.5)\log(p), listput(u, p^e))); vecsort(concat(v, Vec(u))) \\ Charles R Greathouse IV, Nov 20 2012

(Haskell)

import Data.Set (singleton, deleteFindMin, insert)

a000961 n = a000961_list !! (n-1)

a000961_list = 1 : g (singleton 2) (tail a000040_list) where

   g s (p:ps) = m : g (insert (m * a020639 m) $ insert p s') ps

     where (m, s') = deleteFindMin s

-- Reinhard Zumkeller, May 01 2012, Apr 25 2011

(Sage)

def A000961_list(n) :

    R = [1]

    for i in (2..n) :

        if i.is_prime_power() : R.append(i)

    return R

A000961_list(227) # Peter Luschny, Feb 07 2012

CROSSREFS

Cf. A010055, A065515, A095874, A025473.

Cf. indices of record values of A003418; A000668 and A019434 give a member of twin pairs a(n+1)=a(n)+1.

A138929(n) = 2*a(n).

Cf. A000040, A001221, A001477. [From Juri-Stepan Gerasimov, Dec 09 2009]

A028236 (if n = Product (p_j^k_j), a(n) = numerator of Sum 1/p_j^k_j). [From Klaus Brockhaus, Nov 06 2010]

A000015(n) = Min{term : >= n}; A031218(n) = Max{term : <= n}.

Sequence in context: A059046 A144711 A036116 * A128603 A195943 A096165

Adjacent sequences:  A000958 A000959 A000960 * A000962 A000963 A000964

KEYWORD

nonn,easy,core,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Corrected comment and formula referring to Catalan's conjecture M. F. Hasler, Apr 18 2010

STATUS

approved

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Last modified April 25 00:50 EDT 2014. Contains 240991 sequences.