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 A025475 1 and the prime powers p^m where m >= 2, thus excluding the primes. 183
 1, 4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1024, 1331, 1369, 1681, 1849, 2048, 2187, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4096, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889, 7921, 8192 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Also nonprime n such that sigma(n)*phi(n) > (n-1)^2. - Benoit Cloitre, Apr 12 2002 If p is a term of the sequence, then the index n for which a(n) = p is given by n := b(p) := 1 + Sum_{k>=2} PrimePi(p^(1/k)). Here, the sum has floor(log_2(p)) positive terms. For any m > 0, the greatest number n such that a(n) <= m is also given by b(m), thus, b(m) is the number of such prime powers <= m. - Hieronymus Fischer, May 31 2013 That 8 and 9 are the only two consecutive integers in this sequence is known as Catalan's Conjecture and was proved in 2002 by Preda Mihăilescu. - Geoffrey Critzer, Nov 15 2015 LINKS T. D. Noe, Table of n, a(n) for n = 1..10000 Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867v1 [math.NT], May 4, 2013. Preda Mihăilescu, On Catalan's Conjecture, Kuwait Foundation Lecture 30 - April 28, 2003. Eric Weisstein's World of Mathematics, Prime Power. FORMULA The number of terms <= N is O(sqrt(N)*log N). [See Weisstein link] - N. J. A. Sloane, May 27 2022 A005171(a(n))*A010055(a(n)) = 1. - Reinhard Zumkeller, Nov 01 2009 A192280(a(n)) = 0 for n > 1. - Reinhard Zumkeller, Aug 26 2011 A014963(a(n)) - A089026(a(n)) = A014963(a(n)) - 1. - Eric Desbiaux, May 18 2013 From Hieronymus Fischer, May 31 2013: (Start) The greatest number n such that a(n) <= m is given by 1 + Sum_{k>=2} A000720(floor(m^(1/k))). Example 1: m = 10^10 ==> n = 10085; Example 2: m = 10^11 ==> n = 28157; Example 3: m = 10^12 ==> n = 80071; Example 4: m = 10^15 ==> n = 1962690. (End) Sum_{n>=2} 1/a(n) = Sum_{p prime} 1/(p*(p-1)) = A136141. - Amiram Eldar, Oct 11 2020 From Amiram Eldar, Jan 28 2021: (Start) Product_{n>=2} (1 + 1/a(n)) = Product_{k>=2} zeta(k)/zeta(2*k) = 2.0729553047... Product_{n>=2} (1 - 1/a(n)) = A068982. (End) MAPLE with(numtheory); A025475:=proc(q) local n; print(1); for n from 2 to q do if not isprime(n) then if type(phi(n)/(n-phi(n)), integer) then print(n); fi; fi; od; end: A025475(10^6); # Paolo P. Lava, May 23 2013 # alternative implementation isA025475 := proc(n) if n < 1 then false; elif n = 1 then true; elif isprime(n) then false; elif nops(numtheory[factorset](n)) = 1 then true; else false; end if; end proc: A025475 := proc(n) option remember; local a; if n = 1 then 1; else for a from procname(n-1)+1 do if isA025475(a) then return a; end if; end do: end if; end proc: # R. J. Mathar, Jun 06 2013 # another alternative N:= 10^5: # to get all terms <= N Primes:= select(isprime, [2, (2*i+1 \$ i = 1 .. floor((sqrt(N)-1)/2))]): sort([1, seq(seq(p^i, i=2..floor(log[p](N))), p=Primes)]); # Robert Israel, Jul 27 2015 MATHEMATICA A025475 = Select[ Range[ 2, 10000 ], ! PrimeQ[ # ] && Mod[ #, # - EulerPhi[ # ] ] == 0 & ] A025475 = Sort[ Flatten[ Table[ Prime[n]^i, {n, 1, PrimePi[ Sqrt[10^4]]}, {i, 2, Log[ Prime[n], 10^4]}]]] {1}~Join~Select[Range[10^4], And[! PrimeQ@ #, PrimePowerQ@ #] &] (* Michael De Vlieger, Jul 04 2016 *) Join[{1}, Select[Range, PrimePowerQ[#]&&!PrimeQ[#]&]] (* Harvey P. Dale, Oct 29 2023 *) PROG (PARI) for(n=1, 10000, if(sigma(n)*eulerphi(n)*(1-isprime(n))>(n-1)^2, print1(n, ", "))) (PARI) is_A025475(n)={ ispower(n, , &p) && isprime(p) || n==1 } \\ M. F. Hasler, Sep 25 2011 (PARI) list(lim)=my(v=List(), L=log(lim+.5)); forprime(p=2, (lim+.5)^(1/3), for(e=3, L\log(p), listput(v, p^e))); vecsort(concat(Vec(v), apply(n->n^2, primes(primepi(sqrtint(lim\1)))))) \\ Charles R Greathouse IV, Nov 12 2012 (PARI) list(lim)=my(v=List()); for(m=2, logint(lim\=1, 2), forprime(p=2, sqrtnint(lim, m), listput(v, p^m))); Set(v) \\ Charles R Greathouse IV, Aug 26 2015 (Haskell) a025475 n = a025475_list !! (n-1) a025475_list = filter ((== 0) . a010051) a000961_list -- Reinhard Zumkeller, Jun 22 2011 (Python) from sympy import primerange A025475_list, m = , 10*2 m2 = m**2 for p in primerange(1, m): a = p**2 while a < m2: A025475_list.append(a) a *= p A025475_list = sorted(A025475_list) # Chai Wah Wu, Sep 08 2014 CROSSREFS Subsequence of A000961. - Reinhard Zumkeller, Jun 22 2011 Cf. A001597, A000720, A068982, A136141, A193166. Differences give A053707. Cf. A076048 (number of terms < 10^n). 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 Sequence in context: A319163 A134611 A134612 * A246547 A195942 A125643 Adjacent sequences: A025472 A025473 A025474 * A025476 A025477 A025478 KEYWORD nonn,easy,nice AUTHOR David W. Wilson EXTENSIONS Edited by Daniel Forgues, Aug 18 2009 STATUS approved

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Last modified December 6 16:01 EST 2023. Contains 367612 sequences. (Running on oeis4.)