cp's OEIS Frontend

A345531 Smallest prime power greater than the n-th prime.

%I A345531 #68 Nov 23 2024 09:28:28
%S A345531 3,4,7,8,13,16,19,23,25,31,32,41,43,47,49,59,61,64,71,73,79,81,89,97,
%T A345531 101,103,107,109,113,121,128,137,139,149,151,157,163,167,169,179,181,
%U A345531 191,193,197,199,211,223,227,229,233,239,241,243,256,263,269,271,277
%N A345531 Smallest prime power greater than the n-th prime.
%C A345531 Take the family of correlated prime-indexed conjectures appearing in A343249 - A343253, in which an alternative formula for the p-adic order of positive integers is proposed. There, the general p-indexed conjecture says that v_p(n), the p-adic order of n, is given by the formula: v_p(n) = log_p(n / L_p(k0, n)), where L_p(k0, n) is the lowest common denominator of the elements of the set S_p(k0, n) = {(1/n)*binomial(n, k), with 0 < k <= k0 such that k is not divisible by p}. Evidence suggests that the primality of p is a necessary condition in this general conjecture. So, if a composite number q is used instead of a prime p in the proposed formula for the p-adic (now, q-adic) order of n, the first counterexample (failure) is expected to occur for n = q * a(i), where i is the index of the smallest prime that divides q.
%C A345531 The prime-power a(n) is at most the next prime, so this sequence is strictly increasing. See also A366833. - _Gus Wiseman_, Nov 06 2024
%H A345531 Robert Israel, <a href="/A345531/b345531.txt">Table of n, a(n) for n = 1..10000</a>
%H A345531 Dario T. de Castro, <a href="http://math.colgate.edu/~integers/w61/w61.pdf">P-adic Order of Positive Integers via Binomial Coefficients</a>, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 22, Paper A61, 2022.
%F A345531 a(n) = A000015(1+A000040(n)). - _Antti Karttunen_, Jul 19 2021
%F A345531 a(n) = A000015(A008864(n)). - _Omar E. Pol_, Oct 27 2021
%e A345531 a(4) = 8 because the fourth prime number is 7, and the least power of a prime which is greater than 7 is 2^3 = 8.
%p A345531 f:= proc(n) local p,x;
%p A345531   p:= ithprime(n);
%p A345531   for x from p+1 do
%p A345531     if nops(numtheory:-factorset(x)) = 1 then return x fi
%p A345531   od
%p A345531 end proc:
%p A345531 map(f, [$1..100]); # _Robert Israel_, Aug 25 2024
%t A345531 a[i_]:= Module[{j, k, N = 0, tab={}}, tab = Sort[Drop[DeleteDuplicates[Flatten[Table[ If[Prime[j]^k > Prime[i], Prime[j]^k], {j, 1, i+1}, {k, 1, Floor[Log[Prime[j], Prime[i+1]]]}]]], 1]]; N = Take[tab, 1][[1]]; N];
%t A345531 tabseq = Table[a[i],{i, 1, 100}];
%t A345531 (* second program *)
%t A345531 Table[NestWhile[#+1&,Prime[n]+1, Not@*PrimePowerQ],{n,100}] (* _Gus Wiseman_, Nov 06 2024 *)
%o A345531 (PARI)
%o A345531 A000015(n) = for(k=n,oo,if((1==k)||isprimepower(k),return(k)));
%o A345531 A345531(n) = A000015(1+prime(n)); \\ _Antti Karttunen_, Jul 19 2021
%o A345531 (Python)
%o A345531 from itertools import count
%o A345531 from sympy import prime, factorint
%o A345531 def A345531(n): return next(filter(lambda m:len(factorint(m))<=1, count(prime(n)+1))) # _Chai Wah Wu_, Oct 25 2024
%Y A345531 Cf. A000040, A007814, A007949, A008864, A343249, A343250, A112765, A343251, A214411, A343252, A343253.
%Y A345531 Starting with n instead of prime(n): A000015, A031218, A377468, A377780, A377782.
%Y A345531 Opposite (greatest prime-power less than): A065514, A377289, A377781.
%Y A345531 For squarefree instead of prime-power: A112926, opposite A112925.
%Y A345531 The difference from prime(n) is A377281.
%Y A345531 The prime terms have indices A377286(n) - 1.
%Y A345531 First differences are A377703.
%Y A345531 A version for perfect-powers is A378249.
%Y A345531 A000961 and A246655 list the prime-powers, differences A057820.
%Y A345531 A024619 and A361102 list the non-prime-powers, differences A375735.
%Y A345531 Cf. A366833, A053607, A080101, A304521, A366833, A377057, A377283, A377287, A377288, A377432.
%K A345531 nonn
%O A345531 1,1
%A A345531 _Dario T. de Castro_, Jun 20 2021