cp's OEIS Frontend

A080101 Number of prime powers in all composite numbers between n-th prime and next prime.

%I A080101 #31 Mar 26 2025 08:30:35
%S A080101 0,1,0,2,0,1,0,0,2,0,1,0,0,0,1,0,0,1,0,0,0,1,0,0,0,0,0,0,0,2,1,0,0,0,
%T A080101 0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,
%U A080101 0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0
%N A080101 Number of prime powers in all composite numbers between n-th prime and next prime.
%C A080101 The maximum value of terms in the sequence, through the (10^5)th term, is 2. - _Harvey P. Dale_, Aug 24 2014
%C A080101 This is conjectured to be the maximum, see also A366833. - _Gus Wiseman_, Nov 06 2024
%H A080101 Amiram Eldar, <a href="/A080101/b080101.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harvey P. Dale)
%F A080101 a(n) = A366833(n) - 1. - _Gus Wiseman_, Nov 06 2024
%e A080101 There are two prime powers between 2179 = A000040(327) and 2203 = A000040(328): 2187 = 3^7 and 2197 = 13^3, therefore a(327) = 2, A080102(327) = 2187 and A080103(327) = 2197.
%p A080101 a := proc(n) local c, k, p: c, p := 0, ithprime(n): for k from p+1 to nextprime(p)-1 do if nops(numtheory:-factorset(k)) = 1 then c := c+1: fi: od: c: end:
%p A080101 seq(a(n), n = 1 .. 105); # _Lorenzo Sauras Altuzarra_, Jul 08 2022
%t A080101 prpwQ[n_]:=Module[{fi=FactorInteger[n]},Length[fi]==1&&fi[[1,2]]>1]; nn=600;With[{pwrs=Table[If[prpwQ[n],1,0],{n,nn}]},Table[Total[ Take[ pwrs,{Prime[n],Prime[n+1]}]],{n,PrimePi[nn]-1}]] (* _Harvey P. Dale_, Aug 24 2014 *)
%t A080101 Table[Length[Select[Range[Prime[n]+1,Prime[n+1]-1],PrimePowerQ]],{n,30}] (* _Gus Wiseman_, Nov 06 2024 *)
%Y A080101 Cf. A080102, A080103, A025475, A000961.
%Y A080101 For powers of 2 instead of primes we have A244508, see also A013597, A014210, A014234, A304521.
%Y A080101 Adding one gives A366833.
%Y A080101 For non-prime-powers instead of prime-powers we have A368748.
%Y A080101 Positions of positive terms are A377057, primes A053607.
%Y A080101 Positions of 0 are A377286.
%Y A080101 Positions of 1 are A377287.
%Y A080101 Positions of 2 are A377288, primes A053706.
%Y A080101 For perfect-powers (instead of prime-powers) we have A377432.
%Y A080101 A000015 gives the least prime-power >= n, difference A377282.
%Y A080101 A000040 lists the primes, differences A001223.
%Y A080101 A000961 lists the powers of primes, differences A057820, seconds A376596.
%Y A080101 A031218 gives the greatest prime-power <= n, difference A276781.
%Y A080101 A046933(n) counts the interval from A008864(n) to A006093(n+1).
%Y A080101 A065514 gives the greatest prime-power < prime(n), difference A377289.
%Y A080101 A246655 lists the prime-powers not including 1, complement A361102.
%Y A080101 A345531 gives the least prime-power > prime(n), difference A377281.
%Y A080101 Cf. A001597, A002808, A024619, A065890, A182908, A224363, A376597, A377051, A377054, A377436.
%K A080101 nonn
%O A080101 1,4
%A A080101 _Reinhard Zumkeller_, Jan 28 2003