login
Number of prime powers of prime numbers less than 10^n.
1

%I #22 Sep 13 2024 11:29:01

%S 3,7,18,40,88,208,513,1345,3605,9965,27984,79815,230184,669541,

%T 1961782,5781027,17121898,50926700,152037454,455379086,1367866472,

%U 4119420084,12434683582,37613680284,113995431428,346090115288,1052421037283,3205047207003,9774084239631

%N Number of prime powers of prime numbers less than 10^n.

%H Chai Wah Wu, <a href="/A076700/b076700.txt">Table of n, a(n) for n = 1..34</a>

%e a(2) = 7 because 4, 8, 9, 25, 27, 32 and 49 are the only prime powers of prime numbers less than 100.

%t pp = Sort[ Flatten[ Table[ Prime[n]^Prime[i], {n, 1, PrimePi[ Sqrt[10^17]]}, {i, 1, PrimePi[ Floor[ Log[ Prime[n], 10^17]]]}]]]; a = {}; k = 1; Do[ While[ pp[[k]] <= 10^n, k++ ]; a = Append[a, k-1], {n, 1, 17}]

%o (Python)

%o from sympy import primepi, integer_nthroot, primerange

%o def A076700(n): return int(sum(primepi(integer_nthroot(10**n, p)[0]) for p in primerange((10**n).bit_length()))) # _Chai Wah Wu_, Sep 12 2024

%Y Cf. A053810, A075308.

%K nonn

%O 1,1

%A _Zak Seidov_, Oct 26 2002

%E Edited and extended by _Robert G. Wilson v_, Oct 31 2002

%E a(18), a(19) from _Ray Chandler_, Oct 30 2008

%E a(20)-a(29) from _Donovan Johnson_, Mar 12 2013