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Numbers n in whose prime factorization no exponent of any prime(k) exceeds k.
34

%I #43 Mar 12 2021 07:52:25

%S 1,2,3,5,6,7,9,10,11,13,14,15,17,18,19,21,22,23,25,26,29,30,31,33,34,

%T 35,37,38,39,41,42,43,45,46,47,49,50,51,53,55,57,58,59,61,62,63,65,66,

%U 67,69,70,71,73,74,75,77,78,79,82,83,85,86,87,89,90,91,93,94,95,97,98,99,101,102,103,105,106,107,109,110,111,113,114,115,117,118,119,121

%N Numbers n in whose prime factorization no exponent of any prime(k) exceeds k.

%C Numbers not divisible by p^(1+A000720(p)) for any prime p, where A000720(p) gives the index of prime p: 1 for 2, 2 for 3, 3 for 5, and so on.

%C Also Heinz numbers of integer partitions where the multiplicity of i does not exceed i for any i (A052335). Differs from A048103 in lacking {625, 1250, 1875, 3750, 4375, 5625, 6875, 8125, 8750, ...}. - _Gus Wiseman_, Mar 09 2019

%C Asymptotic density is Product_{i>=1} 1-prime(i)^(-1-i) = 0.72102334... - _Amiram Eldar_, Oct 20 2020

%H Antti Karttunen, <a href="/A276078/b276078.txt">Table of n, a(n) for n = 1..10000</a>

%t Select[Range@ 121, Or[# == 1, AllTrue[FactorInteger[#], PrimePi[#1] >= #2 & @@ # &]] &] (* _Michael De Vlieger_, Jun 24 2017 *)

%o (Scheme, with _Antti Karttunen_'s IntSeq-library)

%o (define A276078 (ZERO-POS 1 1 A276077))

%o (Python)

%o from sympy import factorint, primepi

%o def ok(n):

%o f = factorint(n)

%o return all(f[i] <= primepi(i) for i in f)

%o print([n for n in range(1, 151) if ok(n)]) # _Indranil Ghosh_, Jun 24 2017

%o (PARI) isok(n) = my(f=factor(n)); for (k=1, #f~, if (f[k, 2] > primepi(f[k, 1]), return(0))); return (1); \\ _Michel Marcus_, Jun 24 2017

%o (PARI) is(n) = {my(t=1);forprime(p = 2, , t++; pp = p^t; if(n%pp==0, return(0)); if(pp > n, return(1)))} \\ _David A. Corneth_, Jun 24 2017

%o (PARI) upto(n) = {my(v = vector(n,i,1), t=1, res=List()); forprime(p=2, , t++; pp = p^t; if(pp>n, break); for(i=1, n\pp, v[pp*i] = 0)); for(i=1, n, if(v[i]==1, listput(res, i))); res} \\ _David A. Corneth_, Jun 24 2017

%Y Positions of zeros in A276077.

%Y Complement: A276079.

%Y Sequence A276076 sorted into ascending order.

%Y Cf. A000040, A000720.

%Y Subsequence of A048103 from which it differs for the first time at n=451, where a(451) = 626, while A048103(451) = 625, a value missing from here.

%Y Cf. A052335, A056239, A062457, A087153, A109298, A112798, A115584, A117144, A118914, A124010, A197987, A276078.

%Y Cf. A324524, A324525, A324571, A324587, A324588.

%K nonn

%O 1,2

%A _Antti Karttunen_, Aug 18 2016