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A097621 In canonical prime factorization of n replace p^e with its index in A000961.

%I A097621 #44 Jan 19 2025 15:24:00
%S A097621 1,2,3,4,5,6,6,7,8,10,9,12,10,12,15,11,12,16,13,20,18,18,14,21,15,20,
%T A097621 16,24,17,30,18,19,27,24,30,32,20,26,30,35,21,36,22,36,40,28,23,33,24,
%U A097621 30,36,40,25,32,45,42,39,34,26,60,27,36,48,28,50,54,29,48,42,60,30,56,31
%N A097621 In canonical prime factorization of n replace p^e with its index in A000961.
%C A097621 The definition of the sequence has been corrected, given that it uses A095874, the indices in the list A000961 of "powers of primes" starting with A000961(1) = 1, rather than A322981, index of p^e in the list of prime powers A246655, as written in the original definition. See A333235 for the variant of this sequence which uses A322981 and A246655 instead, maybe the more natural choice given that the factorization of integers consists of prime powers > 1. - _M. F. Hasler_, Jun 15 2021
%H A097621 Ivan Neretin, <a href="/A097621/b097621.txt">Table of n, a(n) for n = 1..10000</a>
%F A097621 Multiplicative with: p^e -> A095874(p^e), p prime.
%F A097621 a(A000961(n)) = n; a(a(n)) = A097622(n); a(a(a(n))) = A097623(n);
%F A097621 a(n) <= n; a(n) = n iff 60 mod n = 0: a(A018266(n)) = A018266(n);
%F A097621 a(A097624(n)) = n and a(m) <> n for n < A097624(n).
%e A097621 n=600 = 2^3 * 3 * 5^2 -> A095874(8)*A095874(3)*A095874(25) = 7 * 3 * 15 = 315.
%p A097621 N:= 1000: # to get a(1) to a(N)
%p A097621 Primes:= select(isprime,[2, seq(2*i+1,i=1..(N-1)/2)]):
%p A097621 PP:= sort([1,seq(seq(p^k, k=1..floor(log[p](N))),p=Primes)]):
%p A097621 for n from 1 to nops(PP) do B[PP[n]]:= n od:
%p A097621 seq(mul(B[f[1]^f[2]],f=ifactors(n)[2]),n=1..N); # _Robert Israel_, Sep 02 2015
%t A097621 pp = Select[Range@100, Length[FactorInteger[#]] == 1 &]; a = Table[Times @@ (Position[pp, #][[1, 1]] & /@ (#[[1]]^#[[2]] & /@ FactorInteger[n])), {n, 73}] (* _Ivan Neretin_, Sep 02 2015 *)
%o A097621 (PARI) f(n) = if(isprimepower(n), sum(i=1, logint(n, 2), primepi(sqrtnint(n, i)))+1, n==1); \\ A095874
%o A097621 a(n) = my(fr=factor(n)); for (k=1, #fr~, fr[k,1] = f(fr[k,1]^fr[k,2]); fr[k,2] = 1); factorback(fr); \\ _Michel Marcus_, May 29 2021
%o A097621 A097621(n)=vecprod([A095874(f[1]^f[2])|f<-factor(n)~]) \\ _M. F. Hasler_, Jun 15 2021
%o A097621 (Python)
%o A097621 from math import prod
%o A097621 from sympy import primepi, integer_nthroot, factorint
%o A097621 def A097621(n): return prod(1+int(primepi(m:=p**e)+sum(primepi(integer_nthroot(m,k)[0]) for k in range(2,m.bit_length()))) for p,e in factorint(n).items()) # _Chai Wah Wu_, Jan 19 2025
%Y A097621 Cf. A000961 (powers of primes), A246655 (prime powers), A003963, A018266, A095874 (index of n = p^e in A000961).
%Y A097621 Cf. A097622, A097623, A097624.
%Y A097621 Cf. A322981 (index of n = p^e in A246655), A333235 (variant of this sequence).
%K A097621 nonn,mult
%O A097621 1,2
%A A097621 _Reinhard Zumkeller_, Aug 17 2004
%E A097621 Definition corrected by _M. F. Hasler_, Jun 16 2021
%E A097621 Example corrected by _Ray Chandler_, Jun 30 2021