A100509 Primitive prime power perfect numbers: A096290, with multiples of earlier numbers excluded.
216, 1149984, 365727111552, 1590733122516339863159808000
Offset: 1
Keywords
Crossrefs
Cf. A096290.
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a(12) = 2^2 = 4 because 4 is the only divisor of the divisors set 1, 2 = 2^1, 3 = 3^1, 4 = 2^2, 6 = 2^1 * 3^1, 12 = 2^2 * 3^1 for which all the exponents are prime. a(9) = 9 because 9 is the only divisor of the set 1, 3 = 3^1, 9 = 3^2 for which all the exponents are prime.
Array[DivisorSum[#, # &, AllTrue[FactorInteger[#][[All, -1]], PrimeQ] &] &, 96] (* Michael De Vlieger, Nov 17 2017 *)
vecproduct(v) = { my(m=1); for(i=1,#v,m *= v[i]); m; };
A293449(n) = vecproduct(apply(e -> isprime(e), factorint(n)[, 2]));
A126849(n) = if(1==n,n,sumdiv(n,d,(d>1)*A293449(d)*d)); \\ Antti Karttunen, Nov 17 2017
first(n) = {my(res = vector(n)); res[1] = 1; forprime(p = 2, sqrtint(n), forprime(e = 2, logint(n, p), for(k = 1, n \ (p^e), res[k*p^e] += p^e))); res} \\ David A. Corneth, Nov 17 2017
01PPsigma(2^5*3^3)=(1+2+2^2+2^3+2^5)*(1+3+3^2+3^3)=1880 All exponents of the terms are 0 or 1 or prime.
1PPsigma(2^5*3^3)=(2+2^2+2^3+2^5)*(3+3^2+3^3)=1794
NPPSigma(2^5*7^4)=(1+2+2^4)*(1+7+7^4)=45771 13770=2*3^4*5*17 so NPPSigma(2*3^4*5*17)=(1+2^1)*(1+3^1+3^4)*(1+5^1)*(1+17^1)=2*13770. Factorizations : 2*3, 2^4*3*5*19, 13770=2*3^4*5*17, 2^6*3*7*83, ...
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