A082814 -id:A082814 - cp's OEIS frontend

A082813 Prime numbers of form a^m + b^m + c^m + ..., where abc* ... is the prime factorization of m.

539633, 43909277783870034878569768760415886733743786946105343887995366054267119200102384004474562849

Offset: 1

N. J. A. Sloane, following a suggestion of G. L. Honaker, Jr., May 24 2003

No more terms with m < 822. - W. Edwin Clark, May 24 2003
No more terms with m < 10000. - Joint effort of Hugo Pfoertner and Joshua Zucker, May 30 2003
No more terms with m < 25500. - Martin Renner, Jan 31 2012

			a(1) = 539633 = 2^12 + 2^12 + 3^12 and 2*2*3 = 12.
a(2) is 2^88 + 2^88 + 2^88 + 11^88.
a(3) is 3^207 + 3^207 + 23^207.

Hugo Pfoertner, Table of n, a(n) for n = 1..3
G. L. Honaker, Jr., 539633
Hugo Pfoertner, Program and input files for PFGW.

Cf. A082814 for values of m.

a(2) and a(3) from Joshua Zucker, May 24 2003. Their primality was confirmed by Hugo Pfoertner, May 24 2003.

A082872 a^n + b^n + c^n + ..., where abc* ... is the prime factorization of n.

Original entry on oeis.org

1, 4, 27, 32, 3125, 793, 823543, 768, 39366, 9766649, 285311670611, 539633, 302875106592253, 678223089233, 30531927032, 262144, 827240261886336764177, 775103122, 1978419655660313589123979, 95367433737777, 558545874543637210

Offset: 1

Jason Earls, May 25 2003

n*log_10(2) + log_10(log_2(n)) <= length(a(n)) <= n*log_10(n). - Martin Renner, Jan 18 2012

If m = p^k is a power of a prime then a(n) = sum(p^m,i=1..k) = k*p^m is composite. - Martin Renner, Jan 31 2013

			a(6) = a(2*3) = 2^6 + 3^6 = 793.
a(8) = a(2*2*2) = 2^8 + 2^8 + 2^8 = 768.

T. D. Noe, Table of n, a(n) for n = 1..100

Cf. A082813, A082814, A051674.

A082872 := proc(n)
    local ps;
    if n= 1 then
        1;
    else
        ps := ifactors(n)[2] ;
        add( op(2,p)*op(1,p)^n,p=ps) ;
    end if;
end proc: # R. J. Mathar, Mar 12 2014

Table[f = FactorInteger[n]; Total[Flatten[Table[Table[f[[i, 1]], {f[[i, 2]]}], {i, Length[f]}]]^n], {n, 25}] (* T. D. Noe, Feb 01 2013 *)
Table[Total[Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[n]]^n],{n,30}] (* Harvey P. Dale, Jun 10 2016 *)

A082876 Number of prime divisors (counted with multiplicity) of numbers of form a^n + b^n + c^n + ..., where abc* ... is the prime factorization of n.

Original entry on oeis.org

0, 2, 3, 5, 5, 2, 7, 9, 10, 3, 11, 1, 13, 5, 7, 18, 17, 4, 19, 3, 7, 7, 23, 3, 26, 6, 28, 3, 29, 4, 31, 33, 8, 5, 11, 6, 37, 7, 9, 3, 41, 5, 43, 5, 4, 7, 47, 5, 50, 8, 14, 7, 53, 5, 11, 4, 8, 9, 59, 4, 61, 9, 5, 66, 11, 4, 67, 7, 11, 11, 71, 7, 73, 9, 4

Offset: 1

Jason Earls, May 25 2003

A082872(2) and A082872(6) are semiprimes. Where is the next?

Cf. A001222, A082813, A082814, A082872.

a(n) = if(n<2, 0, bigomega(sum(i=1, matsize(f=factor(n))[1], f[i, 1]^n*f[i, 2]))); \\ Jinyuan Wang, Apr 01 2020

a(74)-a(75) from Jinyuan Wang, Apr 01 2020

cp's OEIS Frontend

A082813 Prime numbers of form a^m + b^m + c^m + ..., where abc* ... is the prime factorization of m.

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A082872 a^n + b^n + c^n + ..., where abc* ... is the prime factorization of n.

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A082876 Number of prime divisors (counted with multiplicity) of numbers of form a^n + b^n + c^n + ..., where abc* ... is the prime factorization of n.

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cp's OEIS Frontend

A082813 Prime numbers of form a^m + b^m + c^m + ..., where a*b*c* ... is the prime factorization of m.

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Author

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Crossrefs

Extensions

A082872 a^n + b^n + c^n + ..., where a*b*c* ... is the prime factorization of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A082876 Number of prime divisors (counted with multiplicity) of numbers of form a^n + b^n + c^n + ..., where a*b*c* ... is the prime factorization of n.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Extensions

A082813 Prime numbers of form a^m + b^m + c^m + ..., where abc* ... is the prime factorization of m.

A082872 a^n + b^n + c^n + ..., where abc* ... is the prime factorization of n.

A082876 Number of prime divisors (counted with multiplicity) of numbers of form a^n + b^n + c^n + ..., where abc* ... is the prime factorization of n.