A082872 -id:A082872 - cp's OEIS frontend

A239283 n^(p1) + n^(p2) + n^(p3) + ... where (p1)(p2)(p3)*.... is the prime factorization of n (with multiplicity).

0, 4, 27, 32, 3125, 252, 823543, 192, 1458, 100100, 285311670611, 2016, 302875106592253, 105413700, 762750, 1024, 827240261886336764177, 11988, 1978419655660313589123979, 3200800, 1801097802, 584318301411812, 20880467999847912034355032910567, 15552, 19531250

Offset: 1

R. J. Mathar, Mar 14 2014

The definition of the terms swaps the roles of the primes in the base and their exponents of A082872.

Contains A051674 as a subsequence at the prime positions n= 2, 3, 5, 7,.... Michel Marcus, Mar 14 2014

			a(8) = a(2*2*2) = 8^2 + 8^2 + 8^2 = 192.

A239283 := proc(n)
    local ps;
    ps := ifactors(n)[2] ;
    add( op(2,p)*n^op(1,p),p=ps) ;
end proc:
seq(A239283(n),n=1..22) ;

a(n) = my(f = factor(n)); sum(i=1, #f~, f[i,2]*n^f[i, 1]); \\ Michel Marcus, Mar 14 2014

a(n) = sum_i [e_i*n^(p_i)], where n=product_i (p_i)^(e_i) 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.

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

A108705 Numbers n such that a^n + b^n + c^n + ... is a square, where abc* ... is the prime factorization of n.

Original entry on oeis.org

1, 2, 16, 27, 512, 3125, 65536, 531441

Offset: 1

Jason Earls, Jun 20 2005

No more terms through 630000. - Ryan Propper, May 12 2006

			16 is in the sequence because 16 = 2^4 and 2^16 + 2^16 + 2^16 + 2^16 = 262144
= 512^2.

Cf. A082872.

Two more terms from Ryan Propper, May 12 2006

cp's OEIS Frontend

A239283 n^(p1) + n^(p2) + n^(p3) + ... where (p1)(p2)(p3)*.... is the prime factorization of n (with multiplicity).

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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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Author

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PARI

Extensions

A108705 Numbers n such that a^n + b^n + c^n + ... is a square, where abc* ... is the prime factorization of n.

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Author

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Examples

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Extensions

cp's OEIS Frontend

A239283 n^(p1) + n^(p2) + n^(p3) + ... where (p1)*(p2)*(p3)*.... is the prime factorization of n (with multiplicity).

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Examples

Programs

Maple

PARI

Formula

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.

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Author

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Comments

Crossrefs

Programs

PARI

Extensions

A108705 Numbers n such that a^n + b^n + c^n + ... is a square, where a*b*c* ... is the prime factorization of n.

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Author

Keywords

Comments

Examples

Crossrefs

Extensions

A239283 n^(p1) + n^(p2) + n^(p3) + ... where (p1)(p2)(p3)*.... is the prime factorization of n (with multiplicity).

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.

A108705 Numbers n such that a^n + b^n + c^n + ... is a square, where abc* ... is the prime factorization of n.