A082300 -id:A082300 - cp's OEIS frontend

A082299 Greatest common divisor of n and its sum of prime factors (with repetition).

1, 2, 3, 4, 5, 1, 7, 2, 3, 1, 11, 1, 13, 1, 1, 8, 17, 2, 19, 1, 1, 1, 23, 3, 5, 1, 9, 1, 29, 10, 31, 2, 1, 1, 1, 2, 37, 1, 1, 1, 41, 6, 43, 1, 1, 1, 47, 1, 7, 2, 1, 1, 53, 1, 1, 1, 1, 1, 59, 12, 61, 1, 1, 4, 1, 2, 67, 1, 1, 14, 71, 12, 73, 1, 1, 1, 1, 6, 79, 1, 3, 1, 83, 14, 1, 1, 1, 1, 89, 1, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Apr 08 2003

nonn

For n > 4, a(n) = n iff n is prime.

			a(100) = GCD(2*2*5*5,2+2+5+5) = GCD(2*2*5,2*7) = 2;
a(200) = GCD(2*2*2*5*5,2+2+2+5+5) = GCD(2*2*2*5,2*2*2*2) = 8.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from Harvey P. Dale)

Cf. A001414, A082300 (positions of ones), A082343, A082344.

Cf. also A099635, A099636.

Table[GCD[Total[Times@@@FactorInteger[n]],n],{n,100}] (* Harvey P. Dale, Dec 27 2015 *)

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.
A082299(n) = gcd(n, A001414(n)); \\ Antti Karttunen, Feb 01 2021

a(n) = gcd(n, A001414(n)).

a(n) = n / A082344(n) = A001414(n) / A082343(n). - Antti Karttunen, Feb 01 2021

A275665 Numbers n such that n and sopf(n) are relatively prime, where sopf(n) (A008472) is the sum of the distinct primes dividing n.

Original entry on oeis.org

1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100, 104, 106, 108, 111, 112, 115, 116, 117, 118, 119, 122, 123, 124, 129, 133, 134, 135, 136, 141, 142, 143, 144, 145, 146, 147, 148, 152, 153, 155, 158, 159, 160, 161, 162, 164, 165

Offset: 1

Charles R Greathouse IV, Aug 04 2016

nonn

Hall shows that the density of this sequence is 6/Pi^2, so a(n) ~ (Pi^2/6)n.

Differs from A267114, from A030231, and from A007774 (shifted by one index) first at n=93. - R. J. Mathar, Aug 22 2016

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
R. B. Hall, On the probability that n and f(n) are relatively prime, Acta Arithmetica 17 (1970), pp. 169-183.

Cf. A008472, A082300, A267114, A030231, A007774.

Select[Range@ 165, CoprimeQ[#, Total@ FactorInteger[#][[All, 1]]] &] (* Michael De Vlieger, Aug 06 2016 *)

sopf(n)=vecsum(factor(n)[,1])
is(n)=gcd(sopf(n),n)==1

cp's OEIS Frontend

A082299 Greatest common divisor of n and its sum of prime factors (with repetition).

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