A099635 -id:A099635 - cp's OEIS frontend

A099637 Numbers such that gcd(Sum,n) = A099635 and gcd(Sum,Product) = A099636 are not identical. Sum and Product here are the sum and product of all distinct prime factors of n.

84, 132, 168, 228, 234, 252, 260, 264, 276, 308, 336, 340, 372, 396, 456, 468, 504, 516, 520, 528, 532, 552, 558, 564, 580, 588, 616, 644, 672, 680, 684, 702, 708, 740, 744, 756, 792, 804, 820, 828, 836, 852, 855, 868, 884, 912, 936, 948, 996, 1008, 1012, 1032

Offset: 1

Labos Elemer, Oct 28 2004

nonn

Of the first million integers, 75811 (of which 6300 are odd) belong to this sequence. - Robert G. Wilson v, Nov 04 2004

All terms have at least 3 distinct prime factors, and at least 4 prime factors counted with multiplicity. - Robert Israel, Aug 05 2024

			84 is here because its factor list = {2,3,7} and sum = 2 + 3 + 7 = 12, product = 2*3*7 = 42, gcd(12,84) = 12, gcd(12,42) = 6 != 12.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A099634, A099635, A099636.

filter:= proc(n) local F,s,p,t;
  F:= numtheory:-factorset(n);
  s:= convert(F,`+`);
  p:= convert(F,`*`);
  igcd(s,n) <> igcd(s,p)
end proc:
select(filter, [$1..2000]); # Robert Israel, Aug 05 2024

Mathematica
```
<Robert G. Wilson v, Nov 04 2004 *)
```

More terms from Robert G. Wilson v, Nov 04 2004

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

Original entry on oeis.org

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

A099636 a(n) = gcd(sum of distinct prime factors of n, product of distinct prime factors of n).

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Oct 28 2004

nonn

			n=84: a(84) = gcd(2*3*7, 2+3+7) = gcd(42, 12) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A007947, A008472, A082299, A099634, A340677, A340678.
Differs from related A099635 for the first time at n=84, where a(84) = 6, while A099635(84) = 12.
Differs from A014963 for the first time at n=30, where a(30) = 10, while A014963(30) = 1.

PrimeFactors[n_Integer] := Flatten[ Table[ # [[1]], {1}] & /@ FactorInteger[n]]; f[n_] := Block[{pf = PrimeFactors[n]}, GCD[Plus @@ pf, Times @@ pf]]; Table[ f[n], {n, 93}] (* Robert G. Wilson v, Nov 04 2004 *)

A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A008472(n) = vecsum(factor(n)[, 1]); \\ From A008472
A099636(n) = gcd(A007947(n), A008472(n));

From Antti Karttunen, Feb 01 2021: (Start)
a(n) = gcd(A007947(n), A008472(n)).
a(n) = A007947(n) / A340677(n) = A008472(n) / A340678(n).
(End)

Name clarified by Antti Karttunen, Feb 01 2021

cp's OEIS Frontend

A099637 Numbers such that gcd(Sum,n) = A099635 and gcd(Sum,Product) = A099636 are not identical. Sum and Product here are the sum and product of all distinct prime factors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A099636 a(n) = gcd(sum of distinct prime factors of n, product of distinct prime factors of n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions