A082344 -id:A082344 - 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

A082343 Numerator of sopfr(n)/n, where sopfr=A001414 is the sum of prime factors (with repetition).

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 3, 2, 7, 1, 7, 1, 9, 8, 1, 1, 4, 1, 9, 10, 13, 1, 3, 2, 15, 1, 11, 1, 1, 1, 5, 14, 19, 12, 5, 1, 21, 16, 11, 1, 2, 1, 15, 11, 25, 1, 11, 2, 6, 20, 17, 1, 11, 16, 13, 22, 31, 1, 1, 1, 33, 13, 3, 18, 8, 1, 21, 26, 1, 1, 1, 1, 39, 13, 23, 18, 3, 1, 13, 4, 43, 1, 1, 22, 45, 32, 17

Offset: 1

Reinhard Zumkeller, Apr 09 2003

Denominator is A082344(n) = n/A082299(n).

			n=200: (2+2+2+5+5)/(2*2*2*5*5) = 16/(2*2*2*5*5) = (2*2*2*2)/(2*2*2*5*5) = 2/25, therefore a(200)=2, A082344(200)=25.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001414, A082299, A082344.

sopfr[n_] := If[n == 1, 0, Total[Times @@@ FactorInteger[n]]];
a[n_] := Numerator[sopfr[n]/n];
Array[a, 100] (* Jean-François Alcover, Dec 03 2021 *)

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.
A082299(n) = gcd(n, A001414(n));
A082343(n) = A001414(n)/A082299(n); \\ Antti Karttunen, Mar 04 2018

a(n) = A001414(n)/A082299(n).

A340677 a(n) = A007947(n) / gcd(A007947(n), A008472(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 10, 1, 6, 1, 14, 15, 1, 1, 6, 1, 10, 21, 22, 1, 6, 1, 26, 1, 14, 1, 3, 1, 1, 33, 34, 35, 6, 1, 38, 39, 10, 1, 7, 1, 22, 15, 46, 1, 6, 1, 10, 51, 26, 1, 6, 55, 14, 57, 58, 1, 3, 1, 62, 21, 1, 65, 33, 1, 34, 69, 5, 1, 6, 1, 74, 15, 38, 77, 13, 1, 10, 1, 82, 1, 7, 85, 86, 87, 22, 1, 3, 91, 46, 93

Offset: 1

Antti Karttunen, Feb 01 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000961 (positions of ones), A006881, A007947, A008472, A099636, A340678.

Cf. also A082344.

A007947(n) = factorback(factorint(n)[, 1]);
A008472(n) = vecsum(factor(n)[, 1]);
A340677(n) = { my(u=A007947(n)); (u/gcd(u, A008472(n))); };

a(n) = A007947(n) / A099636(n) = A007947(n) / gcd(A007947(n), A008472(n)).
a(n) = 1 iff n is power of prime (A000961). - Bernard Schott, Feb 01 2021
a(A006881(n)) = A006881(n). - Bernard Schott and Antti Karttunen, Feb 01 2021

A381249 Indices of records in k/A001414(k), k>=2.

Original entry on oeis.org

2, 6, 8, 9, 12, 15, 16, 18, 24, 27, 32, 36, 40, 45, 48, 54, 60, 64, 72, 80, 81, 90, 96, 108, 120, 128, 135, 144, 160, 162, 180, 192, 216, 240, 243, 270, 288, 320, 324, 360, 384, 405, 432, 480, 486, 540, 576, 640, 648, 720, 729, 810, 864, 960, 972, 1024, 1080, 1152

Offset: 1

Clark Kimberling, Apr 19 2025

nonn

Except for initial 2, this is a subsequence of A381972.

			f(2) = 1 < f(6) = 6/5 < f(8) = 4/3 < f(9) = 3/2 < f(12) = 12/7, where f(k) = k/A001414(k).

Cf. A001414, A082299, A082343, A082344, A381972.

z = 800; g[n_] := FactorInteger[n];
f[n_] := Map[First, g[n]] . Map[Last, g[n]];
mx = -1; k = 2; u = {}; While[k < z, a = k/f[k]; If[a > mx, mx = a; AppendTo[u, k]]; k++]; u

sopfr(n) = (n=factor(n))[, 1]~*n[, 2];
lista(nn) = my(r=oo, list=List()); for (n=2, nn, my(x=sopfr(n)/n); if (x < r, listput(list, n); r = x)); Vec(list); \\ Michel Marcus, Apr 27 2025

More terms from Michel Marcus, Apr 27 2025

A381972 Numbers k>=3 such that k/A001414(k) > (k-1)/A001414(k-1).

Original entry on oeis.org

6, 8, 9, 12, 14, 15, 16, 18, 20, 24, 27, 30, 32, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 77, 78, 80, 81, 84, 87, 88, 90, 95, 96, 98, 100, 102, 104, 105, 108, 110, 112, 114, 117, 119, 120, 123, 124, 125, 126

Offset: 1

Clark Kimberling, Mar 16 2025

nonn

1 <= a(n+1) - a(n) <= 10 for n = 2..3000000.

			f(2) = 1 < f(6) = 6/5 < f(8) = 4/3 < f(9) = 3/2 < f(12) = 12/7, where f(k) = k/A001414(k).

Cf. A001414, A082299, A082343, A082344, A381249.

z = 200; f[n_] := FactorInteger[n];
g[n_] := Map[First, f[n]] . Map[Last, f[n]];
h[n_] := If[n/g[n] > (n - 1)/g[n - 1], n, 0];
Rest[Union[Table[h[n], {n, 2, z}]]]

Definition corrected by Clark Kimberling, May 08 2025

cp's OEIS Frontend

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

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A082343 Numerator of sopfr(n)/n, where sopfr=A001414 is the sum of prime factors (with repetition).

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A340677 a(n) = A007947(n) / gcd(A007947(n), A008472(n)).

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A381249 Indices of records in k/A001414(k), k>=2.

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A381972 Numbers k>=3 such that k/A001414(k) > (k-1)/A001414(k-1).

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