A381639 -id:A381639 - cp's OEIS frontend

A381640 Numbers k such that f(k) > f(m) for all m < k, where f(k) = A381638(k)/A381639(k).

1, 6, 30, 105, 210, 2310, 15015, 30030, 255255, 510510, 1939938, 3233230, 4849845, 9699690, 111546435, 223092870, 3234846615, 6469693230

Offset: 1

Amiram Eldar, Mar 03 2025

Called "f-champion numbers" by Erdős and Nicolas (1989).
All the terms are squarefree numbers.
The primorial numbers, A002110(k), are all terms for k large enough.

Paul Erdős and Jean-Louis Nicolas, Grandes valeurs de fonctions liées aux diviseurs premiers consécutifs d'un entier, in: Jean-Marie de Koninck and Claude Levesque (eds.), Théorie des nombres / Number Theory, Proceedings of the International Number Theory Conference held at Université Laval, July 5-18, 1987, De Gruyter, 1989; alternative link.

Cf. A002110, A005117, A381638, A381639, A381642.

f[n_] := Module[{p = FactorInteger[n][[;; , 1]]}, Total[Most[p]/Rest[p]]]; seq[lim_] := Module[{fi, fmax = -1, s = {}}, Do[fi = f[i]; If[fi > fmax, fmax = fi; AppendTo[s, i]], {i, 1, lim}]; s]; seq[12000]

f(n) = {my(p = factor(n)[,1]); sum(i = 1, #p-1, p[i]/p[i+1]);}
list(lim) = {my(fm = -1, f1); for(k = 1, lim, f1 = f(k); if(f1 > fm, print1(k, ", "); fm = f1));}

A381642 Numbers k such that F(k) > F(m) for all m < k, where F(k) = A381641(k)/A381639(k).

Original entry on oeis.org

1, 6, 10, 14, 22, 26, 34, 38, 42, 46, 58, 62, 66, 78, 102, 114, 130, 170, 190, 230, 290, 310, 370, 406, 410, 430, 434, 470, 518, 574, 602, 658, 742, 826, 854, 938, 994, 1022, 1106, 1162, 1218, 1302, 1554, 1722, 1806, 1974, 2226, 2478, 2562, 2706, 2814, 2838, 2982

Offset: 1

Amiram Eldar, Mar 03 2025

nonn

Called "F-champion numbers" by Erdős and Nicolas (1989).

All the terms are squarefree numbers.

Amiram Eldar, Table of n, a(n) for n = 1..3000
Paul Erdős and Jean-Louis Nicolas, Grandes valeurs de fonctions liées aux diviseurs premiers consécutifs d'un entier, in: Jean-Marie de Koninck and Claude Levesque (eds.), Théorie des nombres / Number Theory, Proceedings of the International Number Theory Conference held at Université Laval, July 5-18, 1987, De Gruyter, 1989; alternative link.

Cf. A005117, A381639, A381640, A381641.

f[n_] := Module[{p = FactorInteger[n][[;; , 1]]}, Total[1 - Most[p]/Rest[p]]]; seq[lim_] := Module[{fi, fmax = -1, s = {}}, Do[fi = f[i]; If[fi > fmax, fmax = fi; AppendTo[s, i]], {i, 1, lim}]; s]; seq[3000]

f(n) = {my(p = factor(n)[,1]); sum(i = 1, #p-1, 1 - p[i]/p[i+1]);}
list(lim) = {my(fm = -1, f1); for(k = 1, lim, f1 = f(k); if(f1 > fm, print1(k, ", "); fm = f1));}

A381638 Numerators of Sum_{i=1..omega(n)-1} p_{i}/p_{i+1}, where omega(n) = A001221(n) and p_1 < p_2 < ... p_omega(n) are the distinct prime factors of n; a(1) = 0.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 2, 3, 0, 0, 2, 0, 2, 3, 2, 0, 2, 0, 2, 0, 2, 0, 19, 0, 0, 3, 2, 5, 2, 0, 2, 3, 2, 0, 23, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 5, 2, 3, 2, 0, 19, 0, 2, 3, 0, 5, 31, 0, 2, 3, 39, 0, 2, 0, 2, 3, 2, 7, 35, 0, 2, 0, 2, 0, 23, 5

Offset: 1

Amiram Eldar, Mar 03 2025

			Fractions begin with 0, 0, 0, 0, 0, 2/3, 0, 0, 0, 2/5, 0, 2/3, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Erdős and Jean-Louis Nicolas, Grandes valeurs de fonctions liées aux diviseurs premiers consécutifs d'un entier, in: Jean-Marie de Koninck and Claude Levesque (eds.), Théorie des nombres / Number Theory, Proceedings of the International Number Theory Conference held at Université Laval, July 5-18, 1987, De Gruyter, 1989; alternative link.

Cf. A000961, A001221, A381639 (denominators), A381640, A381641.

a[n_] := Module[{p = FactorInteger[n][[;; , 1]]}, Numerator[Total[Most[p]/Rest[p]]]]; Array[a, 100]

a(n) = {my(p = factor(n)[,1]); numerator(sum(i = 1, #p-1, p[i]/p[i+1]));}

a(n) = 0 if and only if n is a power of a prime (A000961).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{q prime} (1/q^2) * Sum_{primes p < q} Product_{primes r, p < r < q} (1-1/r). This sum converges slowly: for primes q that are not exceeding 10^9, 10^10, 10^11, and 10^12, the sums are 0.5399..., 0.5447..., 0.5487..., and 0.5520..., respectively.

A381641 Numerators of Sum_{i=1..omega(n)-1} (1 - p_{i}/p_{i+1}), where omega(n) = A001221(n) and p_1 < p_2 < ... p_omega(n) are the distinct prime factors of n; a(1) = 0.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 5, 2, 0, 0, 1, 0, 3, 4, 9, 0, 1, 0, 11, 0, 5, 0, 11, 0, 0, 8, 15, 2, 1, 0, 17, 10, 3, 0, 19, 0, 9, 2, 21, 0, 1, 0, 3, 14, 11, 0, 1, 6, 5, 16, 27, 0, 11, 0, 29, 4, 0, 8, 35, 0, 15, 20, 31, 0, 1, 0, 35, 2, 17, 4, 43, 0, 3, 0

Offset: 1

Amiram Eldar, Mar 03 2025

			Fractions begin with 0, 0, 0, 0, 0, 1/3, 0, 0, 0, 3/5, 0, 1/3, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Erdős and Jean-Louis Nicolas, Grandes valeurs de fonctions liées aux diviseurs premiers consécutifs d'un entier, in: Jean-Marie de Koninck and Claude Levesque (eds.), Théorie des nombres / Number Theory, Proceedings of the International Number Theory Conference held at Université Laval, July 5-18, 1987, De Gruyter, 1989; alternative link.

Cf. A000961, A001221, A381638, A381639 (denominators).

a[n_] := Module[{p = FactorInteger[n][[;; , 1]]}, Numerator[Total[1 - Most[p]/Rest[p]]]]; Array[a, 100]

a(n) = {my(p = factor(n)[,1]); numerator(sum(i = 1, #p-1, 1 - p[i]/p[i+1]));}

a(n) = 0 if and only if n is a power of a prime (A000961).

a(n)/A381639(n) = A001221(n) - 1 - A381638(n)/A381639(n).

cp's OEIS Frontend

A381640 Numbers k such that f(k) > f(m) for all m < k, where f(k) = A381638(k)/A381639(k).

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A381642 Numbers k such that F(k) > F(m) for all m < k, where F(k) = A381641(k)/A381639(k).

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A381638 Numerators of Sum_{i=1..omega(n)-1} p_{i}/p_{i+1}, where omega(n) = A001221(n) and p_1 < p_2 < ... p_omega(n) are the distinct prime factors of n; a(1) = 0.

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A381641 Numerators of Sum_{i=1..omega(n)-1} (1 - p_{i}/p_{i+1}), where omega(n) = A001221(n) and p_1 < p_2 < ... p_omega(n) are the distinct prime factors of n; a(1) = 0.

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