A305444 -id:A305444 - cp's OEIS frontend

A173557 a(n) = Product_{primes p dividing n} (p-1).

1, 1, 2, 1, 4, 2, 6, 1, 2, 4, 10, 2, 12, 6, 8, 1, 16, 2, 18, 4, 12, 10, 22, 2, 4, 12, 2, 6, 28, 8, 30, 1, 20, 16, 24, 2, 36, 18, 24, 4, 40, 12, 42, 10, 8, 22, 46, 2, 6, 4, 32, 12, 52, 2, 40, 6, 36, 28, 58, 8, 60, 30, 12, 1, 48, 20, 66, 16, 44, 24, 70, 2, 72, 36

Offset: 1

José María Grau Ribas, Feb 21 2010

This is A023900 without the signs. - T. D. Noe, Jul 31 2013
Numerator of c_n = Product_{odd p| n} (p-1)/(p-2). Denominator is A305444. The initial values c_1, c_2, ... are 1, 1, 2, 1, 4/3, 2, 6/5, 1, 2, 4/3, 10/9, 2, 12/11, 6/5, 8/3, 1, 16/15, ... [Yamasaki and Yamasaki]. - N. J. A. Sloane, Jan 19 2020
Kim et al. (2019) named this function the absolute Möbius divisor function. - Amiram Eldar, Apr 08 2020

			300 = 3*5^2*2^2 => a(300) = (3-1)*(2-1)*(5-1) = 8.

Alois P. Heinz, Table of n, a(n) for n = 1..65536 (first 1000 terms from T. D. Noe)
Daeyeoul Kim, Umit Sarp, and Sebahattin Ikikardes, Certain combinatoric convolution sums arising from Bernoulli and Euler Polynomials, Miskolc Mathematical Notes, No. 20, Vol. 1 (2019): pp. 311-330.
Daeyeoul Kim, Umit Sarp, and Sebahattin Ikikardes, Iterating the Sum of Möbius Divisor Function and Euler Totient Function, Mathematics, Vol. 7, No. 11 (2019), pp. 1083-1094.
Yamasaki, Yasuo, and Aiichi Yamasaki, On the Gap Distribution of Prime Numbers, Kyoto University Research Information Repository, October 1994. MR1370273 (97a:11141).

Cf. A023900, A141564, A027748, A305444, A307868.

a173557 1 = 1
a173557 n = product $ map (subtract 1) $ a027748_row n
-- Reinhard Zumkeller, Jun 01 2015

[EulerPhi(n)/(&+[(Floor(k^n/n)-Floor((k^n-1)/n)): k in [1..n]]): n in [1..100]]; // Vincenzo Librandi, Jan 20 2020

A173557 := proc(n) local dvs; dvs := numtheory[factorset](n) ; mul(d-1,d=dvs) ; end proc: # R. J. Mathar, Feb 02 2011
# second Maple program:
a:= n-> mul(i[1]-1, i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Aug 27 2018

a[n_] := Module[{fac = FactorInteger[n]}, If[n==1, 1, Product[fac[[i, 1]]-1, {i, Length[fac]}]]]; Table[a[n], {n, 100}]

a(n) = my(f=factor(n)[,1]); prod(k=1, #f, f[k]-1); \\ Michel Marcus, Oct 31 2017

for(n=1, 100, print1(direuler(p=2, n, (1 - 2*X + p*X)/(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Jun 18 2020

apply( {A173557(n)=vecprod([p-1|p<-factor(n)[,1]])}, [1..77]) \\ M. F. Hasler, Aug 14 2021

from math import prod
from sympy import primefactors
def A173557(n): return prod(p-1 for p in primefactors(n)) # Chai Wah Wu, Sep 08 2023

;; With memoization-macro definec.
(definec (A173557 n) (if (= 1 n) 1 (* (- (A020639 n) 1) (A173557 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

a(n) = A003958(n) iff n is squarefree. a(n) = |A023900(n)|.
Multiplicative with a(p^e) = p-1, e >= 1. - R. J. Mathar, Mar 30 2011
a(n) = phi(rad(n)) = A000010(A007947(n)). - Enrique Pérez Herrero, May 30 2012
a(n) = A000010(n) / A003557(n). - Jason Kimberley, Dec 09 2012
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 - 2p^(-s) + p^(1-s)). The Dirichlet inverse is multiplicative with b(p^e) = (1 - p) * (2 - p)^(e - 1) = Sum_k A118800(e, k) * p^k. - Álvar Ibeas, Nov 24 2017
a(1) = 1; for n > 1, a(n) = (A020639(n)-1) * a(A028234(n)). - Antti Karttunen, Nov 28 2017
From Vaclav Kotesovec, Jun 18 2020: (Start)
Dirichlet g.f.: zeta(s) * zeta(s-1) / zeta(2*s-2)  * Product_{p prime} (1 - 2/(p + p^s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A307868 = Product_{p prime} (1 - 2/(p*(p+1))) = 0.471680613612997868... (End)
a(n) = (-1)^A001221(n)*A023900(n). - M. F. Hasler, Aug 14 2021

Definition corrected by M. F. Hasler, Aug 14 2021

Incorrect formula removed by Pontus von Brömssen, Aug 15 2021

A307410 Numerators of the product in the singular series.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 5, 1, 1, 3, 9, 1, 11, 5, 3, 1, 15, 1, 17, 3, 5, 9, 21, 1, 3, 11, 1, 5, 27, 3, 29, 1, 9, 15, 5, 1, 35, 17, 11, 3, 39, 5, 41, 9, 3, 21, 45, 1, 5, 3, 15, 11, 51, 1, 27, 5, 17, 27, 57, 3, 59, 29, 5, 1, 11, 9, 65, 15, 21, 5, 69, 1, 71, 35, 3, 17, 3, 11, 77, 3, 1, 39, 81, 5, 45

Offset: 1

Mats Granvik, Apr 07 2019

Differs from A305444 at n = 35, 65, 70, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
John Omielan, How do you compute the singular series?, Mathematics Stack Exchange.
Terence Tao, Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges. See the next formula after equation 2.

Cf. A005596, A005597, A305444, A380839 (denominators).

f:= proc(n) numer(mul((p-2)/(p-1),p=select(type,numtheory:-factorset(n),odd))) end proc:
map(f, [$1..100]); # Robert Israel, Apr 07 2019

Table[Times@@(DeleteDuplicates[DeleteCases[DeleteCases[Exp[MangoldtLambda[Divisors[h]]], 1],2]] - 2)/Times@@(DeleteDuplicates[DeleteCases[DeleteCases[Exp[MangoldtLambda[Divisors[h]]], 1], 2]] - 1), {h, 1, 85}]
Numerator[%]
f[p_, e_] := If[p == 2, 1, (p-2)/(p-1)]; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Mar 03 2025 *)

a(n) = my(f=factor(n)[,1]~); numerator(prod(k=1, #f, if (f[k]>2, (f[k]-2)/(f[k]-1), 1))); \\ Michel Marcus, Apr 07 2019

a(n) = numerator of Product_{p|n;p>2}(p-2)/(p-1) where p is a prime number.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A380839(k) = 2 * Product_{p prime} (1-1/(p^2-p)) = 2 * A005596 = 0.7479116272384045761094... . - Amiram Eldar, Mar 03 2025

A380839 Numerators of J(n) = Product_{p|n, p odd prime} (p - 1)/(p - 2).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 1, 2, 4, 10, 2, 12, 6, 8, 1, 16, 2, 18, 4, 12, 10, 22, 2, 4, 12, 2, 6, 28, 8, 30, 1, 20, 16, 8, 2, 36, 18, 24, 4, 40, 12, 42, 10, 8, 22, 46, 2, 6, 4, 32, 12, 52, 2, 40, 6, 36, 28, 58, 8, 60, 30, 12, 1, 16, 20, 66, 16, 44, 8, 70, 2, 72, 36

Offset: 1

Artur Jasinski, Feb 05 2025

This sequence is similar to A173557 but differences occurs for indices n=35,65,70,...
Coefficients J(n)=a(n)/A307410(n) occurs in many formulas on density of primes with gap 2*n.
Sylvester was the first who uses these coefficients at 1871.

			1, 1, 2, 1, 4/3, 2, 6/5, 1, 2, 4/3, 10/9, 2, 12/11, ...
a(35) = 8 because 35 = 5 * 7 and then product is ((5-1)/(5-2))*((7-1)/(7-2)) = 8/5.

Amiram Eldar, Table of n, a(n) for n = 1..10000
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70 (e.g. formula 4.11 p.32).
J. J. Sylvester, On the partition of an even number into two primes, Proc. London Math. Soc., ser. I, vol.4 (1871), pp. 4-6 (Math. Papers, vol. 2, pp. 709-711).
Marek Wolf, Applications of statistical mechanics in number theory, Physica A 274 (1999) 149-157 (formula (9) p. 154).

Cf. A167864, A173557, A305444, A307410 (denominators).

j = {}; Do[prod = 1; Do[If[PrimeQ[n] && IntegerQ[d/n], prod = prod (n - 1)/(n - 2)], {n, 3, d}]; AppendTo[j, prod], {d, 1, 74}]; Numerator[j]
f[p_, e_] := If[p == 2, 1, (p-1)/(p-2)]; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Mar 03 2025 *)

a(n) = my(f=factor(n)[,1]); numerator(prod(k=1, #f, if ((p=f[k])>2, (p-1)/(p-2), 1))); \\ Michel Marcus, Feb 05 2025

a(n) = numerator(A173557(n)/A305444(n)).
a(p^n) = p - 1 for prime p.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A307410(k) = Product_{prime p > 2} (1 + 1/(p*(p-2))) = 1.51478012... (A167864). - Amiram Eldar, Mar 03 2025

A380139 Prime gaps between 10^m and 10^(m+1), m>=0, sorted first by falling number of occurrences and then by rising gap size, written as an irregular triangle.

Original entry on oeis.org

2, 1, 4, 4, 6, 2, 8, 6, 4, 2, 10, 8, 12, 14, 18, 20, 6, 2, 4, 10, 12, 8, 14, 18, 16, 22, 24, 20, 30, 28, 26, 34, 32, 36, 6, 2, 4, 12, 10, 8, 18, 14, 16, 20, 22, 24, 30, 28, 26, 36, 32, 34, 40, 38, 42, 52, 44, 50, 46, 54, 58, 48, 56, 60, 62, 64, 72

Offset: 1

Hugo Pfoertner based on an idea by Richard Stephen Donovan, Jan 23 2025

A gap between two primes p1 and p2 is assumed to belong to the range [10^m .. 10^(m+1)[ if 10^m <= (p1+p2)/2 < 10^(m+1). Thus the gap between 7 and 11 is counted in the interval 1 .. 10. Gaps symmetric to 10^k occur for k = 17, 45, ... .

			The triangle begins, with corresponding counts in [...]:
  [2, 1, 1]
   2, 1, 4,
  [7, 7, 6, 1]
   4, 6, 2, 8,
  [37, 32, 27, 16, 14,  8,  7,  1,  1]
    6,  4,  2, 10,  8, 12, 14, 18, 20
  [255, 170, 162, 103, 98, 86, 47, 39, 33, 16, 15, 14, 11,  5,  3,  3,  1,  1]
    6,   2,   4,   10, 12,  8, 14, 18, 16, 22, 24, 20, 30, 28, 26, 34, 32, 36,
  [1641, 1018, 1013, 860, 797, 672, 474, 430, 306, 223, 207, 191, 135, 93, 85, ...]
     6,    2,    4,   12,  10,  8,   18,  14,  16,  20,  22,  24,  30, 28, 26, ...
  [11609, 7040, 6945, 6928, 6163, 4796, 4395, 3749, 2542, 2476, 2164, 1949, ...]
     6,    12,    2,    4,   10,    8,   18,   14,   16,   24,   20,   22,  ...
  6, 12, 2, 4, 10, 18, 8, 14, 24, 16, 30, 20, 22, 28, 26, 36, 42, 34, ...
  6, 12, 4, 2, 10, 18, 8, 14, 24, 30, 16, 20, 22, 28, 26, 36, 42, 34, ...
  6, 12, 10, 4, 2, 18, 8, 14, 24, 30, 16, 20, 22, 28, 36, 26, 42, 34, ...
  6, 12, 18, 10, 2, 4, 8, 24, 30, 14, 20, 16, 22, 36, 28, 26, 42, 34, ...

Hugo Pfoertner, Table of n, a(n) for n = 1..583

Cf. A001223, A028334, A038460, A354604.

Cf. A005597, A173557, A305444 for the asymptotic behavior of gap sizes.

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A173557 a(n) = Product_{primes p dividing n} (p-1).

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A307410 Numerators of the product in the singular series.

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A380839 Numerators of J(n) = Product_{p|n, p odd prime} (p - 1)/(p - 2).

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A380139 Prime gaps between 10^m and 10^(m+1), m>=0, sorted first by falling number of occurrences and then by rising gap size, written as an irregular triangle.

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