A065465 -id:A065465 - cp's OEIS frontend

A335762 Decimal expansion of Product_{p prime} (1 + p/((p-1)*(p+1)^2)).

1, 4, 5, 0, 0, 3, 2, 1, 4, 5, 3, 6, 2, 1, 2, 0, 8, 3, 1, 6, 0, 8, 3, 9, 5, 8, 8, 7, 1, 8, 9, 2, 2, 3, 4, 2, 2, 3, 2, 5, 0, 6, 2, 1, 1, 7, 4, 4, 7, 1, 6, 7, 1, 4, 4, 6, 5, 2, 4, 3, 8, 8, 3, 6, 7, 0, 9, 4, 1, 6, 3, 3, 7, 2, 9, 3, 8, 0, 8, 3, 0, 7, 6, 8, 1, 3, 5, 8, 7, 0, 3, 6, 5, 5, 6, 3, 9, 1, 4, 6, 5, 5, 8, 5, 5

Offset: 1

Amiram Eldar, Jun 21 2020

The asymptotic mean of A367987. - Amiram Eldar, Dec 23 2023

			1.450032145362120831608395887189223422325062117447167...

Cf. A000082, A001615, A013661, A065465, A065484, A367987.

$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{-1, 1, 2, 0, -1}, {0, 2, -3, 6, -5}, m]; RealDigits[Exp[NSum[Indexed[c, n]*(PrimeZetaP[n])/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

prodeulerrat(1 + p/((p-1)*(p+1)^2)) \\ Amiram Eldar, Mar 18 2021

Equals Sum_{k>=1} 1/A000082(k) = Sum_{k>=1} 1/(k * A001615(k)).

Equals A013661 * A065465. - Amiram Eldar, Dec 23 2023

More digits from Vaclav Kotesovec, Sep 19 2020

A375032 The maximum odd exponent in the prime factorization of n, or 0 if no such exponent exists.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 3, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 3, 0, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 0, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 0, 1, 1, 1, 3, 1

Offset: 1

Amiram Eldar, Jul 28 2024

The asymptotic density of the occurrences of 0's is 0 (the asymptotic density of squares).
The asymptotic density of the occurrences of 1's is d(0) = Product_{p prime} (1 - 1/(p^2*(p+1))) = 0.881513... (A065465, asymptotic density of A335275).
The asymptotic density of the occurrences of 2*k+1, for k = 1, 2, ..., is d(k) = Product_{p prime} (1 - 1/(p^(2*k+2)*(p+1))) - Product_{p prime} (1 - 1/(p^(2*k)*(p+1))).

Cf. A000290, A051903, A065465, A162642, A335275, A350389, A375033.

a[n_] := Max[0, Max[Select[FactorInteger[n][[;; , 2]], OddQ]]]; a[1] = 0; Array[a, 100]

a(n) = {my(e = select(x -> (x % 2), factor(n)[,2])); if(#e == 0, 0, vecmax(e));}

max(a(n), A375033(n)) = A051903(n).
a(n) = 0 if and only if n is a square (A000290).
a(n) = 1 if and only if n is in A335275 \ A000290.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=0} (2*k+1) * d(k) = 1.30000522546018852138..., where d(k) is defined in the Comments section above.
a(n) = A051903(A350389(n)). - Amiram Eldar, Aug 17 2024

A055654 Difference between n and the result of "Phi-summation" over unitary divisors of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 3, 2, 0, 0, 3, 0, 0, 0, 7, 0, 4, 0, 5, 0, 0, 0, 9, 4, 0, 8, 7, 0, 0, 0, 15, 0, 0, 0, 15, 0, 0, 0, 15, 0, 0, 0, 11, 10, 0, 0, 21, 6, 8, 0, 13, 0, 16, 0, 21, 0, 0, 0, 15, 0, 0, 14, 31, 0, 0, 0, 17, 0, 0, 0, 37, 0, 0, 12, 19, 0, 0, 0, 35, 26, 0, 0, 21, 0, 0, 0, 33, 0, 20, 0, 23

Offset: 1

Labos Elemer, Jun 07 2000

Squarefree numbers are roots of a(n)=0 equation, while Min n for which a(n)=k is k^2. See also A000188, A008833.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000188, A008833, A055653, A053570, A053571, A055631, A055632, A055653, A065465.

a055654 n = a055654_list !! (n-1)
a055654_list = zipWith (-) [1..] a055653_list
-- Reinhard Zumkeller, Mar 11 2012

Table[n - DivisorSum[n, EulerPhi[#] &, CoprimeQ[#, n/#] &], {n, 92}] (* Michael De Vlieger, Oct 26 2017 *)
f[p_, e_] := p^e - p^(e-1) + 1; a[1] = 0; a[n_] := n - Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 04 2024 *)

a(n) = n - sumdiv(n, d, if (gcd(d, n/d)==1, eulerphi(d))); \\ Michel Marcus, Oct 27 2017

a(n) = {my(f = factor(n)); n - prod(k = 1, #f~, f[k,1]^f[k,2] - f[k,1]^(f[k,2] - 1) + 1);} \\ Amiram Eldar, Oct 04 2024

a(n) = n - Sum_{u|n, gcd(u,n/u) = 1} phi(u), i.e. when u is a unitary divisor of n.
a(n) = n - A055653(n). - Sean A. Irvine, Mar 30 2022
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 - A065465 = 0.11848616... . - Amiram Eldar, Oct 04 2024

A056196 Numbers n such that A055229(n) = 2.

Original entry on oeis.org

8, 24, 32, 40, 56, 72, 88, 96, 104, 120, 128, 136, 152, 160, 168, 184, 200, 224, 232, 248, 264, 280, 288, 296, 312, 328, 344, 352, 360, 376, 384, 392, 408, 416, 424, 440, 456, 472, 480, 488, 504, 512, 520, 536, 544, 552, 568, 584, 600, 608, 616, 632, 640

Offset: 1

Labos Elemer, Aug 02 2000

nonn

By definition, the largest square divisor and squarefree part of these numbers have GCD = 2.
Different from A036966. E.g., 81 is not here because A055229(81) = 1.
Numbers of the form 2^(2*k+1) * m, where k >= 1 and m is an odd number whose prime factorization contains only exponents that are either 1 or even. The asymptotic density of this sequence is (1/12) * Product_{p odd prime} (1-1/(p^2*(p+1))) = A065465 / 11 = 0.08013762179319734335... - Amiram Eldar, Dec 04 2020, Nov 25 2022

			88 is here because 88 has squarefree part 22, largest square divisor 4, and A055229(88) = gcd(22, 4) = 2.

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

Cf. A036966, A055229, A065465.

filter:= proc(n) local T;
    T:= select(t -> (t[2]::odd and t[2]>1), ifactors(n)[2]);
    nops(T) = 1 and T[1][1]=2;
end proc:
select(filter, [$1..1000]); # Robert Israel, Sep 21 2015

f[n_] := Block[{p = FactorInteger@ n, a}, a = Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ p); GCD[a, n/a]]; Position[Array[f, 640], 2] // Flatten (* Michael De Vlieger, Sep 22 2015, after Jean-François Alcover at A055229 *)

isok(n) = my(c=core(n)); gcd(c, n/c) == 2; \\ after PARI in A055229; Michel Marcus, Sep 20 2015

Edited by Robert Israel, Sep 21 2015

A072780 a(n) = sigma_2(n) + phi(n) * sigma(n) - 2n^2, which is A072779(n) - 2n^2.

Original entry on oeis.org

0, 0, 0, 3, 0, 2, 0, 17, 7, 2, 0, 34, 0, 2, 2, 77, 0, 41, 0, 82, 2, 2, 0, 178, 21, 2, 82, 154, 0, 76, 0, 325, 2, 2, 2, 411, 0, 2, 2, 450, 0, 124, 0, 370, 188, 2, 0, 786, 43, 115, 2, 514, 0, 428, 2, 858, 2, 2, 0, 948, 0, 2, 356, 1333, 2, 268, 0, 874, 2, 156, 0, 2047, 0, 2, 220

Offset: 1

T. D. Noe, Jul 15 2002

This sequence is interesting because (1) a(n) >= 0, with equality only when n is prime (or 1) and (2) a(n) = 2 if and only if n is the product of two distinct primes. Note for twin primes: let n = m^2 - 1, then m-1 and m+1 are twin primes if and only if a(n) = 2. Note for the Goldbach conjecture: let n = m^2 - r^2, then m-r and m+r are primes that add to 2m if and only if a(n) = 2.

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Divisor Function.
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A000010, A000203, A001157, A051709, A072779.

Cf. A002117, A065465.

Table[DivisorSigma[2, n]+EulerPhi[n]DivisorSigma[1, n]-2n^2, {n, 100}]

a(n)=sigma(n,2)+eulerphi(n)*sigma(n)-2*n^2 \\ Charles R Greathouse IV, May 15 2013

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(3) + Product_{p prime} (1 - 1/(p^2*(p+1))) - 2 = A002117 + A065465 - 2 = 0.083570742884... . - Amiram Eldar, Dec 03 2023

A076998 Difference between cubefree and squarefree components of n.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 2, 6, 0, 0, 6, 0, 0, 0, 2, 0, 12, 0, 10, 0, 0, 0, 6, 20, 0, 6, 14, 0, 0, 0, 2, 0, 0, 0, 30, 0, 0, 0, 10, 0, 0, 0, 22, 30, 0, 0, 6, 42, 40, 0, 26, 0, 12, 0, 14, 0, 0, 0, 30, 0, 0, 42, 2, 0, 0, 0, 34, 0, 0, 0, 30, 0, 0, 60, 38, 0, 0, 0, 10, 6, 0, 0, 42, 0, 0, 0, 22, 0, 60, 0, 46

Offset: 1

Jon Perry, Nov 28 2002

			a(4)=2 as cubefree(4)=4 and squarefree(4)=2. 4-2=2

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

Cf. A005117, A007947, A007948.

Cf. A065463, A065465.

a[n_] := Module[{f = FactorInteger[n]}, Times @@ (First[#]^Min[Last[#], 2] & /@ f) - Times @@ (First[#] & /@ f)]; Array[a, 100] (* Amiram Eldar, Sep 24 2023 *)

rad(n)=local(p,i); p=factor(n)[,1]; prod(i=1,length(p),p[i])
rad2(n)=local(p,pn,i); p=factor(n)[,1]; pn=factor(n)[,2]; prod(i=1,length(p),p[i]^min(2,pn[i]))
for (k=1,100,print1(rad2(k)-rad(k)", "))

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^min(f[i,2],2)) - vecprod(f[,1]);} \\ Amiram Eldar, Sep 24 2023

a(n) = A007948(n) - A007947(n). - Antti Karttunen, Jul 21 2018
From Amiram Eldar, Sep 24 2023: (Start)
a(n) >= 0, with equality if and only if n is squarefree (A005117).
Sum_{k=1..n} a(k) ~ (1/2) * c * n^2, where c = A065465 - A065463 = 0.17707163872600518419... . (End)

A116393 Decimal expansion of Product(1 - 1/(p+1)^3), p prime >= 2.

Original entry on oeis.org

9, 4, 0, 3, 0, 0, 4, 1, 4, 5, 7, 0, 3, 4, 5, 0, 7, 0, 6, 1, 9, 4, 6, 5, 3, 8, 5, 3, 7, 3, 2, 3, 9, 4, 4, 9, 9, 9, 8, 5, 7, 4, 0, 1, 4, 3, 6, 9, 1, 3, 0, 0, 7, 2, 5, 7, 0, 3, 1, 3, 3, 8, 3, 0, 5, 9, 4, 7, 4, 6, 2, 1, 0, 4, 5, 5, 4, 5, 7, 8, 0, 1, 2, 6, 7, 0, 4

Offset: 0

N. J. A. Sloane, Apr 15 2006

Calculated from A065465 * A065467^3 * X / A065466^2 where X = 1.03108637675008536... = product[ 1+ (4p^13 +13p^12 +10p^11 -3p^10 -9p^9 -12p^8 -9p^7 +6p^5 +4p^4 +2p^3 +p^2 -p -1) /(p^5+p^4-1)^3 / (p^3+p^2-1) / (p+1)] over prime p >=2. - R. J. Mathar, Sep 10 2007

			0.9403004...

Cf. A065472.

prodeulerrat(1 - 1/(p+1)^3) \\ Amiram Eldar, Nov 29 2020

More terms from John W. Layman and Zak Seidov, Apr 20 2006
More terms from R. J. Mathar, Sep 10 2007
More terms from Amiram Eldar, Nov 29 2020

A135178 a(n) = p^3 + p^2 where p = prime(n).

Original entry on oeis.org

12, 36, 150, 392, 1452, 2366, 5202, 7220, 12696, 25230, 30752, 52022, 70602, 81356, 106032, 151686, 208860, 230702, 305252, 362952, 394346, 499280, 578676, 712890, 922082, 1040502, 1103336, 1236492, 1306910, 1455666, 2064512, 2265252

Offset: 1

Omar E. Pol, Nov 25 2007

nonn

			a(4)=392 because the 4th prime number is 7, 7^3=343, 7^2=49 and 343+49=392.

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Cf. A000040 (p), A001248 (p^2), A030078 (p^3).

Cf. A065465.

[ p^3 + p^2: p in PrimesUpTo(200)]; // Vincenzo Librandi, Dec 14 2010

A135178:= n -> map(p -> p^(2)+p^(3), ithprime(n)):
seq(A135178(n), n=1..32); #  Jani Melik, Jan 25 2010

Table[p=Prime[n];p^2+p^3,{n,100}] (* Vladimir Joseph Stephan Orlovsky, Mar 21 2011*)
#^3+#^2&/@Prime[Range[40]] (* Harvey P. Dale, May 07 2023 *)

Product_{n>=1} (1 - 1/a(n)) = A065465. - Amiram Eldar, Jan 23 2021

Sum 1/a(n) = A382562.

A365347 The sum of divisors of the smallest number whose square is divisible by n.

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 7, 4, 18, 12, 12, 14, 24, 24, 7, 18, 12, 20, 18, 32, 36, 24, 28, 6, 42, 13, 24, 30, 72, 32, 15, 48, 54, 48, 12, 38, 60, 56, 42, 42, 96, 44, 36, 24, 72, 48, 28, 8, 18, 72, 42, 54, 39, 72, 56, 80, 90, 60, 72, 62, 96, 32, 15, 84, 144, 68, 54

Offset: 1

Amiram Eldar, Sep 02 2023

The number of divisors of the smallest number whose square is divisible by n is A322483(n).

The sum of divisors of the smallest square divisible by n is A365346(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A019554, A322483, A365346.

Cf. A002117, A065465.

f[p_, e_] := (p^((e + Mod[e, 2])/2 + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^((f[i,2] + f[i,2]%2)/2 + 1) - 1)/(f[i,1] - 1));}

a(n) = sigma(n/core(n, 1)[2]); \\ Michel Marcus, Sep 02 2023

a(n) = A000203(A019554(n)).
Multiplicative with a(p^e) = (p^(e + 1 + (e mod 2)) - 1)/(p - 1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * zeta(3) * Product_{p prime} (1 - 1/(p^2*(p+1))) = (1/2) * A002117 * A065465 = 0.529814898136... .

A367407 a(n) = sqrt(A367406(n)).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 4, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 12, 26, 9, 29, 30, 31, 8, 33, 34, 35, 37, 38, 39, 20, 41, 42, 43, 46, 47, 51, 53, 18, 55, 28, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 44, 89, 91, 93, 94, 95, 24, 97

Offset: 1

Amiram Eldar, Nov 17 2023

A permutation of the positive integers.

Cf. A064549, A268335, A367406, A367417, A367419.

Cf. A065463, A065465, A253905.

s[n_] := Sqrt[n * Times @@ FactorInteger[n][[;;, 1]]]; s /@ Select[Range[100], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]

b(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, f[i,1]^(f[i,2]+1), 0));}
lista(kmax) = {my(b1); for(k = 1, kmax, b1 = b(k); if(b1 > 0, print1(sqrtint(b1), ", ")));}

a(n) = sqrt(A064549(A268335(n))).
a(n) = sqrt(A268335(n)*A367417(n)).
a(n) = A268335(n)/A367419(n).
Sum_{k=1..n} a(k) = c * n^2 / 2, where c = (zeta(3)/(zeta(2)*d^2)) * Product_{p prime} (1 - 1/(p^2*(p+1))) = A253905 * A065465 / d^3 = 1.29812028442810841122..., and d = A065463 is the asymptotic density of the exponentially odd numbers (A268335).

cp's OEIS Frontend

A335762 Decimal expansion of Product_{p prime} (1 + p/((p-1)*(p+1)^2)).

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A375032 The maximum odd exponent in the prime factorization of n, or 0 if no such exponent exists.

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A055654 Difference between n and the result of "Phi-summation" over unitary divisors of n.

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A056196 Numbers n such that A055229(n) = 2.

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A072780 a(n) = sigma_2(n) + phi(n) * sigma(n) - 2*n^2, which is A072779(n) - 2*n^2.

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A076998 Difference between cubefree and squarefree components of n.

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PARI

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A116393 Decimal expansion of Product(1 - 1/(p+1)^3), p prime >= 2.

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A072780 a(n) = sigma_2(n) + phi(n) * sigma(n) - 2n^2, which is A072779(n) - 2n^2.