A065474 -id:A065474 - cp's OEIS frontend

A070258 Smallest of 3 consecutive numbers each divisible by a square.

48, 98, 124, 242, 243, 342, 350, 423, 475, 548, 603, 724, 774, 844, 845, 846, 1024, 1250, 1274, 1323, 1375, 1420, 1448, 1519, 1664, 1674, 1680, 1681, 1682, 1848, 1862, 1924, 2007, 2023, 2056, 2106, 2150, 2223, 2275, 2348, 2366, 2523, 2527, 2574, 2644

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 09 2002

nonn

The sequence includes an infinite family of arithmetic progressions. Such AP's can be constructed to each term, with large differences [like e.g. square of primorials, A061742]. It is necessary to solve suitable systems of linear Diophantine equations. E.g.: subsequences of triples of terms = {900a+548, 900a+549, 900a+550} = {4(225f+137), 9(100f+61), 25(36f+22)}; starting terms in this sequence = {548, 1448, 2348, ...}; difference = A002110(3)^2. - Labos Elemer, Nov 25 2002
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 2, 16, 180, 1868, 18649, 186335, 1863390, 18634236, 186340191, ... . Apparently, the asymptotic density of this sequence exists and equals 0.01863... . - Amiram Eldar, Jan 18 2023
The asymptotic density of this sequence is 1 - 3/zeta(2) + 3 * Product_{p prime} (1 - 2/p^2) - Product_{p prime} (1 - 3/p^2) = 1 - 3 * A059956 + 3 * A065474 - A206256 = 0.018634010349844827414... . - Amiram Eldar, Sep 12 2024

Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 48, p. 18, Ellipses, Paris, 2008.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Subsequence of A013929 and A068781.
Cf. A002110, A061742, A235578.
Cf. A059956, A065474, A206256.

f[n_] := Union[ Transpose[ FactorInteger[n]] [[2]]] [[ -1]]; a = 0; b = 1; Do[c = f[n]; If[a> 1 && b > 1 && c > 1, Print[n - 2]]; a = b; b = c, {n, 3, 10^6}]
Flatten[Position[Partition[SquareFreeQ/@Range[3000],3,1],?(Union[#] == {False}&),{1},Heads->False]] (* _Harvey P. Dale, May 24 2014 *)
f@n_ := Flatten@  Position[Partition[SquareFreeQ /@ Range@2000, n, 1], Table[False, {n}]]; f@3 (* Hans Rudolf Widmer, Aug 30 2022 *)

a(n) = A235578(n) - 1. - Amiram Eldar, Feb 09 2021

More terms from Jason Earls and Robert G. Wilson v, May 10 2002

Offset corrected by Amiram Eldar, Feb 09 2021

A206256 Decimal expansion of Product_{p prime} (1 - 3/p^2).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Feb 05 2012, based on a posting by Warren Smith to the Math Fun Mailing List, Feb 04 2012

For a randomly selected number k, this is the probability that k, k+1, k+2 all are squarefree.

			0.1254869809058...

Leon Mirsky, Note on an asymptotic formula connected with r-free integers, The Quarterly Journal of Mathematics, Vol. os-18, No. 1 (1947), pp. 178-182.
Leon Mirsky, Arithmetical pattern problems relating to divisibility by rth powers, Proceedings of the London Mathematical Society, Vol. s2-50, No. 1 (1949), pp. 497-508.

Cf. A059956, A065474, A007675, A335131, A370600.

# See A175640 using efact := 1-3/p^2. - R. J. Mathar, Mar 22 2012

$MaxExtraPrecision = 500; m = 500; c = LinearRecurrence[{0, 3}, {0, -6}, m]; RealDigits[(1/4) * Exp[NSum[Indexed[c, n]*(PrimeZetaP[n] - 1/2^n)/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]] (* Amiram Eldar, Oct 01 2019 *)

prodeulerrat(1 - 3/p^2) \\ Amiram Eldar, Mar 16 2021

More terms from Amiram Eldar, Oct 01 2019

More terms from Vaclav Kotesovec, Dec 17 2019

A280892 Squareful numbers with both neighbors squarefree.

Original entry on oeis.org

4, 12, 16, 18, 20, 32, 36, 40, 52, 54, 56, 60, 68, 72, 84, 88, 90, 92, 96, 104, 108, 112, 128, 132, 140, 144, 150, 156, 160, 162, 164, 180, 184, 192, 196, 198, 200, 204, 212, 216, 220, 228, 232, 234, 236, 240, 248, 250, 252, 256, 264, 268, 270, 272, 284, 292, 294, 300, 304, 306, 308, 312

Offset: 1

Juri-Stepan Gerasimov, Jan 16 2017

Subsequence of A013929 and A067874.

The asymptotic density of this sequence is Product_{p prime} (1 - 2/p^2) - Product_{p prime} (1 - 3/p^2) = 0.197147118033435... (Mossinghoff et al., 2021). - Amiram Eldar, Nov 11 2021, Mar 21 2024

			4 is in this sequence because 4 = 2^2 is nonsquarefree and both 4 - 1 = 3 and 4 + 1 = 5 are squarefree.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Michael J. Mossinghoff, Tomás Oliveira e Silva, and Tim Trudgian, The distribution of k-free numbers, Mathematics of Computation, Vol. 90, No. 328 (2021), pp. 907-929; arXiv preprint, arXiv:1912.04972 [math.NT], 2019-2020.

Cf. A005117, A013929, A067874, A228649, A272799, A281192.

Cf. A065474, A206256.

[n: n in [2..300] | not IsSquarefree(n) and IsSquarefree(n-1) and IsSquarefree(n+1)];

Mean/@SequencePosition[Table[If[SquareFreeQ[n],1,0],{n,400}],{1,0,1}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 19 2020 *)

isok(n) = !issquarefree(n) && issquarefree(n-1) && issquarefree(n+1); \\ Michel Marcus, Jun 18 2017

Definition corrected by Jon E. Schoenfield, Jun 18 2017

A067874 Positive integers x satisfying x^2 - D*y^2 = 1 for a unique integer D.

Original entry on oeis.org

2, 4, 6, 12, 14, 16, 18, 20, 22, 30, 32, 34, 36, 38, 40, 42, 52, 54, 56, 58, 60, 66, 68, 70, 72, 78, 84, 86, 88, 90, 92, 94, 96, 102, 104, 106, 108, 110, 112, 114, 128, 130, 132, 138, 140, 142, 144, 150, 156, 158, 160, 162, 164, 166, 178, 180, 182, 184, 186, 192, 194, 196, 198

Offset: 1

Lekraj Beedassy, Feb 25 2002

D is unique iff x^2 - 1 is squarefree, in which case it follows with necessity that D=x^2-1 and y=1.
All terms are even. A014574 is a subsequence.
Conjecture: All terms of A002110 > 1 are a subsequence. - Griffin N. Macris, Apr 11 2016
All n such that n+1 and n-1 are in A056911. - Robert Israel, Apr 12 2016
The asymptotic density of this sequence is Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474). - Amiram Eldar, Feb 25 2021

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A002110, A005117, A014574, A056911, A065474, A226993, A272799, A280892, A379971 (characteristic function).

Subsequence of A379965.

[n: n in [1..110] | IsSquarefree(n-1) and IsSquarefree(n+1)]; // Juri-Stepan Gerasimov, Jan 17 2017

select(t -> numtheory:-issqrfree(t^2-1), [seq(n,n=2..1000,2)]); # Robert Israel, Apr 12 2016

Select[Range[200], SquareFreeQ[#^2-1]&] (* Vladimir Joseph Stephan Orlovsky, Oct 26 2009 *)

from itertools import count, islice
from sympy import factorint
def A067874_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda k:max(factorint(k-1).values(),default=1)==1 and max(factorint(k+1).values())==1, count(max(startvalue+(startvalue&1),2),2))
A067874_list = list(islice(A067874_gen(),20)) # Chai Wah Wu, Apr 24 2024

a(n) = 2*A272799(n). - Juri-Stepan Gerasimov, Jan 17 2017

Corrected and extended by Max Alekseyev, Apr 26 2009

Further edited by Max Alekseyev, Apr 28 2009

A083542 a(n) = phi(n+1)*phi(n), product of totients of two consecutive integers.

Original entry on oeis.org

1, 2, 4, 8, 8, 12, 24, 24, 24, 40, 40, 48, 72, 48, 64, 128, 96, 108, 144, 96, 120, 220, 176, 160, 240, 216, 216, 336, 224, 240, 480, 320, 320, 384, 288, 432, 648, 432, 384, 640, 480, 504, 840, 480, 528, 1012, 736, 672, 840, 640, 768, 1248, 936, 720, 960, 864, 1008

Offset: 1

Labos Elemer, May 21 2003

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

Cf. A000010, A002378, A058515, A065474, A066813, A330319 (partial sums).

Cf. A083481, A083539, A092517.

a083542 n = a000010 n * a000010 (n + 1)
a083542_list = zipWith (*) (tail a000010_list) a000010_list
-- Reinhard Zumkeller, Apr 22 2012

a:= n-> (p-> p(n)*p(n+1))(numtheory[phi]):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 21 2022

Times @@ EulerPhi@ # & /@ Partition[Range@ 58, 2, 1] (* Michael De Vlieger, Mar 25 2017 *)
Times@@@Partition[EulerPhi[Range[60]],2,1] (* Harvey P. Dale, Oct 29 2019 *)

a(n) = eulerphi(n) * eulerphi(n+1); \\ Amiram Eldar, Jul 10 2024

a(n) = A000010(A002378(n)). - Amiram Eldar, Jul 10 2024
Sum_{k=1..n} a(k) = c * n^3 / 3 + O((n*log(n))^2), where c = Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474). - Amiram Eldar, Dec 09 2024
a(n) = A058515(n)*A066813(n). - Amiram Eldar, May 07 2025

A057475 Number of k, 1 <= k <= n, such that gcd(n,k) = gcd(n+1,k) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 3, 2, 4, 3, 4, 5, 3, 4, 8, 5, 6, 7, 5, 5, 10, 7, 7, 9, 8, 8, 12, 7, 8, 15, 10, 9, 11, 8, 12, 17, 11, 9, 16, 11, 12, 19, 11, 11, 22, 15, 14, 17, 13, 15, 24, 17, 14, 17, 15, 17, 28, 15, 16, 29, 17, 18, 24, 15, 20, 31, 21, 15, 24, 23, 24, 35, 19, 19, 28, 18, 24, 31, 22

Offset: 1

Leroy Quet, Sep 27 2000

nonn

Number of numbers between 1 and n-1 coprime to n(n+1).

It is conjectured that every positive integer appears. - Jon Perry, Dec 12 2002

			a(8) = 3 because 1, 5 and 7 are all relatively prime to both 8 and 9.
a(9) counts those numbers coprime to 90, i.e., 1 and 7, hence a(9) = 2.

Cf. A000040, A000010, A065474, A118854, A124738, A124739, A124740, A124741.

Cf. A057828, A002378.

[#[k:k in [1..n]| Gcd(n,k) eq Gcd(n+1,k) and Gcd(n,k) eq 1]: n in [1..80]]; // Marius A. Burtea, Oct 15 2019

A057475 := proc(n)
    local a,k ;
    a :=  0;
    for k from 1 to n do
        if igcd(k,n) = 1 and igcd(k,n+1)=1 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A057475(n),n=1..80) ; # R. J. Mathar, May 13 2025

a[ n_ ] := Length @ Select[ Range[ n ], GCD[ n, # ] == GCD[ n + 1, # ] == 1 & ]; Table[ a[ n ], {n, 80} ] (* Ray Chandler, Dec 06 2006 *)

newphi(v)=local(vl,fl,np); vl=length(v); np=0; for (s=1,v[1],fl=false; for (r=1,vl,if (gcd(s,v[r])>1,fl=true; break)); if (fl==false,np++)); np
v=vector(2); for (i=1,500,v[1]=i; v[2]=i+1; print1(newphi(v)","))

From Reinhard Zumkeller, May 02 2006: (Start)
a(A000040(n)-1) = A000010(A000040(n)-1);
a(A000040(n)) = A000010(A000040(n)+1)-1;
a(A118854(n)-1) = a(A118854(n)). (End)
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474). - Amiram Eldar, Dec 10 2024
a(n) = A057828(A002378(n)). - Ridouane Oudra, May 30 2025

A065469 Decimal expansion of Product_{p prime} (1 - 1/(p^2-1)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

			0.53071182047204479497294377247...

G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]

Cf. A000010, A000203, A007434, A008683, A013661, A065474, A072101.

$MaxExtraPrecision = 800; digits = 98; terms = 800; P[n_] := PrimeZetaP[n]; LR = LinearRecurrence[{0, 3, 0, -2}, {0, 0, -2, 0}, terms + 10]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)

prodeulerrat(1 - 1/(p^2-1)) \\ Amiram Eldar, Mar 13 2021

Product of A013661 by A065474. - R. J. Mathar, Mar 26 2011
From Amiram Eldar, Jan 14 2022: (Start)
Equals Sum_{k>=1} mu(k)/(phi(k)*sigma(k)), where mu is the Möbius function (A008683), phi is the Euler totient function (A000010) and sigma(k) is the sum of divisors of k (A000203).
Equals Sum_{k>=1} mu(k)/J_2(k), where J_2 is Jordan's totient function (A007434). (End)

A281192 Numbers with no squarefree neighbors.

Original entry on oeis.org

17, 19, 26, 49, 51, 53, 55, 89, 91, 97, 99, 125, 127, 149, 151, 161, 163, 170, 197, 199, 233, 235, 241, 243, 244, 249, 251, 269, 271, 293, 295, 305, 307, 337, 339, 341, 343, 349, 351, 362, 377, 379, 413, 415, 424, 449, 451, 476, 485, 487, 489, 491, 521, 523, 530, 549, 551, 557, 559, 577

Offset: 1

Juri-Stepan Gerasimov, Jan 16 2017

Includes all k == 17 or 19 (mod 36), also 2*p^2-1 and 2*p^2+1 for odd primes p. - Robert Israel, Jan 17 2017
This sequence has density around 0.106.... - Charles R Greathouse IV, Jan 23 2017
More accurately, the asymptotic density of this sequence is 1 - 2/zeta(2) + Product_{p prime} (1 - 2/p^2) = 1 - 2 * A059956 + A065474 = 0.1067798952... - Amiram Eldar, Feb 25 2021

			17 is in this sequence because 17 - 1 = 16 = 2^4 and 17 + 1 = 18 = 2*3^2 are not squarefree.

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

Supersequence of A075432 and A235578.

Cf. A005117, A059956, A065474, A280892.

[n: n in [2..600] | not IsSquarefree(n-1) and not IsSquarefree(n+1)];

select(t -> not numtheory:-issqrfree(t-1) and not numtheory:-issqrfree(t+1), [$1..1000]); # Robert Israel, Jan 17 2017

Select[Range[600], !SquareFreeQ[# - 1] && !SquareFreeQ[# + 1] &] (* Vincenzo Librandi, Jan 17 2017 *)

is(n)=!issquarefree(n-1) && !issquarefree(n+1) \\ Charles R Greathouse IV, Jan 23 2017

A002472 Number of pairs x,y such that y-x=2, (x,n)=1, (y,n)=1 and 1 <= x <= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 4, 3, 3, 9, 2, 11, 5, 3, 8, 15, 3, 17, 6, 5, 9, 21, 4, 15, 11, 9, 10, 27, 3, 29, 16, 9, 15, 15, 6, 35, 17, 11, 12, 39, 5, 41, 18, 9, 21, 45, 8, 35, 15, 15, 22, 51, 9, 27, 20, 17, 27, 57, 6, 59, 29, 15, 32, 33, 9, 65, 30, 21, 15, 69, 12, 71, 35, 15, 34, 45, 11, 77, 24, 27

Offset: 1

N. J. A. Sloane

This is the function phi(n, 2) defined in Alder. - Michel Marcus, Nov 14 2017

			For n = 4, the condition gcd(x,4) = gcd(x+2,4) = 1 is satisfied by exactly two positive integers x not exceeding n, namely, by x = 1 and x = 3. Therefore a(4) = 2.

V. A. Golubev, Sur certaines fonctions multiplicatives et le problème des jumeaux. Mathesis 67 (1958), 11-20.
V. A. Golubev, Nombres de Mersenne et caractères du nombre 2. Mathesis 67 (1958), 257-262.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Henry L. Alder, A Generalization of the Euler phi-Function, The American Mathematical Monthly, Vol. 65, No. 9 (Nov., 1958), pp. 690-692.
V. A. Golubev, A generalization of the functions phi(n) and pi(x), Časopis pro pěstování matematiky 78 (1953), 47-48.
V. A. Golubev, Exact formulas for the number of twin primes and other generalizations of the function pi(x), Časopis pro pěstování matematiky 87 (1962), 296-305.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Cf. A000010 (phi(n,0)), A058026 (phi(n,1)), A065474.
Similar generalizations of Euler's totient for prime k-tuples: this sequence (k=2), A319534 (k=3), A319516 (k=4), A321029 (k=5), A321030 (k=6).
Cf. A160467, A000265.

a002472 n = length [x | x <- [1..n], gcd n x == 1, gcd n (x + 2) == 1]
-- Reinhard Zumkeller, Mar 23 2012

with(numtheory): seq(add(mobius(d)*phi(2*n)/phi(2*d), d in divisors(n)), n=1..100); # Ridouane Oudra, Aug 20 2024

a[n_] := If[ Head[ r=Reduce[ GCD[x, n] == 1 && GCD[x+2, n] == 1 && 1 <= x <= n, x, Integers]] === Or, Length[r], 1]; Table[a[n], {n, 1, 81}] (* Jean-François Alcover, Nov 22 2011 *)
(* Second program (5 times faster): *)
a[n_] := Sum[Boole[GCD[n, x] == 1 && GCD[n, x+2] == 1], {x, 1, n}];
Array[a, 81] (* Jean-François Alcover, Jun 19 2018, after Michel Marcus *)
f[p_, e_] := If[p == 2, p^(e-1), (p-2)*p^(e-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 22 2020 *)

a(n)=my(k=valuation(n,2),f=factor(n>>k));prod(i=1,#f[,1],(f[i,1]-2)*f[i,1]^(f[i,2]-1))<Charles R Greathouse IV, Nov 22 2011

a(n) = sum(x=1, n, (gcd(n,x) == 1) && (gcd(n, x+2) == 1)); \\ Michel Marcus, Nov 14 2017

Multiplicative with a(p^e) = p^(e-1) if p = 2; (p-2)*p^(e-1) if p > 2. - David W. Wilson, Aug 01 2001
a(n) = Sum_{k=1..n} [GCD(2*n-k,n) * GCD(k+2,n) = 1], where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Sep 29 2021
Sum_{k=1..n} a(k) ~ c * n^2, where c = (3/4) * Product_{p prime} (1 - 2/p^2) = (3/4) * A065474 = 0.2419755742... . - Amiram Eldar, Oct 23 2022
From Ridouane Oudra, Aug 20 2024: (Start)
a(n) = phi(2*n) * Sum_{d|n} mu(d)/phi(2*d).
a(n) = - phi(n) * Sum_{d|n} mu(2*d)/phi(d).
a(n) = A160467(n)*A058026(A000265(n)).
a(2*n+1) = A070554(n).
a(2^m*(2*n+1)) = 2^(m-1)*A070554(n), with m>0. (End)

More terms from David W. Wilson

A065093 Convolution of A000010 with itself.

Original entry on oeis.org

1, 2, 5, 8, 16, 20, 36, 44, 68, 76, 120, 124, 188, 196, 276, 272, 404, 380, 544, 532, 716, 668, 968, 860, 1184, 1120, 1472, 1332, 1896, 1624, 2204, 2036, 2656, 2352, 3284, 2752, 3684, 3356, 4324, 3744, 5192, 4312, 5720, 5180, 6540, 5628, 7768, 6388, 8476

Offset: 1

Vladeta Jovovic, Nov 11 2001

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
A. E. Ingham, Some asymptotic formulae in the theory of numbers, Journal of the London Mathematical Society, Vol. s1-2, No. 3 (1927), pp. 202-208.

Cf. A000010, A000385, A055507, A065474, A330319.

Column k=2 of A340995.

Table[Sum[EulerPhi[j]*EulerPhi[n-j], {j, 1, n-1}], {n, 2, 50}] (* Vaclav Kotesovec, Aug 18 2021 *)

{ for (n=1, 1000, a=sum(k=1, n, eulerphi(k)*eulerphi(n+1-k)); write("b065093.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 06 2009

a(n) = Sum_{k=1..n} phi(k)*phi(n+1-k), where phi is Euler totient function (A000010).
G.f.: (1/x)*(Sum_{k>=1} mu(k)*x^k/(1 - x^k)^2)^2. - Ilya Gutkovskiy, Jan 31 2017
a(n) ~ (n^3/6) * c * Product_{primes p|n+1} ((p^3-2*p+1)/(p*(p^2-2))), where c = Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474) (Ingham, 1927). - Amiram Eldar, Jul 13 2024

cp's OEIS Frontend

A070258 Smallest of 3 consecutive numbers each divisible by a square.

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A206256 Decimal expansion of Product_{p prime} (1 - 3/p^2).

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A280892 Squareful numbers with both neighbors squarefree.

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A067874 Positive integers x satisfying x^2 - D*y^2 = 1 for a unique integer D.

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A083542 a(n) = phi(n+1)*phi(n), product of totients of two consecutive integers.

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A057475 Number of k, 1 <= k <= n, such that gcd(n,k) = gcd(n+1,k) = 1.

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A065469 Decimal expansion of Product_{p prime} (1 - 1/(p^2-1)).

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