keyword:mult - cp's OEIS frontend

A003958 If n = Product p(k)^e(k) then a(n) = Product (p(k)-1)^e(k).

1, 1, 2, 1, 4, 2, 6, 1, 4, 4, 10, 2, 12, 6, 8, 1, 16, 4, 18, 4, 12, 10, 22, 2, 16, 12, 8, 6, 28, 8, 30, 1, 20, 16, 24, 4, 36, 18, 24, 4, 40, 12, 42, 10, 16, 22, 46, 2, 36, 16, 32, 12, 52, 8, 40, 6, 36, 28, 58, 8, 60, 30, 24, 1, 48, 20, 66, 16, 44, 24, 70, 4, 72, 36, 32, 18, 60, 24, 78, 4, 16

Offset: 1

Marc LeBrun

Completely multiplicative.

Dirichlet inverse of A097945. - R. J. Mathar, Aug 29 2011

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)
Vaclav Kotesovec, Graph - the asymptotic ratio (10^7 terms).
Index to divisibility sequences

Cf. A003959, A168065, A168066, A027746, A006093, A027748, A124010.

a003958 1 = 1
a003958 n = product $ map (subtract 1) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012, Mar 02 2012

a:= n-> mul((i[1]-1)^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 13 2017

DirichletInverse[f_][1] = 1/f[1]; DirichletInverse[f_][n_] := DirichletInverse[f][n] = -1/f[1]*Sum[ f[n/d]*DirichletInverse[f][d], {d, Most[ Divisors[n]]}]; muphi[n_] := MoebiusMu[n]*EulerPhi[n]; Table[ DirichletInverse[ muphi][n], {n, 1, 81}] (* Jean-François Alcover, Dec 12 2011, after R. J. Mathar *)
a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 1)^fi[[All, 2]])); Table[a[n], {n, 1, 50}] (* G. C. Greubel, Jun 10 2016 *)

a(n)=if(n<1,0,direuler(p=2,n,1/(1-p*X+X))[n]) /* Ralf Stephan */

from math import prod
from sympy import factorint
def a(n): return prod((p-1)**e for p, e in factorint(n).items())
print([a(n) for n in range(1, 82)]) # Michael S. Branicky, Feb 27 2022

Multiplicative with a(p^e) = (p-1)^e. - David W. Wilson, Aug 01 2001
a(n) = A000010(n) iff n is squarefree (see A005117). - Reinhard Zumkeller, Nov 05 2004
a(n) = abs(A125131(n)). - Tom Edgar, May 26 2014
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4 / (315 * zeta(3)) = 1/(2*A082695) = 0.25725505075419... - Vaclav Kotesovec, Jun 14 2020
Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(1-s) + p^(-s)). - Ilya Gutkovskiy, Feb 27 2022
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{primes p} (1 + (p^(1-s) - 2) / (1 - p + p^s)), (with a product that converges for s=2). - Vaclav Kotesovec, Feb 11 2023

Definition reedited (from formula) by Daniel Forgues, Nov 17 2009

A057521 Powerful (1) part of n: if n = Product_i (pi^ei) then a(n) = Product_{i : ei > 1} (pi^ei); if n=bc^2d^3 then a(n)=c^2*d^3 when b is minimized.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 8, 9, 1, 1, 4, 1, 1, 1, 16, 1, 9, 1, 4, 1, 1, 1, 8, 25, 1, 27, 4, 1, 1, 1, 32, 1, 1, 1, 36, 1, 1, 1, 8, 1, 1, 1, 4, 9, 1, 1, 16, 49, 25, 1, 4, 1, 27, 1, 8, 1, 1, 1, 4, 1, 1, 9, 64, 1, 1, 1, 4, 1, 1, 1, 72, 1, 1, 25, 4, 1, 1, 1, 16, 81, 1, 1, 4, 1, 1, 1, 8, 1, 9, 1, 4, 1, 1, 1, 32, 1

Offset: 1

Henry Bottomley, Sep 01 2000

			a(40) = 8 since 40 = 2^3 * 5 so the powerful part is 2^3 = 8.

Antti Karttunen, Table of n, a(n) for n = 1..16383 (first 1000 terms from T. D. Noe)
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (100000000 terms).
Christian Krause, LODA, an assembly language, a computational model and a tool for mining integer sequences.
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seq., Vol. 6 (2003), Article 03.3.4.
Index entries for sequences related to powerful numbers.

Cf. A001694, A003415, A003557, A055231, A064549.

A057521 := proc(n)
    local a,d,e,p;
    a := 1;
    for d in ifactors(n)[2] do
        e := d[1] ;
        p := d[2] ;
        if e > 1 then
            a := a*p^e ;
        end if;
    end do:
    return a;
end proc: # R. J. Mathar, Jun 09 2016

rad[n_] := Times @@ First /@ FactorInteger[n]; a[n_] := n/Denominator[n/rad[n]^2]; Table[a[n], {n, 1, 97}] (* Jean-François Alcover, Jun 20 2013 *)
f[p_, e_] := If[e > 1, p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 21 2020 *)

a(n)=my(f=factor(n));prod(i=1,#f~,if(f[i,2]>1,f[i,1]^f[i,2],1)) \\ Charles R Greathouse IV, Aug 13 2013

a(n) = my(f=factor(n)); for (i=1, #f~, if (f[i,2]==1, f[i,1]=1)); factorback(f); \\ Michel Marcus, Jan 29 2021

from sympy import factorint, prod
def a(n): return 1 if n==1 else prod(1 if e==1 else p**e for p, e in factorint(n).items())
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 19 2017

from math import prod
from sympy import factorint
def A057521(n): return n//prod(p for p, e in factorint(n).items() if e == 1) # Chai Wah Wu, Nov 14 2022

a(n) = n / A055231(n).
Multiplicative with a(p)=1 and a(p^e)=p^e for e>1. - Vladeta Jovovic, Nov 01 2001
From Antti Karttunen, Nov 22 2017: (Start)
a(n) = A064549(A003557(n)).
A003557(a(n)) = A003557(n).
(End)
a(n) = gcd(n, A003415(n)^k), for all k >= 2. [This formula was found in the form k=3 by Christian Krause's LODA miner. See Ufnarovski and Åhlander paper, Theorem 5 on p. 4 for why this holds] - Antti Karttunen, Mar 09 2021
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 + 1/p^s - 1/ p^(s-1) + 1/p^(2*s-2) - 1/p^(2*s-1)). - Amiram Eldar, Sep 18 2023
From Vaclav Kotesovec, Apr 09 2025, simplified May 11 2025: (Start)
Dirichlet g.f.: zeta(2*s-2) * Product_{p prime} (1 - 1/p^(3*s-2) + 1/p^(3*s-3) + 1/p^s).
Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = Product_{p prime} (1 + 2/p^(3/2) - 1/p^(5/2)) = 3.51955505841710664719752940369857817... = A328013. (End)

A048272 Number of odd divisors of n minus number of even divisors of n.

Original entry on oeis.org

1, 0, 2, -1, 2, 0, 2, -2, 3, 0, 2, -2, 2, 0, 4, -3, 2, 0, 2, -2, 4, 0, 2, -4, 3, 0, 4, -2, 2, 0, 2, -4, 4, 0, 4, -3, 2, 0, 4, -4, 2, 0, 2, -2, 6, 0, 2, -6, 3, 0, 4, -2, 2, 0, 4, -4, 4, 0, 2, -4, 2, 0, 6, -5, 4, 0, 2, -2, 4, 0, 2, -6, 2, 0, 6, -2, 4, 0, 2, -6, 5, 0, 2, -4, 4, 0, 4, -4, 2, 0, 4, -2, 4

Offset: 1

Adam Kertesz

abs(a(n)) = (1/2) * (number of pairs (i,j) satisfying n = i^2 - j^2 and -n <= i,j <= n). - Benoit Cloitre, Jun 14 2003
As A001227(n) is the number of ways to write n as the difference of 3-gonal numbers, a(n) describes the number of ways to write n as the difference of e-gonal numbers for e in {0,1,4,8}. If pe(n):=(1/2)*n*((e-2)*n+(4-e)) is the n-th e-gonal number, then 4*a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e=1, 2*a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e in {0,4} and for a(n) itself is a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e=8. (Same for e=-1 see A035218.) - Volker Schmitt (clamsi(AT)gmx.net), Nov 09 2004
An argument by Gareth McCaughan suggests that the average of this sequence is log(2). - Hans Havermann, Feb 10 2013 [Supported by a graph. - Vaclav Kotesovec, Mar 01 2023]
From Keith F. Lynch, Jan 20 2024: (Start)
a(n) takes every possible integer value, positive, negative, and zero. Proof: For all nonnegative integers k, a(3^k) = 1+k, a(2^k) = 1-k.
a(n) takes every possible integer value except 1 and -1 infinitely many times. Proof: a(o^(k-1)) = k and a(4*o^(k-1)) = -k for all positive integers k and odd primes o, of which there are infinitely many.  a(n) = 0 iff n = 2 (mod 4).  a(n) = 1 iff n = 1.  a(n) = -1 iff n = 4.
a(n) takes prime value p only for n = o^(p-1), where o is any odd prime.
Terms have a simple pattern that repeats with a period of 4: Positive, zero, positive, negative.
(End)
Inverse Möbius transform of (-1)^(n+1). - Wesley Ivan Hurt, Jun 22 2024

			a(20) = -2 because 20 = 2^2*5^1 and (1-2)*(1+1) = -2.
G.f. = x + 2*x^3 - x^4 + 2*x^5 + 2*x^7 - 2*x^8 + 3*x^9 + 2*x^11 - 2*x^12 + ...

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), first formula.
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 97, 7(ii).

T. D. Noe, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n, for n = 1..1000000
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1921), 75-113; Coll. Papers II, pp. 303-341.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function tau_{o-e}(n).
Index entries for sequences mentioned by Glaisher.

Cf. A048298. A diagonal of A060184.
First differences of A059851.
Cf. A001227, A035218, A094572, A099774, A112329, A183063, A000005.
Indices of records: A053624 (highs), A369151 (lows).

a048272 n = a001227 n - a183063 n  -- Reinhard Zumkeller, Jan 21 2012

[&+[(-1)^(d+1):d in Divisors(n)] :n in [1..95] ]; // Marius A. Burtea, Aug 10 2019

add(x^n/(1+x^n), n=1..60): series(%,x,59);
A048272 := proc(n)
    local a;
    a := 1 ;
    for pfac in ifactors(n)[2] do
        if pfac[1] = 2 then
            a := a*(1-pfac[2]) ;
        else
            a := a*(pfac[2]+1) ;
        end if;
    end do:
    a ;
end proc: # Schmitt, sign corrected R. J. Mathar, Jun 18 2016
# alternative Maple program:
a:= n-> -add((-1)^d, d=numtheory[divisors](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 28 2018

Rest[ CoefficientList[ Series[ Sum[x^k/(1 - (-x)^k), {k, 111}], {x, 0, 110}], x]] (* Robert G. Wilson v, Sep 20 2005 *)
dif[n_]:=Module[{divs=Divisors[n]},Count[divs,?OddQ]-Count[ divs, ?EvenQ]]; Array[dif,100] (* Harvey P. Dale, Aug 21 2011 *)
a[n]:=Sum[-(-1)^d,{d,Divisors[n]}] (* Steven Foster Clark, May 04 2018 *)
f[p_, e_] := If[p == 2, 1 - e, 1 + e]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Jun 09 2022 *)

{a(n) = if( n<1, 0, -sumdiv(n, d, (-1)^d))}; /* Michael Somos, Jul 22 2006 */

N=17; default(seriesprecision,N); x=z+O(z^(N+1))
c=sum(j=1,N,j*x^j); \\ log case
s=-log(prod(j=1,N,(1+x^j)^(1/j)));
s=serconvol(s,c)
v=Vec(s) \\ Joerg Arndt, May 03 2008

a(n)=my(o=valuation(n,2),f=factor(n>>o)[,2]);(1-o)*prod(i=1,#f,f[i]+1) \\ Charles R Greathouse IV, Feb 10 2013

a(n)=direuler(p=1,n,if(p==2,(1-2*X)/(1-X)^2,1/(1-X)^2))[n] /* Ralf Stephan, Mar 27 2015 */

{a(n) = my(d = n -> if(frac(n), 0, numdiv(n))); if( n<1, 0, if( n%4, 1, -1) * (d(n) - 2*d(n/2) + 2*d(n/4)))}; /* Michael Somos, Aug 11 2017 */

Coefficients in expansion of Sum_{n >= 1} x^n/(1+x^n) = Sum_{n >= 1} (-1)^(n-1)*x^n/(1-x^n). Expand Sum 1/(1+x^n) in powers of 1/x.
If n = 2^p1*3^p2*5^p3*7^p4*11^p5*..., a(n) = (1-p1)*Product_{i>=2} (1+p_i).
Multiplicative with a(2^e) = 1 - e and a(p^e) = 1 + e if p > 2. - Vladeta Jovovic, Jan 27 2002
a(n) = (-1)*Sum_{d|n} (-1)^d. - Benoit Cloitre, May 12 2003
Moebius transform is period 2 sequence [1, -1, ...]. - Michael Somos, Jul 22 2006
G.f.: Sum_{k>0} -(-1)^k * x^(k^2) * (1 + x^(2*k)) / (1 - x^(2*k)) [Ramanujan]. - Michael Somos, Jul 22 2006
Equals A051731 * [1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Nov 07 2007
From Reinhard Zumkeller, Jan 21 2012: (Start)
a(n) = A001227(n) - A183063(n).
a(A008586(n)) < 0; a(A005843(n)) <= 0; a(A016825(n)) = 0; a(A042968(n)) >= 0; a(A005408(n)) > 0. (End)
a(n) = Sum_{k=0..n} A081362(k)*A015723(n-k). - Mircea Merca, Feb 26 2014
abs(a(n)) = A112329(n) = A094572(n) / 2. - Ray Chandler, Aug 23 2014
From Peter Bala, Jan 07 2015: (Start)
Logarithmic g.f.: log( Product_{n >= 1} (1 + x^n)^(1/n) ) = Sum_{n >= 1} a(n)*x^n/n.
a(n) = A001227(n) - A183063(n). By considering the logarithmic generating functions of these three sequences we obtain the identity
( Product_{n >= 0} (1 - x^(2*n+1))^(1/(2*n+1)) )^2 = Product_{n >= 1} ( (1 - x^n)/(1 + x^n) )^(1/n). (End)
Dirichlet g.f.: zeta(s)*eta(s) = zeta(s)^2*(1-2^(-s+1)). - Ralf Stephan, Mar 27 2015
a(2*n - 1) = A099774(n). - Michael Somos, Aug 12 2017
From Paul D. Hanna, Aug 10 2019: (Start)
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (x^(n+1) - x^k)^(n-k)  =  Sum_{n>=0} a(n)*x^(2*n).
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (x^(n+1) + x^k)^(n-k) * (-1)^k  =  Sum_{n>=0} a(n)*x^(2*n). (End)
a(n) = 2*A000005(2n) - 3*A000005(n). - Ridouane Oudra, Oct 15 2019
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A000005(k) = 2*log(2)-1. - Amiram Eldar, Mar 01 2023

New definition from Vladeta Jovovic, Jan 27 2002

A048767 If n = Product (p_j^k_j) then a(n) = Product ( prime(k_j)^pi(p_j) ) where pi is A000720.

Original entry on oeis.org

1, 2, 4, 3, 8, 8, 16, 5, 9, 16, 32, 12, 64, 32, 32, 7, 128, 18, 256, 24, 64, 64, 512, 20, 27, 128, 25, 48, 1024, 64, 2048, 11, 128, 256, 128, 27, 4096, 512, 256, 40, 8192, 128, 16384, 96, 72, 1024, 32768, 28, 81, 54, 512, 192, 65536, 50, 256, 80, 1024, 2048

Offset: 1

Naohiro Nomoto

If the prime power factors p^e of n are replaced by prime(e)^pi(p), then the prime terms q in the sequence pertain to 2^m with m > 1, since pi(2) = 1. - Michael De Vlieger, Apr 25 2017

Also the Heinz number of the integer partition obtained by applying the map described in A217605 (which interchanges the parts with their multiplicities) to the integer partition with Heinz number n, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The image of this map (which is the union of this sequence) is A130091. - Gus Wiseman, May 04 2019

			For n=6, 6 = (2^1)*(3^1), a(6) = ([first prime]^pi(2))*([first prime]^pi(3)) = (2^1)*(2^2) = 8.
From _Gus Wiseman_, May 04 2019: (Start)
For n = 1..20, the prime indices of n together with the prime indices of a(n) are the following:
   1: {} {}
   2: {1} {1}
   3: {2} {1,1}
   4: {1,1} {2}
   5: {3} {1,1,1}
   6: {1,2} {1,1,1}
   7: {4} {1,1,1,1}
   8: {1,1,1} {3}
   9: {2,2} {2,2}
  10: {1,3} {1,1,1,1}
  11: {5} {1,1,1,1,1}
  12: {1,1,2} {1,1,2}
  13: {6} {1,1,1,1,1,1}
  14: {1,4} {1,1,1,1,1}
  15: {2,3} {1,1,1,1,1}
  16: {1,1,1,1} {4}
  17: {7} {1,1,1,1,1,1,1}
  18: {1,2,2} {1,2,2}
  19: {8} {1,1,1,1,1,1,1,1}
  20: {1,1,3} {1,1,1,2}
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Stephan Wagner, The Number of Fixed Points of Wilf's Partition Involution, The Electronic Journal of Combinatorics, 20(4) (2013), #P13.

Cf. A008477, A048768, A048769, A064988.

Cf. A056239, A098859, A109298, A112798, A118914, A130091, A217605, A320348, A325326, A325337.

A048767 := proc(n)
    local a,p,e,f;
    a := 1 ;
    for f in ifactors(n)[2] do
        p := op(1,f) ;
        e := op(2,f) ;
        a := a*ithprime(e)^numtheory[pi](p) ;
    end do:
    a ;
end proc: # R. J. Mathar, Nov 08 2012

Table[{p, k} = Transpose@ FactorInteger[n]; Times @@ (Prime[k]^PrimePi[p]), {n, 58}] (* Ivan Neretin, Jun 02 2016 *)
Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; e >= 0 :> Prime[e]^PrimePi[p]] &, 65] (* Michael De Vlieger, Apr 25 2017 *)

a(1)=1 prepended by Alois P. Heinz, Jul 26 2015

A013954 a(n) = sigma_6(n), the sum of the 6th powers of the divisors of n.

Original entry on oeis.org

1, 65, 730, 4161, 15626, 47450, 117650, 266305, 532171, 1015690, 1771562, 3037530, 4826810, 7647250, 11406980, 17043521, 24137570, 34591115, 47045882, 65019786, 85884500, 115151530, 148035890, 194402650, 244156251, 313742650, 387952660, 489541650, 594823322, 741453700

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).
Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001
Inverse Mobius transform of A001014. - R. J. Mathar, Oct 13 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sigma(n).

Cf. A000203, A001014, A001157-A001160, A013665, A013955-A013972, A017665-A017712.

[DivisorSigma(6,n): n in [1..30]]; // Bruno Berselli, Apr 10 2013

A013954 := proc(n)
        numtheory[sigma][6](n) ;
end proc: # R. J. Mathar, Oct 13 2011

lst={};Do[AppendTo[lst,DivisorSigma[6,n]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
DivisorSigma[6,Range[30]] (* Harvey P. Dale, May 11 2025 *)

a(n)=sigma(n,6) \\ Charles R Greathouse IV, Apr 28 2011

[sigma(n,6)for n in range(1,24)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^6*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^5)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(6*e+6)-1)/(p^6-1).
Dirichlet g.f.: zeta(s)*zeta(s-6).
Sum_{k=1..n} a(k) = zeta(7) * n^7 / 7 + O(n^8). (End)

A055615 a(n) = n * mu(n), where mu is the Möbius function A008683.

Original entry on oeis.org

1, -2, -3, 0, -5, 6, -7, 0, 0, 10, -11, 0, -13, 14, 15, 0, -17, 0, -19, 0, 21, 22, -23, 0, 0, 26, 0, 0, -29, -30, -31, 0, 33, 34, 35, 0, -37, 38, 39, 0, -41, -42, -43, 0, 0, 46, -47, 0, 0, 0, 51, 0, -53, 0, 55, 0, 57, 58, -59, 0, -61, 62, 0, 0, 65, -66, -67, 0, 69, -70, -71, 0

Offset: 1

Michael Somos, Jun 04 2000

Dirichlet inverse of n (A000027).
Absolute values give n if n is squarefree, otherwise 0.
a(n) is multiplicative because both mu(n) and n are. - Mitch Harris, Jun 09 2005
a(n) is multiplicative with a(p^1) = -p, a(p^e) = 0 if e > 1. - David W. Wilson, Jun 12 2005
Negative of the Moebius number of the dihedral group of order 2n. - Eric M. Schmidt, Jul 28 2013

			G.f. = x - 2*x^2 - 3*x^3 - 5*x^5 + 6*x^6 - 7*x^7 + 10*x^10 - 11*x^11 - 13*x^13 + ...

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
Mats Granvik, Primes approximated by eigenvalues.
Mats Granvik, Mobius function times n approximated by eigenvalues.

Moebius transform of A023900.
Cf. A000027 (Dirichlet inverse), A061669 (sum with it).
Cf. A062004.
Cf. A013929 (positions of 0's), A068340 (partial sums), A261869 (first differences), A261890 (second differences).
Cf. A008683, A334657, A334659, A334660.

a055615 n = a008683 n * n  -- Reinhard Zumkeller, Sep 04 2015

[n*MoebiusMu(n): n in [1..80]]; // Vincenzo Librandi, Nov 19 2014

with(numtheory): A055615:=n->n*mobius(n): seq(A055615(n), n=1..100); # Wesley Ivan Hurt, Nov 18 2014

Table[n MoebiusMu[n], {n,80}] (* Harvey P. Dale, May 26 2011 *)

PARI
```
{a(n) = if( n<1, 0, n * moebius(n))};
```

{a(n) = if( n<1, 0, direuler(p=2, n, 1 - p*X)[n])};

from sympy import mobius
def A055615(n): return n*mobius(n) # Chai Wah Wu, Apr 01 2023

[n*moebius(n) for n in (1..100)] # G. C. Greubel, May 24 2022

a(n) = n * A008683(n).
Dirichlet g.f.: 1/zeta(s-1).
Multiplicative with a(p^e) = -p*0^(e-1), e>0 and p prime. - Reinhard Zumkeller, Jul 17 2003
Conjectures: lim b->1+ Sum n=1..inf a(n)*b^(-n) = -12 and lim b->1- Sum n=1..inf a(n)*b^n = -12 (+ indicates that b decreases to 1, - indicates it increases to 1), both considering that zeta(-1) = -1/12 and calculations (more generally mu(n)*n^s is Abel summable to zeta(-s)). - Gerald McGarvey, Sep 26 2004
Dirichlet generating function for the absolute value: zeta(s-1)/zeta(2s-2). - Franklin T. Adams-Watters, Sep 11 2005
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} k*A(x^k). - Ilya Gutkovskiy, May 11 2019
Sum_{k=1..n} abs(a(k)) ~ 3*n^2/Pi^2. - Amiram Eldar, Feb 02 2024

A225546 Tek's flip: Write n as the product of distinct factors of the form prime(i)^(2^(j-1)) with i and j integers, and replace each such factor with prime(j)^(2^(i-1)).

Original entry on oeis.org

1, 2, 4, 3, 16, 8, 256, 6, 9, 32, 65536, 12, 4294967296, 512, 64, 5, 18446744073709551616, 18, 340282366920938463463374607431768211456, 48, 1024, 131072, 115792089237316195423570985008687907853269984665640564039457584007913129639936, 24, 81, 8589934592, 36, 768

Offset: 1

Paul Tek, May 10 2013

This is a multiplicative self-inverse permutation of the integers.
A225547 gives the fixed points.
From Antti Karttunen and Peter Munn, Feb 02 2020: (Start)
This sequence operates on the Fermi-Dirac factors of a number. As arranged in array form, in A329050, this sequence reflects these factors about the main diagonal of the array, substituting A329050[j,i] for A329050[i,j], and this results in many relationships including significant homomorphisms.
This sequence provides a relationship between the operations of squaring and prime shift (A003961) because each successive column of the A329050 array is the square of the previous column, and each successive row is the prime shift of the previous row.
A329050 gives examples of how significant sets of numbers can be formed by choosing their factors in relation to rows and/or columns. This sequence therefore maps equivalent derived sets by exchanging rows and columns. Thus odd numbers are exchanged for squares, squarefree numbers for powers of 2 etc.
Alternative construction: For n > 1, form a vector v of length A299090(n), where each element v[i] for i=1..A299090(n) is a product of those distinct prime factors p(i) of n whose exponent e(i) has the bit (i-1) "on", or 1 (as an empty product) if no such exponents are present. a(n) is then Product_{i=1..A299090(n)} A000040(i)^A048675(v[i]). Note that because each element of vector v is squarefree, it means that each exponent A048675(v[i]) present in the product is a "submask" (not all necessarily proper) of the binary string A087207(n).
This permutation effects the following mappings:
A000035(a(n)) = A010052(n), A010052(a(n)) = A000035(n). [Odd numbers <-> Squares]
A008966(a(n)) = A209229(n), A209229(a(n)) = A008966(n). [Squarefree numbers <-> Powers of 2]
(End)
From Antti Karttunen, Jul 08 2020: (Start)
Moreover, we see also that this sequence maps between A016825 (Numbers of the form 4k+2) and A001105 (2*squares) as well as between A008586 (Multiples of 4) and A028983 (Numbers with even sum of the divisors).
(End)

			  7744  = prime(1)^2^(2-1)*prime(1)^2^(3-1)*prime(5)^2^(2-1).
a(7744) = prime(2)^2^(1-1)*prime(3)^2^(1-1)*prime(2)^2^(5-1) = 645700815.

Cf. A225547 (fixed points) and the subsequences listed there.
Transposes A329050, A329332.
An automorphism of positive integers under the binary operations A059895, A059896, A059897, A306697, A329329.
An automorphism of A059897 subgroups: A000379, A003159, A016754, A122132.
Permutes lists where membership is determined by number of Fermi-Dirac factors: A000028, A050376, A176525, A268388.
Also permutes: A036554, A059404, A066427, A066428, A072774, A331593.
Sequences f that satisfy f(a(n)) = f(n): A048675, A064179, A064547, A097248, A302777, A331592.
Pairs of sequences (f,g) that satisfy a(f(n)) = g(a(n)): (A000265,A008833), (A000290,A003961), (A005843,A334747), (A006519,A007913), (A008586,A334748).
Pairs of sequences (f,g) that satisfy a(f(n)) = g(n), possibly with offset change: (A000040,A001146), (A000079,A019565).
Pairs of sequences (f,g) that satisfy f(a(n)) = g(n), possibly with offset change: (A000035, A010052), (A008966, A209229), (A007814, A248663), (A061395, A299090), (A087207, A267116), (A225569, A227291).
Cf. A331287 [= gcd(a(n),n)].
Cf. A331288 [= min(a(n),n)], see also A331301.
Cf. A331309 [= A000005(a(n)), number of divisors].
Cf. A331590 [= a(a(n)*a(n))].
Cf. A331591 [= A001221(a(n)), number of distinct prime factors], see also A331593.
Cf. A331740 [= A001222(a(n)), number of prime factors with multiplicity].
Cf. A331733 [= A000203(a(n)), sum of divisors].
Cf. A331734 [= A033879(a(n)), deficiency].
Cf. A331735 [= A009194(a(n))].
Cf. A331736 [= A000265(a(n)) = a(A008833(n)), largest odd divisor].
Cf. A335914 [= A038040(a(n))].
A self-inverse isomorphism between pairs of A059897 subgroups: (A000079,A005117), (A000244,A062503), (A000290\{0},A005408), (A000302,A056911), (A000351,A113849 U {1}), (A000400,A062838), (A001651,A252895), (A003586,A046100), (A007310,A000583), (A011557,A113850 U {1}), (A028982,A042968), (A053165,A065331), (A262675,A268390).
A bijection between pairs of sets: (A001248,A011764), (A007283,A133466), (A016825, A001105), (A008586, A028983).
Cf. also A336321, A336322 (compositions with another involution, A122111).

Array[If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &, 28] (* Michael De Vlieger, Jan 21 2020 *)

A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
a(n) = {my(f=factor(n)); for (i=1, #f~, my(p=f[i,1]); f[i,1] = A019565(f[i,2]); f[i,2] = 2^(primepi(p)-1);); factorback(f);} \\ Michel Marcus, Nov 29 2019

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A225546(n) = if(1==n,1,my(f=factor(n),u=#binary(vecmax(f[, 2])),prods=vector(u,x,1),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),prods[i] *= f[k,1])); m<<=1); prod(i=1,u,prime(i)^A048675(prods[i]))); \\ Antti Karttunen, Feb 02 2020

from math import prod
from sympy import prime, primepi, factorint
def A225546(n): return prod(prod(prime(i) for i, v in enumerate(bin(e)[:1:-1],1) if v == '1')**(1<Chai Wah Wu, Mar 17 2023

Multiplicative, with a(prime(i)^j) = A019565(j)^A000079(i-1).
a(prime(i)) = 2^(2^(i-1)).
From Antti Karttunen and Peter Munn, Feb 06 2020: (Start)
a(A329050(n,k)) = A329050(k,n).
a(A329332(n,k)) = A329332(k,n).
Equivalently, a(A019565(n)^k) = A019565(k)^n. If n = 1, this gives a(2^k) = A019565(k).
a(A059897(n,k)) = A059897(a(n), a(k)).
The previous formula implies a(n*k) = a(n) * a(k) if A059895(n,k) = 1.
a(A000040(n)) = A001146(n-1); a(A001146(n)) = A000040(n+1).
a(A000290(a(n))) = A003961(n); a(A003961(a(n))) = A000290(n) = n^2.
a(A000265(a(n))) = A008833(n); a(A008833(a(n))) = A000265(n).
a(A006519(a(n))) = A007913(n); a(A007913(a(n))) = A006519(n).
A007814(a(n)) = A248663(n); A248663(a(n)) = A007814(n).
A048675(a(n)) = A048675(n) and A048675(a(2^k * n)) = A048675(2^k * a(n)) = k + A048675(a(n)).
(End)
From Antti Karttunen and Peter Munn, Jul 08 2020: (Start)
For all n >= 1, a(2n) = A334747(a(n)).
In particular, for n = A003159(m), m >= 1, a(2n) = 2*a(n). [Note that A003159 includes all odd numbers]
(End)

Name edited by Peter Munn, Feb 14 2020

"Tek's flip" prepended to the name by Antti Karttunen, Jul 08 2020

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

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, 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

A003973 Inverse Möbius transform of A003961; a(n) = sigma(A003961(n)), where A003961 shifts the prime factorization of n one step towards the larger primes.

Original entry on oeis.org

1, 4, 6, 13, 8, 24, 12, 40, 31, 32, 14, 78, 18, 48, 48, 121, 20, 124, 24, 104, 72, 56, 30, 240, 57, 72, 156, 156, 32, 192, 38, 364, 84, 80, 96, 403, 42, 96, 108, 320, 44, 288, 48, 182, 248, 120, 54, 726, 133, 228, 120, 234, 60, 624, 112, 480, 144, 128, 62, 624, 68

Offset: 1

Marc LeBrun

Sum of the divisors of the prime shifted n, or equally, sum of the prime shifted divisors of n. - Antti Karttunen, Aug 17 2020

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization.
Index entries for sequences related to sigma(n).

Cf. A000203, A000290 (positions of odd terms), A003961, A007814, A048673, A108228, A151800, A295664, A336840.
Permutation of A008438.
Used in the definitions of the following sequences: A326042, A336838, A336841, A336844, A336846, A336847, A336848, A336849, A336850, A336851, A336852, A336856, A336931, A336932.
Cf. also A003972.

b[1] = 1; b[p_?PrimeQ] := b[p] = Prime[ PrimePi[p] + 1]; b[n_] := b[n] = Times @@ (b[First[#]]^Last[#] &) /@ FactorInteger[n]; a[n_] := Sum[ b[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]  (* Jean-François Alcover, Jul 18 2013 *)

aPrime(p,e)=my(q=nextprime(p+1));(q^(e+1)-1)/(q-1)
a(n)=my(f=factor(n));prod(i=1,#f~,aPrime(f[i,1],f[i,2])) \\ Charles R Greathouse IV, Jul 18 2013

A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); }; \\ Antti Karttunen, Aug 06 2020

from math import prod
from sympy import factorint, nextprime
def A003973(n): return prod(((q:=nextprime(p))**(e+1)-1)//(q-1) for p,e in factorint(n).items()) # Chai Wah Wu, Jul 05 2022

Multiplicative with a(p^e) = (q^(e+1)-1)/(q-1) where q = nextPrime(p). - David W. Wilson, Sep 01 2001
From Antti Karttunen, Aug 06-12 2020: (Start)
a(n) = Sum_{d|n} A003961(d) = Sum_{d|A003961(n)} d.
a(n) = A000203(A003961(n)) = A000593(A003961(n)).
a(n) = 2*A336840(n) - A000005(n) = 2*Sum_{d|n} (A048673(d) - (1/2)).
a(n) = A008438(A108228(n)) = A008438(A048673(n)-1).
a(n) = A336838(n) * A336856(n).
a(n) is odd if and only if n is a square.
(End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} p^3/((p+1)*(p^2-nextprime(p))) = 3.39513795..., where nextprime is A151800. - Amiram Eldar, Dec 08 2022, May 30 2025

More terms from David W. Wilson, Aug 29 2001

Secondary name added by Antti Karttunen, Aug 06 2020

A022998 If n is odd then n, otherwise 2n.

Original entry on oeis.org

0, 1, 4, 3, 8, 5, 12, 7, 16, 9, 20, 11, 24, 13, 28, 15, 32, 17, 36, 19, 40, 21, 44, 23, 48, 25, 52, 27, 56, 29, 60, 31, 64, 33, 68, 35, 72, 37, 76, 39, 80, 41, 84, 43, 88, 45, 92, 47, 96, 49, 100, 51, 104, 53, 108, 55, 112, 57, 116, 59, 120, 61, 124, 63, 128, 65, 132, 67

Offset: 0

N. J. A. Sloane

Also for n > 0: numerator of Sum_{i=1..n} 2/(i*(i+1)), denominator=A026741. - Reinhard Zumkeller, Jul 25 2002
For n > 2: a(n) = gcd(A143051((n-1)^2), A143051(1+(n-1)^2)) = A050873(A000290(n-1), A002522(n-1)). - Reinhard Zumkeller, Jul 20 2008
Partial sums give the generalized octagonal numbers A001082. - Omar E. Pol, Sep 10 2011
Multiples of 4 and odd numbers interleaved. - Omar E. Pol, Sep 25 2011
The Pisano period lengths modulo m appear to be A066043(m). - R. J. Mathar, Oct 08 2011
The partial sums a(n)/A026741(n+1) given by R. Zumkeller in a comment above are 2*n/(n+1) (telescopic sum), and thus converge to 2. - Wolfdieter Lang, Apr 09 2013
a(n) = numerator(H(n,1)), where H(n,1) = 2*n/(n+1) is the harmonic mean of 1 and n. a(n) = 2*n/gcd(2n, n+1) = 2*n/gcd(n+1,2). a(n) = A227041(n,1), n>=1. - Wolfdieter Lang, Jul 04 2013
a(n) = numerator of the mean (2n/(n+1), after reduction), of the compositions of n; denominator is given by A001792(n-1). - Clark Kimberling, Mar 11 2014
A strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n,m)) for all natural numbers n and m. The sequence of convergents of the 2-periodic continued fraction [0; 1, -4, 1, -4, ...] = 1/(1 - 1/(4 - 1/(1 - 1/(4 - ...)))) begins [0/1, 1/1, 4/3, 3/2, 8/5, 5/3, 12/7, ...]. The present sequence is the sequence of numerators. The sequence of denominators of the continued fraction convergents [1, 1, 3, 2, 5, 3, 7, ...] is A026741, also a strong divisibility sequence. Cf. A203976. - Peter Bala, May 19 2014
a(n) is also the length of the n-th line segment of a rectangular spiral on the infinite square grid. The vertices of the spiral are the generalized octagonal numbers. - Omar E. Pol, Jul 27 2018
a(n) is the number of petals of the Rhodonea curve r = a*cos(n*theta) or r = a*sin(n*theta). - Matt Westwood, Nov 19 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Rose
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A026741, A059026, A203976, A345082.
Column 4 of A195151. - Omar E. Pol, Sep 25 2011
Cf. A000034, A001082 (partial sums).
Cf. A227041 (first column). - Wolfdieter Lang, Jul 04 2013
Row 2 of A349593. A385555, A385556, A385557, A385558, A385559, and A385560 are respectively rows 3, 4, 5-6, 7, 8, and 9-10.

a022998 n = a000034 (n + 1) * n
a022998_list = zipWith (*) [0..] $ tail a000034_list
-- Reinhard Zumkeller, Mar 31 2012

[((-1)^n+3)*n/2: n in [0..70]]; // Vincenzo Librandi, Sep 17 2011

A022998 := proc(n) if type(n,'odd') then n ; else 2*n; end if; end proc: # R. J. Mathar, Mar 10 2011

Table[n (3 + (-1)^n)/2, {n, 0, 100}] (* Wesley Ivan Hurt, Dec 13 2013 *)
Table[If[OddQ[n],n,2n],{n,0,150}] (* or *) Riffle[ 2*Range[ 0,150,2], Range[ 1,150,2]] (* Harvey P. Dale, Feb 06 2017 *)

PARI
```
a(n)=if(n%2,n,2*n)
```

def A022998(n): return n if n&1 else n<<1 # Chai Wah Wu, Mar 05 2024

[n*(1+((n+1)%2)) for n in (0..80)] # G. C. Greubel, Jul 31 2022

Denominator of (n+1)*(n-1)*(2*n+1)/(2*n) (for n > 0).
a(n+1) = lcm(n, n+2)/n + lcm(n, n+2)/(n+2) for all n >= 1. - Asher Auel, Dec 15 2000
Multiplicative with a(2^e) = 2^(e+1), a(p^e) = p^e, p > 2.
G.f. x*(1 + 4*x + x^2)/(1-x^2)^2. - Ralf Stephan, Jun 10 2003
a(n) = 3*n/2 + n*(-1)^n/2 = n*(3 + (-1)^n)/2. - Paul Barry, Sep 04 2003
a(n) = A059029(n-1) + 1 = A043547(n+2) - 2.
a(n)*a(n+3) = -4 + a(n+1)*a(n+2).
a(n) = n*(((n+1) mod 2) + 1) = n^2 + 2*n - 2*n*floor((n+1)/2). - William A. Tedeschi, Feb 29 2008
a(n) = denominator((n+1)/(2*n)) for n >= 1; A026741(n+1) = numerator((n+1)/(2*n)) for n >= 1. - Johannes W. Meijer, Jun 18 2009
a(n) = 2*a(n-2) - a(n-4).
Dirichlet g.f. zeta(s-1)*(1+2^(1-s)). - R. J. Mathar, Mar 10 2011
a(n) = n * (2 - n mod 2) = n * A000034(n+1). - Reinhard Zumkeller, Mar 31 2012
a(n) = floor(2*n/(1 + (n mod 2))). - Wesley Ivan Hurt, Dec 13 2013
From Ilya Gutkovskiy, Mar 16 2017: (Start)
E.g.f.: x*(2*sinh(x) + cosh(x)).
It appears that a(n) is the period of the sequence k*(k + 1)/2 mod n. (End) [This is correct; see A349593. - Jianing Song, Jul 03 2025]
a(n) = Sum_{d | n} A345082(d). - Peter Bala, Jan 13 2024

More terms from Michael Somos, Aug 07 2000

cp's OEIS Frontend

A003958 If n = Product p(k)^e(k) then a(n) = Product (p(k)-1)^e(k).

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A057521 Powerful (1) part of n: if n = Product_i (pi^ei) then a(n) = Product_{i : ei > 1} (pi^ei); if n=b*c^2*d^3 then a(n)=c^2*d^3 when b is minimized.

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A048272 Number of odd divisors of n minus number of even divisors of n.

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A048767 If n = Product (p_j^k_j) then a(n) = Product ( prime(k_j)^pi(p_j) ) where pi is A000720.

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A013954 a(n) = sigma_6(n), the sum of the 6th powers of the divisors of n.

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A055615 a(n) = n * mu(n), where mu is the Möbius function A008683.

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A057521 Powerful (1) part of n: if n = Product_i (pi^ei) then a(n) = Product_{i : ei > 1} (pi^ei); if n=bc^2d^3 then a(n)=c^2*d^3 when b is minimized.