A048671 -id:A048671 - cp's OEIS frontend

A014963 Exponential of Mangoldt function M(n): a(n) = 1 unless n is a prime or prime power, in which case a(n) = that prime.

1, 2, 3, 2, 5, 1, 7, 2, 3, 1, 11, 1, 13, 1, 1, 2, 17, 1, 19, 1, 1, 1, 23, 1, 5, 1, 3, 1, 29, 1, 31, 2, 1, 1, 1, 1, 37, 1, 1, 1, 41, 1, 43, 1, 1, 1, 47, 1, 7, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 1, 2, 1, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79, 1, 3, 1, 83, 1, 1, 1, 1, 1, 89, 1, 1, 1, 1, 1, 1

Offset: 1

Marc LeBrun

There are arbitrarily long runs of ones (Sierpiński). - Franz Vrabec, Sep 26 2005
a(n) is the smallest positive integer such that n divides Product_{k=1..n} a(k), for all positive integers n. - Leroy Quet, May 01 2007
For n>1, resultant of the n-th cyclotomic polynomial with the 1st cyclotomic polynomial x-1. - Ralf Stephan, Aug 14 2013
A368749(n) is the smallest prime p such that the interval [a(p), a(q)] contains n 1's; q = nextprime(p), n >= 0. - David James Sycamore, Mar 21 2024

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Section 17.7.
I. Vardi, Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 146-147, 152-153 and 249, 1991.

T. D. Noe, Table of n, a(n) for n = 1..10000
Peter Luschny and Stefan Wehmeier, The lcm(1,2,...,n) as a product of sine values sampled over the points in Farey sequences, arXiv:0909.1838 [math.CA], 2009.
Greg Martin, A product of Gamma function values at fractions with the same denominator, arXiv:0907.4384 [math.CA], 2009.
Carl McTague, On the Greatest Common Divisor of C(q*n,n), C(q*n,2*n), ...C(q*n,q*n-q), arXiv:1510.06696 [math.CO], 2015.
A. Nowicki, Strong divisibility and LCM-sequences, arXiv:1310.2416 [math.NT], 2013.
A. Nowicki, Strong divisibility and LCM-sequences, Am. Math. Mnthly 122 (2015), 958-966.
W. Sierpiński, On the numbers [1,2,...n], (Polish) Wiadom. Mat. (2) 9 1966 9-10.
Eric Weisstein's World of Mathematics, Mangoldt Function.
Eric Weisstein's World of Mathematics, Sylvester Cyclotomic Number.
Index entries for sequences related to lcm's

Apart from initial 1, same as A020500. With ones replaced by zeros, equal to A120007.
Cf. A003418, A007947, A008683, A008472, A008578, A048671 (= n/a(n)), A072107 (partial sums), A081386, A081387, A099636, A100994, A100995, A140255 (inverse Mobius transform), A140254 (Mobius transform), A297108, A297109, A340675, A000027, A348846, A368749.
First column of A140256. Row sums of triangle A140581.
Cf. also A140579, A140580 (= n*a(n)).

a014963 1 = 1
a014963 n | until ((> 0) . (`mod` spf)) (`div` spf) n == 1 = spf
          | otherwise = 1
          where spf = a020639 n
-- Reinhard Zumkeller, Sep 09 2011

a := n -> if n < 2 then 1 else numtheory[factorset](n); if 1 < nops(%) then 1 else op(%) fi fi; # Peter Luschny, Jun 23 2009
A014963 := n -> n/ilcm(op(numtheory[divisors](n) minus {1,n}));
seq(A014963(i), i=1..69); # Peter Luschny, Mar 23 2011
# The following is Nowicki's LCM-Transform - N. J. A. Sloane, Jan 09 2024
LCMXFM:=proc(a)  local p,q,b,i,k,n:
if whattype(a) <> list then RETURN([]); fi:
n:=nops(a):
b:=[a[1]]: p:=[a[1]];
for i from 2 to n do q:=[op(p),a[i]]; k := lcm(op(q))/lcm(op(p));
b:=[op(b),k]; p:=q;; od:
RETURN(b); end:
# Alternative, to be called by 'seq' as shown, not for a single n.
a := proc(n) option remember; local i; global f; f := ifelse(n=1, 1, f*n);
iquo(f, mul(a(i)^iquo(n, i), i=1..n-1)) end: seq(a(n), n=1..95); # Peter Luschny, Apr 05 2025

a[n_?PrimeQ] := n; a[n_/;Length[FactorInteger[n]] == 1] := FactorInteger[n][[1]][[1]]; a[n_] := 1; Table[a[n], {n, 95}] (* Alonso del Arte, Jan 16 2011 *)
a[n_] := Exp[ MangoldtLambda[n]]; Table[a[n], {n, 95}] (* Jean-François Alcover, Jul 29 2013 *)
Ratios[LCM @@ # & /@ Table[Range[n], {n, 100}]] (* Horst H. Manninger, Mar 08 2024 *)
Table[Which[PrimeQ[n],n,PrimePowerQ[n],Surd[n,FactorInteger[n][[-1,2]]],True,1],{n,100}] (* Harvey P. Dale, Mar 01 2025 *)

A014963(n)=
{
    local(r);
    if( isprime(n), return(n));
    if( ispower(n,,&r) && isprime(r), return(r) );
    return(1);
}  \\ Joerg Arndt, Jan 16 2011

a(n)=ispower(n,,&n);if(isprime(n),n,1) \\ Charles R Greathouse IV, Jun 10 2011

from sympy import factorint
def A014963(n):
    y = factorint(n)
    return list(y.keys())[0] if len(y) == 1 else 1
print([A014963(n) for n in range(1, 71)]) # Chai Wah Wu, Sep 04 2014

def A014963(n) : return simplify(exp(add(moebius(d)*log(n/d) for d in divisors(n))))
[A014963(n) for n in (1..50)]  # Peter Luschny, Feb 02 2012

def a(n):
    if n == 1: return 1
    return prod(1 - E(n)**k for k in ZZ(n).coprime_integers(n+1))
[a(n) for n in range(1, 14)] # F. Chapoton, Mar 17 2020

a(n) = A003418(n) / A003418(n-1) = lcm {1..n} / lcm {1..n-1}. [This is equivalent to saying that this sequence is the LCM-transform (as defined by Nowicki, 2013) of the positive integers. - David James Sycamore, Jan 09 2024.]
a(n) = 1/Product_{d|n} d^mu(d) = Product_{d|n} (n/d)^mu(d). - Vladeta Jovovic, Jan 24 2002
a(n) = gcd( C(n+1,1), C(n+2,2), ..., C(2n,n) ) where C(n,k) = binomial(n,k). - Benoit Cloitre, Jan 31 2003
a(n) = gcd(C(n,1), C(n+1,2), C(n+2,3), ...., C(2n-2,n-1)), where C(n,k) = binomial(n,k). - Benoit Cloitre, Jan 31 2003; corrected by Ant King, Dec 27 2005
Note: a(n) != gcd(A008472(n), A007947(n)) = A099636(n), GCD of rad(n) and sopf(n) (this fails for the first time at n=30), since a(30) = 1 but gcd(rad(30), sopf(30)) = gcd(30,10) = 10.
a(n)^A100995(n) = A100994(n). - N. J. A. Sloane, Feb 20 2005
a(n) = Product_{k=1..n-1, if(gcd(n, k)=1, 1-exp(2*Pi*i*k/n), 1)}, i=sqrt(-1); a(n) = n/A048671(n). - Paul Barry, Apr 15 2005
Sum_{n>=1} (log(a(n))-1)/n = -2*A001620 [Bateman Manuscript Project Vol III, ed. by Erdelyi et al.]. - R. J. Mathar, Mar 09 2008
n*a(n) = A140580(n) = n^2/A048671(n) = A140579 * [1,2,3,...]. - Gary W. Adamson, May 17 2008
a(n) = (2*Pi)^phi(n) / Product_{gcd(n,k)=1} Gamma(k/n)^2 (for n > 1). - Peter Luschny, Aug 08 2009
a(n) = A166140(n) / A166142(n). - Mats Granvik, Oct 08 2009
a(n) = GCD of rows in A167990. - Mats Granvik, Nov 16 2009
a(n) = A010055(n)*(A007947(n) - 1) + 1. - Reinhard Zumkeller, Mar 26 2010
a(n) = 1 + (A007947(n)-1) * floor(1/A001221(n)), for n>1. - Enrique Pérez Herrero, Jun 01 2011
a(n) = Product_{k=1..n-1} if(gcd(k,n)=1, 2*sin(Pi*k/n), 1). - Peter Luschny, Jun 09 2011
a(n) = exp(Sum_{k>=1} A191898(n,k)/k) for n>1 (conjecture). - Mats Granvik, Jun 19 2011
Dirichlet g.f.: Sum_{n>0} e^Lambda(n)/n^s = Zeta(s) + Sum_{p prime} Sum_{k>0} (p-1)/p^(k*s) = Zeta(s) - ppzeta(s) + Sum(p prime, p/(p^s-1)); for a ppzeta definition see A010055. - Enrique Pérez Herrero, Jan 19 2013
a(n) = exp(lim_{s->1} zeta(s)*Sum_{d|n} moebius(d)/d^(s-1)) for n>1. - Mats Granvik, Jul 31 2013
a(n) = gcd_{k=1..n-1} binomial(n,k) for n > 1, see A014410. - Michel Marcus, Dec 08 2015 [Corrected by Jinyuan Wang, Mar 20 2020]
a(n) = 1 + Sum_{k=2..n} (k-1)*A010051(k)*(floor(k^n/n) - floor((k^n - 1)/n)). - Anthony Browne, Jun 16 2016
The Dirichlet series for log(a(n)) = Lambda(n) is given by the logarithmic derivative of the zeta function -zeta'(s)/zeta(s). - Mats Granvik, Oct 30 2016
a(n) = A008578(1+A297109(n)), For all n >= 1, Product_{d|n} a(d) = n. - Antti Karttunen, Feb 01 2021
Product_{k=1..floor(n/2)} Product_{j=1..floor(n/k)} a(j) = n!. - Ammar Khatab, Jan 28 2025

Additional reference from Eric W. Weisstein, Jun 29 2008

A010055 1 if n is a prime power p^k (k >= 0), otherwise 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0

Offset: 1

N. J. A. Sloane

nonn

Characteristic function of unit or prime powers p^k (k >= 1). Characteristic function of prime powers p^k (k >= 0). - Daniel Forgues, Mar 03 2009

See A065515 for partial sums. - Reinhard Zumkeller, Nov 22 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for characteristic functions

Cf. A069513 (1 if n is a prime power p^k (k >= 1), else 0.)
Cf. A268340.
Cf. A100995.
Cf. A001221, A000961, A024619, A065515,

a010055 n = if a001221 n <= 1 then 1 else 0
-- Reinhard Zumkeller, Nov 28 2015, Mar 19 2013, Nov 17 2011

A010055 := proc(n)
    if n =1 then
        1;
    else
        if nops(ifactors(n)[2]) = 1 then
            1;
        else
            0 ;
        end if;
    end if;
end proc: # R. J. Mathar, May 25 2017

A010055[n_]:=Boole[PrimeNu[n]<=1]; A010055/@Range[20] (* Enrique Pérez Herrero, May 30 2011 *)
{1}~Join~Table[Boole@ PrimePowerQ@ n, {n, 2, 105}] (* Michael De Vlieger, Feb 02 2016 *)

PARI
```
for(n=1,120,print1(omega(n)<=1,","))
```

from sympy import primefactors
def A010055(n): return int(len(primefactors(n)) <= 1) # Chai Wah Wu, Mar 31 2023

Dirichlet generating function: 1 + ppzeta(s). Here ppzeta(s) = Sum_{p prime} Sum_{k>=1} 1/(p^k)^s. Note that ppzeta(s) = Sum_{p prime} 1/(p^s-1) = Sum_{k>=1} primezeta(k*s). - Franklin T. Adams-Watters, Sep 11 2005
a(n) = 0^(A119288(n)-1). - Reinhard Zumkeller, May 13 2006
a(A000961(n)) = 1; a(A024619(n)) = 0. - Reinhard Zumkeller, Nov 17 2011
a(n) = if A001221(n) <= 1 then 1, otherwise 0. - Reinhard Zumkeller, Nov 28 2015
If n >= 2, a(n) = A069513(n). - Jeppe Stig Nielsen, Feb 02 2016
Conjecture: a(n) = (n - A048671(n))/A000010(n) for all n > 1. - Velin Yanev, Mar 10 2021 [The conjecture is true. - Andrey Zabolotskiy, Mar 11 2021]

More terms from Charles R Greathouse IV, Mar 12 2008
Edited by Daniel Forgues, Mar 02 2009
Comment re Galois fields moved to A069513 by Franklin T. Adams-Watters, Nov 02 2009

A048050 Chowla's function: sum of divisors of n except for 1 and n.

Original entry on oeis.org

0, 0, 0, 2, 0, 5, 0, 6, 3, 7, 0, 15, 0, 9, 8, 14, 0, 20, 0, 21, 10, 13, 0, 35, 5, 15, 12, 27, 0, 41, 0, 30, 14, 19, 12, 54, 0, 21, 16, 49, 0, 53, 0, 39, 32, 25, 0, 75, 7, 42, 20, 45, 0, 65, 16, 63, 22, 31, 0, 107, 0, 33, 40, 62, 18, 77, 0, 57, 26, 73, 0, 122, 0, 39, 48, 63, 18, 89

Offset: 1

N. J. A. Sloane

a(n) = 0 if and only if n is a noncomposite number (cf. A008578). - Omar E. Pol, Jul 31 2012
If n is semiprime, a(n) = A008472(n). - Wesley Ivan Hurt, Aug 22 2013
If n = p*q where p and q are distinct primes then a(n) = p+q.
If k,m > 1 are coprime, then a(k*m) = a(k)*a(m) + (m+1)*a(k) + (k+1)*a(m) + k + m. - Robert Israel, Apr 28 2015
a(n) is also the total number of parts in the partitions of n into equal parts that contain neither 1 nor n as a part (see example). More generally, a(n) is the total number of parts congruent to 0 mod k in the partitions of k*n into equal parts that contain neither k nor k*n as a part. - Omar E. Pol, Nov 24 2019
Named after the Indian-American mathematician Sarvadaman D. S. Chowla (1907-1995). - Amiram Eldar, Mar 09 2024

			For n = 20 the divisors of 20 are 1,2,4,5,10,20, so a(20) = 2+4+5+10 = 21.
On the other hand, the partitions of 20 into equal parts that contain neither 1 nor 20 as a part are [10,10], [5,5,5,5], [4,4,4,4,4], [2,2,2,2,2,2,2,2,2,2]. There are 21 parts, so a(20) = 21. - _Omar E. Pol_, Nov 24 2019

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 92.

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Lal and A. Forbes, A note on Chowla's function, Math. Comp., 25 (1971), 923-925.
Abdur Rahman Nasir, On a certain arithmetic function, Bull. Calcutta Math. Soc., Vol. 38 (1946), p. 140.
Omar E. Pol, Illustration of initial terms obtained geometrically.

Cf. A000203, A001065, A000593, A002954, A048995, A007956, A048671, A182936.

Cf. A057533, A005276, A027751, A237593, A245092, A244049 (partial sums).

a048050 1 = 0
a048050 n = (subtract 1) $ sum $ a027751_row n
-- Reinhard Zumkeller, Feb 09 2013

A048050:=func< n | n eq 1 or IsPrime(n) select 0 else &+[ a: a in Divisors(n) | a ne 1 and a ne n ] >; [ A048050(n): n in [1..100] ]; // Klaus Brockhaus, Mar 04 2011

A048050 := proc(n) if n > 1 then numtheory[sigma](n)-1-n ; else 0; end if; end proc:

f[n_]:=Plus@@Divisors[n]-n-1; Table[f[n],{n,100}] (*Vladimir Joseph Stephan Orlovsky, Sep 13 2009*)
Join[{0},DivisorSigma[1,#]-#-1&/@Range[2,80]] (* Harvey P. Dale, Feb 25 2015 *)

a(n)=if(n>1,sigma(n)-n-1,0) \\ Charles R Greathouse IV, Nov 20 2012

from sympy import divisors
def a(n): return sum(divisors(n)[1:-1]) # Indranil Ghosh, Apr 26 2017

from sympy import divisor_sigma
def A048050(n): return 0 if n == 1 else divisor_sigma(n)-n-1 # Chai Wah Wu, Apr 18 2021

a(n) = A000203(n) - A065475(n).
a(n) = A001065(n) - 1, n > 1.
For n > 1: a(n) = Sum_{k=2..A000005(n)-1} A027750(n,k). - Reinhard Zumkeller, Feb 09 2013
a(n) = A000203(n) - n - 1, n > 1. - Wesley Ivan Hurt, Aug 22 2013
G.f.: Sum_{k>=2} k*x^(2*k)/(1 - x^k). - Ilya Gutkovskiy, Jan 22 2017

A007956 Product of the proper divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 1, 8, 3, 10, 1, 144, 1, 14, 15, 64, 1, 324, 1, 400, 21, 22, 1, 13824, 5, 26, 27, 784, 1, 27000, 1, 1024, 33, 34, 35, 279936, 1, 38, 39, 64000, 1, 74088, 1, 1936, 2025, 46, 1, 5308416, 7, 2500, 51, 2704, 1, 157464, 55, 175616, 57, 58, 1, 777600000, 1, 62, 3969, 32768, 65

Offset: 1

R. Muller

From Bernard Schott, Feb 01 2019: (Start)
a(n) = 1 iff n = 1 or n is prime.
a(n) = n when n > 1 iff n has exactly four divisors, equally, iff n is either the cube of a prime or the product of two different primes, so iff n belongs to A030513 (very nice proof in Sierpiński).
a(p^3) = 1 * p * p^2 = p^3; a(p*q) = 1 * p * q = p*q.
As a(1) = 1, {1} Union A030513 = A007422, fixed points of this sequence. (End)

			a(18) = 1 * 2 * 3 * 6 * 9 = 324. - _Bernard Schott_, Jan 31 2019

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, p. 57.
Wacław Sierpiński, Elementary Theory of Numbers, Ex. 2 p. 174, Warsaw, 1964.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Zhu Weiyi, On the divisor product sequences, Smarandache Notions J., Vol. 14 (2004), pp. 144-146.

Cf. A000005, A000720, A007955, A048671, A182936, A027751, A066729.
Cf. A007422 (fixed points). A030513 (subsequence).
Cf. A001065 (sums of proper divisors).

a007956 = product . a027751_row
-- Reinhard Zumkeller, Feb 04 2013, Nov 02 2011

A007956 := n -> mul(i,i=op(numtheory[divisors](n) minus {1,n}));
seq(A007956(i), i=1..79); # Peter Luschny, Mar 22 2011

Table[Times@@Most[Divisors[n]], {n, 65}] (* Alonso del Arte, Apr 18 2011 *)
a[n_] := n^(DivisorSigma[0, n]/2 - 1); Table[a[n], {n, 1, 65}] (* Jean-François Alcover, Oct 07 2013 *)

A007956(n) = local(a);a=1;fordiv(n,d,a=a*d);a/n \\ Michael B. Porter, Dec 01 2009

a(n)=my(k); if(issquare(n, &k), k^(numdiv(n)-2), n^(numdiv(n)/2-1)) \\ Charles R Greathouse IV, Oct 15 2015

from math import isqrt
from sympy import divisor_count
def A007956(n): return isqrt(n)**(d-2) if (d:=divisor_count(n))&1 else n**((d>>1)-1) # Chai Wah Wu, Jun 18 2023

a(n) = A007955(n)/n = n^(A000005(n)/2-1) = sqrt(n^(number of factors of n other than 1 and n)).
a(n) = Product_{k=1..A000005(n)-1} A027751(n,k). - Reinhard Zumkeller, Feb 04 2013
a(n) = A240694(n, A000005(n)-1) for n > 1. - Reinhard Zumkeller, Apr 10 2014
Sum_{k=1..n} 1/a(k) ~ pi(n) + log(log(n))^2 + c_1*log(log(n)) + c_2 + O(log(log(n))/log(n)), where pi(n) = A000720(n) and c_1 and c_2 are constants (Weiyi, 2004; Sandor and Crstici, 2004). - Amiram Eldar, Oct 29 2022

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

A025527 a(n) = n!/lcm{1,2,...,n} = (n-1)!/lcm{C(n-1,0), C(n-1,1), ..., C(n-1,n-1)}.

Original entry on oeis.org

1, 1, 1, 2, 2, 12, 12, 48, 144, 1440, 1440, 17280, 17280, 241920, 3628800, 29030400, 29030400, 522547200, 522547200, 10450944000, 219469824000, 4828336128000, 4828336128000, 115880067072000, 579400335360000, 15064408719360000

Offset: 1

Clark Kimberling, Dec 11 1999

nonn

a(n) = a(n-1) iff n is prime. Thus a(1)=a(2)=a(3)=1 is the only triple in this sequence. - Franz Vrabec, Sep 10 2005
a(k) = a(k+1) for k in A006093. - Lekraj Beedassy, Aug 03 2006
Partial products of A048671. - Peter Luschny, Sep 09 2009

			a(5) = 2 as 5!/lcm(1..5) = 120/60 = 2.

Alois P. Heinz, Table of n, a(n) for n = 1..500
Liam Solus, Simplices for Numeral Systems, arXiv:1706.00480 [math.CO], 2017. Mentions this sequence.
Index entries for sequences related to lcm's

Cf. A000142, A002541, A006093, A003418, A048671, A025529, A205957.

A140579 Triangle read by rows, A014963(n) * 0^(n-k); 1<=k<=n.

Original entry on oeis.org

1, 0, 2, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13

Offset: 1

Gary W. Adamson and Mats Granvik, May 17 2008

A140579 * [1, 2, 3,...] = A140580.
(A140579)^(-1) * [1, 2, 3,...] = A048671: (1, 1, 1, 2, 1, 6, 1, 4, 3, 10,...).
A008683 = A140579^(-1) * A140664. - Gary W. Adamson, May 20 2008

			First few rows of the triangle are:
1;
0, 2;
0, 0, 3;
0, 0, 0, 2;
0, 0, 0, 0, 5;
0, 0, 0, 0, 0, 1;
0, 0, 0, 0, 0, 0, 7;
...

G. C. Greubel, Rows n=1..100 of triangle, flattened

Cf. A014963, A140580, A048671.

Cf. A008683, A140664.

Table[If[k != n ,0,Exp[MangoldtLambda[n]]], {n,1,12}, {k,1,n}]//Flatten (* G. C. Greubel, Feb 16 2019 *)

{T(n,k) = if(n==1, 1, gcd(vector(n-1, k, binomial(n, k)))*0^(n-k))};
for(n=1,12, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Feb 16 2019

def T(n,k): return simplify(exp(add(moebius(d)*log(n/d) for d in divisors(n))))*0^(n-k)
[[T(n,k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Feb 16 2019

Triangle read by rows, A014963(n) * 0^(n-k); 1<=k<=n.

Infinite lower triangular matrix with A014963 (1, 2, 3, 2, 5, 1, 7, 2, 3, 1, 11,...) in the main diagonal and the rest zeros.

A075830 Let u(1) = x and u(n+1) = (n^2/u(n)) + 1 for n >= 1; then a(n) is such that u(n) = (b(n)x + c(n))/(a(n)x + d(n)) (in lowest terms) and a(n), b(n), c(n), d(n) are positive integers.

Original entry on oeis.org

0, 1, 1, 5, 7, 47, 37, 319, 533, 1879, 1627, 20417, 18107, 263111, 237371, 52279, 95549, 1768477, 1632341, 33464927, 155685007, 166770367, 156188887, 3825136961, 3602044091, 19081066231, 18051406831, 57128792093, 7751493599

Offset: 1

Benoit Cloitre, Oct 14 2002

nonn

For x real <> 1 - 1/log(2), Lim_{n -> infinity} abs(u(n)-n) = abs((x - 1)/(1 + (x - 1)*log(2))). [Corrected by Petros Hadjicostas, May 18 2020]
From Petros Hadjicostas, May 05 2020: (Start)
Given x > 0, u(n) = (A075827(n)*x + A075828(n))/(a(n)*x + A075829(n)) = (b(n)*x + c(n))/(a(n)*x + d(n)) with gcd(gcd(b(n), c(n)), gcd(a(n), d(n))) = 1 for each n >= 1.
Conjecture 1: Define the sequences (A(n): n >= 1) and (B(n): n >= 1) by A(n+1) = n^2/A(n) + 1 for n >= 2 with A(1) = infinity and A(2) = 1, and B(n+1) = n^2/B(n) + 1 for n >= 3 with B(1) = 0, B(2) = infinity, and B(3) = 1. Then a(n) = denominator(A(n)), b(n) = numerator(A(n)), c(n) = numerator(B(n)), and d(n) = denominator(B(n)) (assuming infinity = 1/0). Also, gcd(a(n), d(n)) = 1.
In 2002, Michael Somos claimed that d(n) = A024168(n-1)/gcd(A024168(n-1), A024168(n)) for n >= 2. In 2006, N. J. A. Sloane claimed that a(n) = A058313(n-1) for n >= 2 while Alexander Adamchuk claimed that d(n) = A058312(n-1) - A058313(n-1) for n >= 2.
Conjecture 2: a(n) = A024167(n-1)/gcd(A024167(n-1), A024167(n)).
Conjecture 3: b(p) = a(p+1) for p = 1 or prime. In general, it seems that b(n) = A048671(n)*a(n+1) for all n for which A048671(n) < n.
Conjecture 4: c(n) = n*(a(n) + d(n)) - b(n) for n >= 1. (End)
All conjectures are proved in the link below except for the second part of Conjecture 3. - Petros Hadjicostas, May 21 2020

Petros Hadjicostas, Proofs of various results about the sequence u(n), 2020.

Apart from the leading term, same as A058313.
Cf. A075827 (= b), A075828 (= c), A075829 (= d).
Cf. A024167, A024168, A048671, A058312.

u(n)=if(n<2,x,(n-1)^2/u(n-1)+1);
a(n)=polcoeff(denominator(u(n)),1,x);

Name edited by Petros Hadjicostas, May 04 2020

A182936 Greatest common divisor of the proper divisors of n, 0 if there are none.

Original entry on oeis.org

0, 0, 0, 2, 0, 1, 0, 2, 3, 1, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 1, 0, 1, 5, 1, 3, 1, 0, 1, 0, 2, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 7, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 3, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1

Offset: 1

Peter Luschny, Mar 22 2011

nonn

Here a proper divisor d of n is a divisor of n such that 1 < d < n.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A014963, A020500, A048671, A034387, A061397, A072107, A380118.

A182936 := n -> igcd(op(numtheory[divisors](n) minus {1,n}));
seq(A182936(i), i=1..79); # Peter Luschny, Mar 22 2011

Join[{0}, Table[GCD@@Most[Rest[Divisors[n]]],{n,2,110}]] (* Harvey P. Dale, May 04 2018 *)
(* From Peter Luschny, Jan 31 2025: (Start) *)
Join[{0}, Table[Exp[MangoldtLambda[n]] - If[PrimeQ[n], n, 0], {n,2,110}]]
(* or *)
Table[Cyclotomic[n, 1] - If[PrimeQ[n], n, 0], {n,1,110}] (* End *)

A182936(n) = { my(divs=divisors(n)); if(#divs<3,0,gcd(vector(numdiv(n)-2,k,divs[k+1]))); }; \\ Antti Karttunen, Sep 23 2017

a(n) = 0 if n is not composite, p if n is a proper power of prime p, and 1 otherwise. - Franklin T. Adams-Watters, Mar 22 2011
Conjecture: Sum_{k=1..n} a(k) = A072107(n) - A034387(n) - 1. - Vaclav Kotesovec, Jan 29 2025
From Peter Luschny, Jan 31 2025: (Start)
a(n) = A014963(n) - A061397(n) for n > 1. In other words, this sequence is the exponential von Mangoldt function restricted to proper divisors of n. See A380118. This implies the above conjecture.
a(n) = A020500(n) - A061397(n). (End)

More terms from Antti Karttunen, Sep 23 2017

A334958 GCD of consecutive terms of the factorial times the alternating harmonic series.

Original entry on oeis.org

1, 1, 1, 2, 2, 12, 12, 48, 144, 1440, 1440, 17280, 17280, 241920, 18144000, 145152000, 145152000, 2612736000, 2612736000, 10450944000, 219469824000, 4828336128000, 4828336128000, 115880067072000, 579400335360000, 15064408719360000, 135579678474240000, 26573616980951040000, 26573616980951040000

Offset: 1

Petros Hadjicostas, May 17 2020

nonn

For n = 1..14, we have a(n) = A025527(n), but a(15) = 18144000 <> 3628800 = A025527(15).
It appears that A025527(n) | a(n) for all n >= 1 and A025527(n) = a(n) for infinitely many n. In addition, it seems that a(n)/a(n-1) = A048671(n) for infinitely many n >= 2. However, I have not established these claims.
This sequence appears in formulas for sequences A075827, A075828, A075829, and A075830 (the first one of which was established in 2002 by Michael Somos).
Conjecture: a(n) = n! * Product_{p <= n} p^min(0, v_p(H'(n))), where the product ranges over primes p, H'(n) = Sum_{k=1..n} (-1)^(k+1)/k, and v_p(r) is the p-adic valuation of rational r (checked for n < 1100).

			A024167(4) = 4!*(1 - 1/2 + 1/3 - 1/4) = 14, A024167(5) = 5!*(1 - 1/2 + 1/3 - 1/4 + 1/5) = 94, A024168(4) = 4!*(1/2 - 1/3 + 1/4) = 10, and A024168(5) = 5!*(1/2 - 1/3 + 1/4 - 1/5) = 26. Then a(4) = gcd(14, 94) = gcd(10, 26) = gcd(14, 4!) = gcd(10, 4!) = gcd(14, 10) = 2.

Cf. A000142, A024167, A024168, A025527, A048671, A058312, A058313, A075827, A075828, A075829, A075830.

Cf. A056612 (similar sequence for the harmonic series).

b:= proc(n) b(n):= (-(-1)^n/n +`if`(n=1, 0, b(n-1))) end:
a:= n-> (f-> igcd(b(n)*f, f))(n!):
seq(a(n), n=1..30);  # Alois P. Heinz, May 18 2020

b[n_] := b[n] = -(-1)^n/n + If[n == 1, 0, b[n-1]];
a[n_] := GCD[b[n] #, #]&[n!];
Array[a, 30] (* Jean-François Alcover, Oct 27 2020, after Alois P. Heinz *)

def A():
    a, b, n = 1, 1, 2
    while True:
        yield gcd(a, b)
        b, a = a, a + b * n * n
        n += 1
a = A(); print([next(a) for  in range(29)]) # _Peter Luschny, May 19 2020

a(n) = gcd(A024167(n+1), A024167(n)) = gcd(A024168(n+1), A024168(n)) = gcd(A024167(n), n!) = gcd(A024168(n), n!) = gcd(A024167(n), A024168(n)).

cp's OEIS Frontend

A140580 a(n) = (n^2)/A048671(n), = n*A014963(n) = A140579 * [1, 2, 3, ...].

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A014963 Exponential of Mangoldt function M(n): a(n) = 1 unless n is a prime or prime power, in which case a(n) = that prime.

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A010055 1 if n is a prime power p^k (k >= 0), otherwise 0.

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A048050 Chowla's function: sum of divisors of n except for 1 and n.

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A007956 Product of the proper divisors of n.

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A025527 a(n) = n!/lcm{1,2,...,n} = (n-1)!/lcm{C(n-1,0), C(n-1,1), ..., C(n-1,n-1)}.

A140580 a(n) = (n^2)/A048671(n), = nA014963(n) = A140579 [1, 2, 3, ...].

A075830 Let u(1) = x and u(n+1) = (n^2/u(n)) + 1 for n >= 1; then a(n) is such that u(n) = (b(n)x + c(n))/(a(n)x + d(n)) (in lowest terms) and a(n), b(n), c(n), d(n) are positive integers.