A140255 -id:A140255 - 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

A140256 Triangle read by columns: Column k is A014963 aerated with groups of (k-1) zeros.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson and Mats Granvik, May 16 2008, Jun 11 2008

If the row number n is prime, the row consists of T(n,1)=n followed by n-2 zeros and followed by T(n,n)=1.
Similar to A138618.
Row products of nonzero terms in row n, equals n. - Mats Granvik, May 22 2016

			First few rows of the triangle are:
   1;
   2, 1;
   3, 0, 1;
   2, 2, 0, 1;
   5, 0, 0, 0, 1;
   1, 3, 2, 0, 0, 1;
   7, 0, 0, 0, 0, 0, 1;
   2, 2, 0, 2, 0, 0, 0, 1;
   3, 0, 3, 0, 0, 0, 0, 0, 1;
   1, 5, 0, 0, 2, 0, 0, 0, 0, 1;
  11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
   1, 1, 2, 3, 0, 2, 0, 0, 0, 0, 0, 1;
  ...
Column 2 = (1, 0, 2, 0, 3, 0, 2, 0, 5, 0, 1, 0, 7, ...).

T. Tao, Simons Lecture I: Structure and randomness in Fourier analysis and number theory.
Wikipedia, Fundamental theorem of arithmetic.

Cf. A140255 (row sums), A014963.

Row products without the zero terms produce A000027. [Mats Granvik, Oct 08 2009]

=if(row()>=column();if(mod(row();column())=0;lookup(roundup(row()/column();0);A000027;A014963);0);"")

t[n_, k_] /; Divisible[n, k] := Exp[ MangoldtLambda[n/k] ]; t[, ] = 0; Table[t[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 28 2013 *)
(* recurrence *)
Clear[t, s, n, k, z, nn];z = 1;nn = 14;t[n_, k_] := t[n, k] = If[k == 1, Zeta[s]*(1 - 1/n^(s - 1)) -Sum[t[n, i]/i^(s - 1), {i, 2, n}], If[Mod[n, k] == 0, t[n/k, 1], 0], 0]; A = Table[Table[Limit[t[n, k], s -> z], {k, 1, n}], {n, 1, nn}]; Flatten[Exp[A]*Table[Table[If[Mod[n, k] == 0, 1, 0], {k, 1, n}], {n, 1, nn}]] (* Mats Granvik, Apr 09 2016, May 22 2016 *)

T(n,k) = A014963(n/k) = A014963(A126988(n,k)) if k|n, T(n,k)=0 otherwise. 1 <= k <= n.
From Mats Granvik, Apr 10 2016, May 22 2016: (Start)
Limit as s -> 1 of the recurrence: Ts(n, k) = if k = 1 then zeta(s)*(1 - 1/n^(s - 1)) -Sum_{i=2..n} Ts(n, i)/(i)^(s - 1) else if n mod k = 0 then Ts(n/k, 1) else 0 else 0.
For n not equal to k: Limit as s -> 1 of the recurrence: Ts(n, k) = if k = 1 then zeta(s) -Sum_{i=2..n} Ts(n, i)/i^(s - 1) else if n mod k = 0 then Ts(n/k, 1) else 0 else 0.
Limit as s -> 1 of the recurrence: Ts(n, k) = if k = 1 then log(n) -Sum_{i=2..n} Ts(n, i)/i^(s - 1) else if n mod k = 0 then Ts(n/k, 1) else 0 else 0. (End)
[The above sentences need a lot of work!  Parentheses might help. - N. J. A. Sloane, Mar 14 2017]

A140254 Mobius transform of A014963.

Original entry on oeis.org

1, 1, 2, 0, 4, -3, 6, 0, 0, -5, 10, 0, 12, -7, -6, 0, 16, 0, 18, 0, -8, -11, 22, 0, 0, -13, 0, 0, 28, 7, 30, 0, -12, -17, -10, 0, 36, -19, -14, 0, 40, 9, 42, 0, 0, -23, 46, 0, 0, 0, -18, 0, 52, 0, -14, 0, -20, -29, 58, 0, 60, -31, 0, 0, -16, 13, 66, 0, -24, 11, 70, 0, 72, -37, 0, 0, -16

Offset: 1

Gary W. Adamson and Mats Granvik, May 16 2008, Jun 29 2008

sign

Conjectures relating to the Mobius sequence A008683:
If mu(n) = 0, a(n) = 0.
If mu(n) = 1, (n>1), a(n) = a negative term.
If mu(n) = -1, a(n) = a positive term.
So except for the first term and zero divided by zero we would have mu(n) = -a(n)/abs(a(n)).
Examples: mu(4) = 0, a(4) = 0; mu(6) = 1, a(6) = (-3); mu(7) = (-1), a(7) = 6.

			a(5) = -3 = (1, -1, -1, 0, 0, 1) dot (1, 2, 3, 2, 5, 1) = (1 - 2 - 3 + 0 + 0 + 1), where (1, -1, -1, 0, 0, 1) = row 5 of triangle A054525 and (1, 2, 3, 2, 5, 1) = the first 5 terms of A014963.

Physics Forums discussion, Moebius function.
Eric. W. Weisstein, Mertens Conjecture.

Cf. A014963, A008683, A140255, A140256.

A054525 as an infinite lower triangular matrix * A014963 as a vector.

More terms from Mats Granvik, Jun 29 2008

A350380 Triangle read by rows in which row n lists A014963(d), the exponential of Mangoldt function, for each divisor d of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 2, 1, 5, 1, 2, 3, 1, 1, 7, 1, 2, 2, 2, 1, 3, 3, 1, 2, 5, 1, 1, 11, 1, 2, 3, 2, 1, 1, 1, 13, 1, 2, 7, 1, 1, 3, 5, 1, 1, 2, 2, 2, 2, 1, 17, 1, 2, 3, 1, 3, 1, 1, 19, 1, 2, 2, 5, 1, 1, 1, 3, 7, 1, 1, 2, 11, 1, 1, 23, 1, 2, 3, 2, 1, 2, 1, 1, 1, 5, 5, 1, 2, 13, 1

Offset: 1

Michel Marcus, Dec 28 2021, following a suggestion from Charles Kusniec

			Triangle begins:
  1;
  1, 2;
  1, 3;
  1, 2, 2;
  1, 5;
  1, 2, 3, 1;
  1, 7;
  1, 2, 2, 2;
  1, 3, 3;
  1, 2, 5, 1;
  ...

Michel Marcus, Table of n, a(n) for n = 1..10006 (rows 1 to 1358, flattened).

Cf. A014963, A027750.

Cf. A000027 (row products), A140255 (row sums).

Table[Exp[MangoldtLambda[Divisors[n]]], {n, 1, 26}] // Flatten (* Amiram Eldar, Dec 28 2021 *)

M(n) = ispower(n, , &n); if(isprime(n), n, 1); \\ A014963
row(n) = apply(M, divisors(n));

a(n) = A014963(A027750(n)).

A303004 Expansion of e.g.f. exp(Sum_{k>=1} M(k)*x^k/k!), where M() is the exponential of Mangoldt function (A014963).

Original entry on oeis.org

1, 1, 3, 10, 39, 186, 962, 5587, 35367, 241216, 1771052, 13827925, 114558314, 1001769237, 9208116647, 88737108635, 893505145271, 9379190223746, 102402586369892, 1160487000658679, 13627075242031720, 165524499516422471, 2076762033563394443, 26877177548737581587

Offset: 0

Ilya Gutkovskiy, Apr 17 2018

nonn

Exponential transform of A014963.

			E.g.f.: A(x) = 1 + x/1! + 3*x^2/2! + 10*x^3/3! + 39*x^4/4! + 186*x^5/5! + 962*x^6/6! + 5587*x^7/7! + ...

Eric Weisstein's World of Mathematics, Mangoldt Function
N. J. A. Sloane, Transforms

Cf. A014963, A140254, A140255.

nmax = 23; CoefficientList[Series[Exp[Sum[Exp[MangoldtLambda[k]] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Exp[MangoldtLambda[k]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]

E.g.f.: exp(Sum_{k>=1} A014963(k)*x^k/k!).

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.

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A140256 Triangle read by columns: Column k is A014963 aerated with groups of (k-1) zeros.

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A140254 Mobius transform of A014963.

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A350380 Triangle read by rows in which row n lists A014963(d), the exponential of Mangoldt function, for each divisor d of n.

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A303004 Expansion of e.g.f. exp(Sum_{k>=1} M(k)*x^k/k!), where M() is the exponential of Mangoldt function (A014963).

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