A100994 -id:A100994 - cp's OEIS frontend

A133232 Triangle T(n,k) read by rows with a minimum number of prime powers A100994 for which the least common multiple of T(n,1),..,T(n,n) is A003418(n).

1, 1, 2, 1, 2, 3, 1, 1, 3, 4, 1, 1, 3, 4, 5, 1, 1, 3, 4, 5, 1, 1, 1, 3, 4, 5, 1, 7, 1, 1, 3, 1, 5, 1, 7, 8, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 11, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 11, 1, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 11, 1, 13, 1, 1, 1, 1, 5, 1, 7, 8, 9, 1, 11, 1

Offset: 1

Mats Granvik, Oct 13 2007

Checked up to 28th row. The rest of the ones in the table are there for the least common multiple to calculate correctly.

			2 occurs 2*1 = 2 times in column 2.
3 occurs 3*2 = 6 times in column 3.
4 occurs 4*1 = 4 times in column 4.
5 occurs 5*4 = 20 times in column 5.
k occurs A133936(k) times in column k. The first rows of the triangle and the least common multiple of the rows are:
lcm{1} = 1
lcm{1, 2} = 2
lcm{1, 2, 3} = 6
lcm{1, 1, 3, 4} = 12
lcm{1, 1, 3, 4, 5} = 60
lcm{1, 1, 3, 4, 5, 1} = 60
lcm{1, 1, 3, 4, 5, 1, 7} = 420
lcm{1, 1, 3, 1, 5, 1, 7, 8} = 840
lcm{1, 1, 1, 1, 5, 1, 7, 8, 9} = 2520

Mats Granvik, Oct 13 2007, Table of n, a(n) for n = 1..406

Cf. A003418, A120112, A014963, A134579.

=if(and(row()>=column();row()A120112));column();1)

=if(and(n>=k; n < A014963*A100994); A100994; 1) - Mats Granvik, Jan 21 2008

A120112 := proc(n) 1-ilcm(seq(i,i=1..n+1))/ilcm(seq(i,i=1..n)) ; end proc:
A133232 := proc(n) if n < k*(1+abs(A120112(k-1))) then k else 1; end if; end proc:
seq(seq(A133232(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Nov 23 2010

b[n_] := b[n] = If[n == 0, 1, LCM @@ Range[n]];
c[n_] := 1 - b[n+1]/b[n];
T[n_, k_] := If[n < k*(1+Abs[c[k-1]]), k, 1];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 01 2021 *)

T(n,k) = if nA120112(k-1)| then k, else 1 (1<=k<=n).

T(n,k) = if n < A014963(k)*A100994(k) then A100994(k), else 1 (1<=k<=n). - Mats Granvik, Jan 21 2008

Indices added to formulas by R. J. Mathar, Nov 23 2010

A133790 A014963*A100994.

Original entry on oeis.org

1, 4, 9, 8, 25, 1, 49, 16, 27, 1, 121, 1, 169, 1, 1, 32, 289, 1, 361, 1, 1, 1, 529, 1, 125, 1, 81, 1, 841, 1, 961, 64, 1, 1, 1, 1, 1369, 1, 1, 1, 1681, 1, 1849, 1, 1, 1, 2209, 1, 343, 1, 1, 1, 2809, 1, 1, 1, 1, 1, 3481, 1, 3721, 1, 1, 128, 1, 1, 4489, 1, 1, 1, 5041, 1, 5329, 1, 1, 1

Offset: 1

Mats Granvik, Jan 21 2008

nonn

Cf. A014963, A100994.

a(n) = A014963*A100994

A133937 A100994-n/A014963.

Original entry on oeis.org

0, 1, 2, 2, 4, -5, 6, 4, 6, -9, 10, -11, 12, -13, -14, 8, 16, -17, 18, -19, -20, -21, 22, -23, 20, -25, 18, -27, 28, -29, 30, 16, -32, -33, -34, -35, 36, -37, -38, -39, 40, -41, 42, -43, -44, -45, 46, -47, 42, -49, -50, -51, 52, -53, -54, -55, -56, -57, 58, -59, 60, -61, -62, 32, -64, -65, 66, -67, -68, -69, 70, -71, 72, -73

Offset: 1

Mats Granvik, Jan 21 2008

sign

Cf. A100994, A014963.

a(n) = A100994-n/A014963.

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.

Original entry on oeis.org

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

A100995 If n is a prime power p^m, m >= 1, then m, otherwise 0.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Nov 26 2004

nonn

Calculate matrix powers: (A175992^1)/1 - (A175992^2)/2 + (A175992^3)/3 - (A175992^4)/4 + ... Then the nonzero values of a(n) are found as reciprocals in the first column. Compare this to the Taylor series for log(1+x) = (x)/1 - (x^2)/2 + (x^3)/3 - (x^4)/4 + ... Therefore it is natural to write 0, 1/1, 1/1, 1/2, 1/1, 0, 1/1, 1/3, 1/2, 0, 1/1, ... Raising n to a such power gives A014963. - Mats Granvik, Gary W. Adamson, Apr 04 2011
The Dirichlet series that generates the reciprocals of this sequence is the logarithm of the Riemann zeta function. - Mats Granvik, Gary W. Adamson, Apr 04 2011
Number of automorphisms of the finite field with n elements, or 0 if the field does not exist. For n=p^k where p is a prime and k is an integer, the automorphism group of the finite field with n elements is a cyclic group of order k generated by the Frobenius endomorphism. - Yancheng Lu, Jan 11 2021

Daniel Forgues, Table of n, a(n) for n = 1..100000

Cf. A028233, A069513, A010055.

a100995 n = f 0 n where
   f e 1 = e
   f e x = if r > 0 then 0 else f (e + 1) x'
           where (x', r) = divMod x p
   p = a020639 n
-- Reinhard Zumkeller, Mar 19 2013

f:= proc(n) local F;
    F:= ifactors(n)[2];
    if nops(F) = 1 then F[1][2]
    else 0
    fi
end proc:
map(f, [$1..100]); # Robert Israel, Jun 09 2015

ppm[n_]:=If[PrimePowerQ[n],FactorInteger[n][[1,2]],0]; Array[ppm,110] (* Harvey P. Dale, Mar 03 2014 *)
a=Table[Limit[Sum[If[Mod[n, k] == 0, MoebiusMu[n/k]/(n/k)^(s - 1)/(1 - 1/n^(s - 1)), 0], {k, 1, n}], s -> 1], {n, 1, 105}];
Numerator[a]*Denominator[a] (* Mats Granvik, Jun 09 2015 *)
a = FullSimplify[Table[MangoldtLambda[n]/Log[n], {n, 1, 105}]]
Numerator[a]*Denominator[a] (* Mats Granvik, Jun 09 2015 *)

{a(n) = my(t); if( n<1, 0, t = factor(n); if( [1,2] == matsize(t), t[1,2], 0))} /* Michael Somos, Aug 15 2012 */

{a(n) = my(t); if( n<1, 0, if( t = isprimepower(n), t))} /* Michael Somos, Aug 15 2012 */

A100994(n) = A014963(n)^a(n);
a(A000961(n)) = A025474(n).
a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(n/d) * bigomega(d). - Ilya Gutkovskiy, Apr 15 2021

Edited by Daniel Forgues and N. J. A. Sloane, Aug 18 2009

a(n) for n >= 2 equals LCM of minimal and maximal prime power factors in prime factorization of n. For n >= 2 holds: a(n)*A100994(n) = A034684(n)*A034699(n). a(n) for n >= 2 it deviates from A000027(n), first different term is a(30)=a(2*3*5), a(30)=lcm(2,5)=10, A000027(30)= 30. Sequence of deviations from A000027(n): 30,42,60,66,70,78,84,90,...

cp's OEIS Frontend

A133232 Triangle T(n,k) read by rows with a minimum number of prime powers A100994 for which the least common multiple of T(n,1),..,T(n,n) is A003418(n).

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A133790 A014963*A100994.

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A133937 A100994-n/A014963.

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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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A100995 If n is a prime power p^m, m >= 1, then m, otherwise 0.

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A133936 Number of times prime powers occur in the columns of tables A133232 and A133233.

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A134579 Column products of tables A133232 and A133233.

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