A048671 -id:A048671 - cp's OEIS frontend

A101085 Diagonal below the main diagonal of A083796, i.e., a(n) = A083796(n, n-1).

2, 2, 5, 4, 8, 6, 11, 8, 24, 10, 17, 12, 20, 14, 135, 16, 26, 18, 29, 20, 252, 22, 35, 24, 90, 26, 153, 28, 44, 30, 47, 32, 594, 34, 665, 36, 56, 38, 819, 40, 62, 42, 65, 44, 1080, 46, 71, 48, 224, 50, 1377, 52, 80, 54, 1595, 56, 1710, 58, 89, 60, 92, 62, 2079, 64, 2210, 66

Offset: 2

David Wasserman, Nov 30 2004

a(n) is the least k such that 1) n divides k + (n-1)(n-2)/2; 2) k > n-2; 3) A048671(n-1) divides k.

Cf. A048671, A083796, A083797.

A380118 a(n) = Sum_{k=1..n} (A014963(k) - A061397(k)).

Original entry on oeis.org

1, 1, 1, 3, 3, 4, 4, 6, 9, 10, 10, 11, 11, 12, 13, 15, 15, 16, 16, 17, 18, 19, 19, 20, 25, 26, 29, 30, 30, 31, 31, 33, 34, 35, 36, 37, 37, 38, 39, 40, 40, 41, 41, 42, 43, 44, 44, 45, 52, 53, 54, 55, 55, 56, 57, 58, 59, 60, 60, 61, 61, 62, 63, 65, 66, 67, 67, 68, 69, 70

Offset: 1

Peter Luschny, Jan 30 2025

nonn

Cf. A014963, A061397, A048671, A010051, A182936, A034387, A072107, A380117.

pSum := L -> ListTools:-PartialSums(L): h := n -> n/A048671(n) - n*A010051(n):
aList := upto -> pSum([seq(h(k), k = 1..upto)]): aList(70);

Accumulate[Table[Exp[MangoldtLambda[n]] - If[PrimeQ[n], n, 0] , {n, 1, 70}]]

a(n) = A072107(n) - A034387(n). - Amiram Eldar, Jan 30 2025

A140651 A140579^(-1) * A000290, the squares starting (1, 4, 9, ...).

Original entry on oeis.org

1, 2, 3, 8, 5, 36, 7, 32, 27, 100, 11, 144, 13, 196, 225, 128, 17, 324, 19, 400, 441, 484, 23, 576, 125, 676, 243, 784, 29, 900, 31, 512, 1089, 1156, 1225, 1296, 37, 1444, 1521, 1600, 41, 1764, 43, 1936, 2025, 2116, 47, 2304, 343, 2500, 2601, 2704, 53, 2916, 3025, 3136, 3249, 3364, 59, 3600

Offset: 1

Gary W. Adamson and Mats Granvik, May 20 2008

nonn

For n > 1, a(n) = n iff n is prime.

a(n) is n times the least common multiple of the proper divisors of n, a(n) = n*A048671(n). - Peter Luschny, Jun 22 2011

			A140579 = an infinite lower triangular matrix with A014963 in the main diagonal and the rest zeros; where A014963 = (1, 2, 3, 2, 5, 1, 7, ...).
a(5) = 5 = (1/A014963(5)) * 25 = (1/5)*25.

Cf. A140579, A014963.

A140651  := n -> n*ilcm(op(numtheory[divisors](n) minus {1,n})); seq(A140651(i),i=1..60); # Peter Luschny, Jun 22 2011

A140579^(-1) * (1, 4, 9, 16, 25, ...).

a(15)-a(60) from Peter Luschny, Jun 22 2011

A189914 a(n) is 2^phi(n) times the least common multiple of the proper divisors of n.

Original entry on oeis.org

1, 2, 2, 4, 8, 16, 24, 64, 64, 192, 160, 1024, 192, 4096, 896, 3840, 2048, 65536, 1152, 262144, 5120, 86016, 22528, 4194304, 6144, 5242880, 106496, 2359296, 114688, 268435456, 7680, 1073741824, 1048576, 34603008, 2228224, 587202560, 147456, 68719476736, 9961472

Offset: 0

Peter Luschny, Jun 22 2011

nonn

The sequence relates arithmetic properties of roots of unity in the complex plane with number theoretic properties of integers. This connection often appears as intriguing identities showing products of specific values of the sine function or the gamma function reducing to simple values (see for instance the first formula below).

Charles R Greathouse IV, Table of n, a(n) for n = 0..3322
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], Sep 2009.
Albert Nijenhuis, Short Gamma Products with Simple Values, The American Mathematical Monthly, Vol. 117, No. 8, Oct 2010, pp. 733-737.
J. Sándor and L. Tóth, A remark on the gamma function, Elemente der Mathematik, 44 (1989), pp. 73-76, Birkhäuser.

A189914 := n -> 2^numtheory[phi](n)*ilcm(op(numtheory[divisors](n) minus {1,n})): seq(A189914(n), n=0..35);

a[n_] := 2^EulerPhi[n] * LCM @@ Most[Divisors[n]]; a[0] = 1; a[1] = 2; Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Jan 22 2014 *)

a(n)=if(n,my(p=n); if(isprime(n)||(ispower(n, , &p)&&isprime(p)), n/p, n)<Charles R Greathouse IV, Jun 24 2011

Let R(n) = {k | gcd(n,k) = 1, k = 1..floor(n/2)} and b(n) = product_{R(n)} sin(Pi*k/n) then a(n) = n / b(n)^2 for n > 1.

a(n) = A066781(n)*A048671(n).

A340679 If n is a power of prime then a(n) = 1, otherwise a(n) = product of the distinct prime factors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 10, 1, 6, 1, 14, 15, 1, 1, 6, 1, 10, 21, 22, 1, 6, 1, 26, 1, 14, 1, 30, 1, 1, 33, 34, 35, 6, 1, 38, 39, 10, 1, 42, 1, 22, 15, 46, 1, 6, 1, 10, 51, 26, 1, 6, 55, 14, 57, 58, 1, 30, 1, 62, 21, 1, 65, 66, 1, 34, 69, 70, 1, 6, 1, 74, 15, 38, 77, 78, 1, 10, 1, 82, 1, 42, 85, 86, 87, 22, 1, 30, 91, 46

Offset: 1

Antti Karttunen, Feb 01 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000961 (positions of ones), A007947, A010055, A014963.

Cf. also A048671, A322591.

Table[If[PrimePowerQ[n],1,Times@@FactorInteger[n][[All,1]]],{n,100}] (* Harvey P. Dale, May 23 2021 *)

A340679(n) = if(isprimepower(n),1,factorback(factorint(n)[, 1]));

If A010055(n) == 1 [when n is a power of prime], a(n) = 1, otherwise a(n) = A007947(n).

a(n) = A007947(n) / A014963(n).

A133689 a(n) = smallest integer that is > n and is a multiple of every proper divisor of n.

Original entry on oeis.org

3, 4, 6, 6, 12, 8, 12, 12, 20, 12, 24, 14, 28, 30, 24, 18, 36, 20, 40, 42, 44, 24, 48, 30, 52, 36, 56, 30, 60, 32, 48, 66, 68, 70, 72, 38, 76, 78, 80, 42, 84, 44, 88, 90, 92, 48, 96, 56, 100, 102, 104, 54, 108, 110, 112, 114, 116, 60

Offset: 2

Leroy Quet, Dec 31 2007

nonn

a(n) = A048671(n) + n.

			The proper divisors of 16 are 1,2,4,8. a(16)=24 is the smallest integer which is both > 16 and is a multiple of 1, of 2, of 4 and of 8.

Cf. A048671.

a = {}; For[n = 2, n < 60, n++, i = n + 1; While[Length[Union[Mod[i, Complement[Divisors[n], {n}]]]] > 1, i++ ]; AppendTo[a, i]]; a (* Stefan Steinerberger, Aug 30 2008 *)

More terms from Stefan Steinerberger, Aug 30 2008

cp's OEIS Frontend

A101085 Diagonal below the main diagonal of A083796, i.e., a(n) = A083796(n, n-1).

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A380118 a(n) = Sum_{k=1..n} (A014963(k) - A061397(k)).

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A140651 A140579^(-1) * A000290, the squares starting (1, 4, 9, ...).

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Maple

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A189914 a(n) is 2^phi(n) times the least common multiple of the proper divisors of n.

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Formula

A340679 If n is a power of prime then a(n) = 1, otherwise a(n) = product of the distinct prime factors of n.

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Author

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Links

Crossrefs

Programs

Mathematica

PARI

Formula

A133689 a(n) = smallest integer that is > n and is a multiple of every proper divisor of n.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions