A051731 -id:A051731 - cp's OEIS frontend

A143152 Inverse Möbius transform of the least prime factor of n: A051731 * A020639.

1, 3, 4, 5, 6, 8, 8, 7, 7, 10, 12, 12, 14, 12, 12, 9, 18, 13, 20, 14, 14, 16, 24, 16, 11, 18, 10, 16, 30, 20, 32, 11, 18, 22, 18, 19, 38, 24, 20, 18, 42, 22, 44, 20, 18, 28, 48, 20, 15, 17, 24, 22, 54, 18, 22, 20, 26, 34, 60, 28, 62, 36, 20, 13, 24, 26, 68, 26, 30, 26, 72, 25, 74

Offset: 1

Gary W. Adamson, Jul 27 2008

nonn

			a(4) = 5 = (1, 1, 0, 1) dot (1, 2, 3, 2) = (1 + 2 + 0 + 2), where (1, 1, 0, 1) = row 4 of triangle A051731 and A010639 = (1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11,...).
Since a(n) = sum of least prime factors of the divisors of n, the divisors of 12 are recorded in triangle row 12 of A127093: (1, 2, 3, 4, 0, 6, 0, 0, 0, 0, 0, 12). Lpf of these terms = row 12 of triangle A143151: (1, 2, 3, 2, 0, 2, 0, 0, 0, 0, 0, 2); sum = 12.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A051731, A020639, A127093, A143151.

read transforms : A020639 := proc(n) local i ; if n = 1 then 1; else for i from 1 do if n mod ithprime(i) = 0 then RETURN(ithprime(i)) ; fi; od: fi; end: a020639 := [seq(A020639(n),n=1..100)] : a143152 := MOBIUSi(a020639) : for i from 1 to nops(a143152) do printf("%d,",op(i,a143152)) ; od: # R. J. Mathar, Aug 11 2008

A020639(n) = if(1==n,n,(factor(n)[1, 1]));
A143152(n) = sumdiv(n,d,A020639(d)); \\ Antti Karttunen, Nov 12 2021

a(p) = (p+1) for prime p.
Inverse Mobius transform of A020639, where A020639(n) = Lpf(n).
Row sums of triangle A143151.
a(n) = Sum_{d|n} A020639(d). - Antti Karttunen, Nov 12 2021

Extended beyond a(14) by R. J. Mathar, Aug 11 2008

Name amended by Antti Karttunen, Nov 12 2021

A143256 Triangle read by rows, matrix multiplication A051731 * A128407 * A127648, 1<=k<=n.

Original entry on oeis.org

1, 1, -2, 1, 0, -3, 1, -2, 0, 0, 1, 0, 0, 0, -5, 1, -2, -3, 0, 0, 6, 1, 0, 0, 0, 0, 0, -7, 1, -2, 0, 0, 0, 0, 0, 0, 1, 0, -3, 0, 0, 0, 0, 0, 0, 1, -2, 0, 0, -5, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -11, 1, -2, -3, 0, 0, 6, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -13, 1, -2, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 14

Offset: 1

Gary W. Adamson, Aug 02 2008

Right border = n*mu(n) = A055615.

Row sums = A023900: (1, -1, -2, -1, -4, 2, -6,...).

			First few rows of the triangle =
1;
1, -2;
1, 0, -3;
1, -2, 0, 0;
1, 0, 0, 0, -5;
1, -2, -3, 0, 0, 6;
1, 0, 0, 0, 0, 0, -7;
...

Robert Israel, Table of n, a(n) for n = 1..9870

Cf. A008683, A055615, A051731, A128407, A127648, A023900.

seq(seq(`if`(i mod j = 0, j*numtheory:-mobius(j),0), j=1..i),i=1..20); # Robert Israel, Sep 07 2014

Table[If[Divisible[n, k], k MoebiusMu[k], 0], {n, 1, 14}, {k, 1, n}] (* Jean-François Alcover, Jun 19 2019 *)

A143256_row = lambda n: [k*moebius(k) if k.divides(n) else 0 for k in (1..n)]
for n in (1..10): print(A143256_row(n)) # Peter Luschny, Jan 05 2018

Triangle read by rows, A051731 * A128407 * A127648, 1<=k<=n

A143317 Triangle read by rows: A051731 * A143239.

Original entry on oeis.org

1, 3, -1, 4, 0, -1, 7, -3, 0, 0, 6, 0, 0, 0, -1, 12, -4, -3, 0, 0, 1, 8, 0, 0, 0, 0, 0, -1, 15, -7, 0, 0, 0, 0, 0, 0, 13, 0, -4, 0, 0, 0, 0, 0, 0, 18, -6, 0, 0, -3, 0, 0, 0, 0, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 28, -12, -7, 0, 0, 3, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1

Offset: 1

Gary W. Adamson, Aug 07 2008

Left border = sigma(n), A000203.
Right border = mu(n), A008683.
Row sums = n.
It appears that T(n,k) = Sum_{d|n} mu(d)*sigma(n/d). - Joerg Arndt, Jul 31 2011

			First few rows of the triangle:
   1;
   3, -1;
   4,  0, -1;
   7, -3,  0,  0;
   6,  0,  0,  0, -1;
  12, -4, -3,  0,  0,  1;
   8,  0,  0,  0,  0,  0, -1;
  15, -7,  0,  0,  0,  0,  0,  0;
  13,  0, -4,  0,  0,  0,  0,  0,  0;
  18, -6,  0,  0, -3,  0,  0,  0,  0,  1;
  ...

Cf. A000203, A008683, A051731, A143239.

a(68) = -12 corrected by Georg Fischer, Jun 05 2023

A143349 Triangle read by rows: A000012 * A054524 = A000012 * A051731 * A128407.

Original entry on oeis.org

1, 2, -1, 3, -1, -1, 4, -2, -1, 0, 5, -2, -1, 0, -1, 6, -3, -2, 0, -1, 1, 7, -3, -2, 0, -1, 1, -1, 8, -4, -2, 0, -1, 1, -1, 0, 9, -4, -3, 0, -1, 1, -1, 0, 0, 10, -5, -3, 0, -2, 1, -1, 0, 0, 1, 11, -5, -3, 0, -2, 1, -1, 0, 0, 1, -1, 12, -6, -4, 0, -2, 2, -1, 0, 0, 1, -1, 0, 13, -6, -4, 0, -2, 2, -1, 0, 0, 1, -1, 0, -1, 14, -7, -4, 0, -2, 2, -2, 0, 0, 1, -1, 0, -1, 1

Offset: 1

Gary W. Adamson, Aug 10 2008

The triangle acts as a transform converting any sequence S(k) into a triangle with row sums = S(k). By way of example, begin with S(k), the primes: (2, 3, 5, 7, 11, ...). Add (0, 1, 2, 3, 4, ...) to the sequence getting (prime(n)+(n-1)) = (2, 4, 7, 10, 15, 18, 23, 36, 31, ...) = sequence Q(k). Then replace column 1 (1, 2, 3, ...) of triangle A143349 with sequence Q(k). This = triangle A143350 with row sums prime(n):
    2;
    4, -1;
    7, -1, -1;
   10, -2, -1,  0;
   ...
The A000012 multiplier takes partial sums of A054524 column terms. A051731 is the inverse Mobius transform and A128407 = an infinite lower triangular matrix with mu(n) in the main diagonal and the rest zeros.

			First few rows of the triangle:
   1;
   2, -1;
   3, -1, -1;
   4, -2, -1,  0;
   5, -2, -1,  0, -1;
   6, -3, -2,  0, -1,  1;
   7, -3, -2,  0, -1,  1, -1;
   8, -4, -2,  0, -1,  1, -1,  0;
   9, -4, -3,  0, -1,  1, -1,  0,  0;
  10, -5, -3,  0, -2,  1, -1,  0,  0,  1;
  11, -5, -3,  0, -2,  1, -1,  0,  0,  1, -1;
  12, -6, -4,  0, -2,  2, -1,  0,  0,  1, -1,  0;
  13, -6, -4,  0, -2,  2, -1,  0,  0,  1, -1,  0, -1;
  14, -7, -4,  0, -2,  2, -2,  0,  0,  1, -1,  0, -1,  1;
  ...

Cf. A000012, A051731, A054524, A128407, A143350.

a(39) ff. corrected by Georg Fischer, Jun 05 2023

A143354 Triangle read by rows, (A051731)^4 * A128407, 1<=k<=n.

Original entry on oeis.org

1, 4, -1, 4, 0, -1, 10, -4, 0, 0, 4, 0, 0, 0, -1, 16, -4, -4, 0, 0, 1, 4, 0, 0, 0, 0, 0, -1, 20, -10, 0, 0, 0, 0, 0, 0, 10, 0, -4, 0, 0, 0, 0, 0, 0, 16, -4, 0, 0, -4, 0, 0, 0, 0, 1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 40, -16, -10, 0, 0, 4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1

Offset: 1

Gary W. Adamson, Aug 10 2008

Left border = A007426.
Right border = mu(n), A008683.
Row sums = A007425: (1, 3, 3, 6, 3, 9,...).

			First few rows of the triangle =
1;
4, -1;
4, 0, -1;
10, -4, 0, 0;
4, 0, 0, 0, 0, -1;
16, -4, -4, 0, 0, 1;
4, 0, 0, 0, 0, 0, -1;
10, -10, 0, 0, 0, 0, 0, 0;
...

Cf. A051731, A128407, A007425, A007426, A008683.

Triangle read by rows, (A051731)^4 * A128407, 1<=k<=n

A143356 A051731 * A006218.

Original entry on oeis.org

1, 4, 6, 12, 11, 23, 17, 32, 29, 41, 30, 66, 38, 61, 61, 82, 53, 104, 61, 115, 92, 107, 77, 170, 98, 132, 124, 170, 104, 216, 114, 201, 158, 183, 158, 287, 143, 210, 193, 293, 161, 318, 171, 291, 266, 266, 189, 418, 218, 335, 269, 357, 220, 426, 271, 429, 309, 354, 250

Offset: 1

Gary W. Adamson, Aug 10 2008

nonn

			a(4) = 12 = sum of row 4 terms of triangle A143355: (7, + 3 + 1 + 1).
a(4) = 12 = (1, 1, 0, 1) dot (1, 3, 5, 8) = (1 + 3 + 0 + 8), where (1, 1, 0, 1) = row 4 of A051731 and A006218 = (1, 3, 5, 8, 10, 14,...).

Cf. A006218, A051731, A143355.

Cf. A135539.

row(n) = my(d=divisors(n)); vector(n, k, #select(x->(x>=k), d)); \\ A135539
a(n) = my(v=row(n)); sum(i=1, n, numdiv(i)*v[i]); \\ Michel Marcus, Jul 26 2022

Inverse Mobius transform (A051731) of A006218. Row sums of triangle A143355.
a(n) = Sum_{i=1..n} tau(i)*A135539(n,i). - Ridouane Oudra, Jul 26 2022
a(n) = Sum_{d|n} A006218(d). - Ridouane Oudra, Jul 27 2022

Corrected typo in A-number in formula; added more terms - R. J. Mathar, Jan 19 2009

A155029 Complement to A051731 with the identity matrix A023531 included.

Original entry on oeis.org

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

Offset: 1

Mats Granvik, Jan 19 2009

			Table begins:
  1;
  0, 1;
  0, 1, 1;
  0, 0, 1, 1;
  0, 1, 1, 1, 1;
  0, 0, 0, 1, 1, 1;
  0, 1, 1, 1, 1, 1, 1;
  0, 0, 1, 0, 1, 1, 1, 1;
  0, 1, 0, 1, 1, 1, 1, 1, 1;

G. C. Greubel, Rows n = 1..30 of the triangle, flattened

Cf. A023531, A051731, A155031.

[k eq n select 1 else (k eq 0 or n mod k eq 0) select 0 else 1: k in [1..n], n in [1..20]]; // G. C. Greubel, Mar 07 2021

Table[If[k==n, 1, If[k==0, 0, If[Mod[n, k]==0, 0, 1]]], {n, 20}, {k, n}]//Flatten (* G. C. Greubel, Mar 07 2021 *)

flatten([[1 if k==n else 0 if (k==0 or n%k==0) else 1 for k in [1..n]] for n in [1..20]]) # G. C. Greubel, Mar 07 2021

T(n, k) = 0 if n==0 (mod k) otherwise 1 with T(n, n) = 1 and T(n, 1) = 0. - G. C. Greubel, Mar 07 2021

A156837 Triangle read by rows, A051731 * A156348.

Original entry on oeis.org

1, 2, 1, 2, 0, 1, 3, 3, 0, 1, 2, 0, 0, 0, 1, 4, 4, 4, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 4, 7, 0, 5, 0, 0, 0, 1, 3, 0, 7, 0, 0, 0, 0, 0, 1, 4, 6, 0, 0, 6, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 12, 14, 11, 0, 7, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson & Mats Granvik, Feb 16 2009

Left border: A000005(n).

Row sums = 3 iff n-th row is prime.

			First few rows of the triangle =
1;
2, 1;
2, 0, 1;
3, 3, 0, 1;
2, 0, 0, 0, 1;
4, 4, 4, 0, 0, 1;
2, 0, 0, 0, 0, 0, 1;
4, 7, 0, 5, 0, 0, 0, 1;
3, 0, 7, 0, 0, 0, 0, 0, 1;
4, 6, 0, 0, 6, 0, 0, 0, 0, 1;
2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
6, 12, 14, 11, 0, 7, 0, 0, 0, 0, 0, 1;
2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
4, 8, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 1;
...

Cf. A156348, A051731, A000005, A156838 (row sums).

A156837 := proc(n,k)
    add(A051731(n,j)*A156348(j,k),j=k..n) ;
end proc: # R. J. Mathar, Mar 03 2013

Triangle read by rows, A051731 * A156348 = inverse Mobius transform of A156348

A158901 A051731 * (1, 1, 2, 3, 4, 5, ...).

Original entry on oeis.org

1, 2, 3, 5, 5, 9, 7, 12, 11, 15, 11, 23, 13, 21, 21, 27, 17, 34, 19, 37, 29, 33, 23, 53, 29, 39, 37, 51, 29, 65, 31, 58, 45, 51, 45, 83, 37, 57, 53, 83, 41, 89, 43, 79, 73, 69, 47, 115, 55, 88, 69, 93, 53, 113, 69, 113, 77, 87, 59, 157, 61, 93, 99, 121, 81, 137, 67, 121, 93, 137

Offset: 1

Gary W. Adamson, Mar 29 2009

a(n) = prime(n) if n is prime but nonprime n's can also have prime a(n).

Equals left border of triangle A158902.

			a(4) = 5 = (1, 1, 0, 1) dot (1, 1, 2, 3) = (1 + 1 + 0 + 3); where (1, 1, 0, 1) = row 4 of triangle A051731.

Cf. A158902.

L := [1,seq(n, n=1..100)] ; read("transforms"); MOBIUSi(L) ; # R. J. Mathar, Apr 02 2009

a(n) = sigma(n) - numdiv(n) + 1; \\ Michel Marcus, Sep 14 2017

A051731 * [1, 1, 2, 3, 4, 5, ...] = inverse Mobius transform of [1, 1, 2, 3, 4, ...].
a(n) = sigma(n) - d(n) + 1. - Juri-Stepan Gerasimov, Aug 30 2009
a(n) = 1 + A065608(n). - R. J. Mathar, Jan 08 2015

A158951 Triangle read by rows, A051731 * A158948.

Original entry on oeis.org

1, 2, 1, 3, 0, 1, 4, 2, 0, 1, 4, 0, 1, 0, 1, 7, 2, 1, 1, 0, 1, 5, 0, 1, 0, 1, 0, 1, 8, 3, 0, 2, 0, 1, 0, 18, 0, 2, 0, 1, 0, 1, 0, 1, 10, 2, 1, 1, 1, 1, 0, 1, 0, 1

Offset: 1

Gary W. Adamson & Mats Granvik, Mar 31 2009

Row sums = sigma(n), A000203. Left column = A079247

			First few rows of the triangle =
1;
2, 1;
3, 0, 1;
4, 2, 0, 1;
4, 0, 1, 0, 1;
7, 2, 1, 1, 0, 1;
5, 0, 1, 0, 1, 0, 1;
8, 3, 0, 2, 0, 1, 0, 1;
8, 0, 2, 0, 1, 0, 1, 0, 1;
10, 2, 1, 1, 1, 1, 0, 1, 0, 1;
7, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
15, 4, 1, 3, 0, 2, 0, 10, 1, 0, 1;
8, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
13, 2, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1;
...

Cf. A158948, A000203, A051731

Triangle read by rows, A051731 * A158948

cp's OEIS Frontend

A143152 Inverse Möbius transform of the least prime factor of n: A051731 * A020639.

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A143256 Triangle read by rows, matrix multiplication A051731 * A128407 * A127648, 1<=k<=n.

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A143317 Triangle read by rows: A051731 * A143239.

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A143349 Triangle read by rows: A000012 * A054524 = A000012 * A051731 * A128407.

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A143354 Triangle read by rows, (A051731)^4 * A128407, 1<=k<=n.

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A143356 A051731 * A006218.

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A155029 Complement to A051731 with the identity matrix A023531 included.

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A156837 Triangle read by rows, A051731 * A156348.

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