A051126 -id:A051126 - cp's OEIS frontend

A048158 Triangular array T read by rows: T(n,k) = n mod k, for k=1,2,...,n, n=1,2,...

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

Offset: 1

Clark Kimberling

Also, rectangular array read by antidiagonals: a(n, k) = n mod k, n >= 0, k >= 1. Cf. A051126, A051127, A051777. - David Wasserman, Oct 01 2008

			Triangle begins
  0;
  0  0;
  0  1  0;
  0  0  1  0;
  0  1  2  1  0;
  0  0  0  2  1  0;
  0  1  1  3  2  1  0;
  0  0  2  0  3  2  1  0;
  0  1  0  1  4  3  2  1  0;
  0  0  1  2  0  4  3  2  1  0;
  0  1  2  3  1  5  4  3  2  1  0;
  0  0  0  0  2  0  5  4  3  2  1  0;
  ...
From _Omar E. Pol_, Feb 21 2014: (Start)
Illustration of the 12th row of triangle:
-----------------------------------
.      k: 1 2 3 4 5 6 7 8 9 10..12
-----------------------------------
.         _ _ _ _ _ _ _ _ _ _ _ _
.        |_| | | | | | | | | | | |
.        |_|_| | | | | | | | | | |
.        |_| |_| | | | | | | | | |
.        |_|_| |_| | | | | | | | |
.        |_| | | |_| | | | | | | |
.        |_|_|_| | |_| | | | | | |
.        |_| | | | | |_| | | | | |
.        |_|_| |_| | |*|_| | | | |
.        |_| |_| | | |* *|_| | | |
.        |_|_| | |_| |* * *|_| | |
.        |_| | | |*| |* * * *|_| |
.        |_|_|_|_|*|_|* * * * *|_|
.
Row 12 is 0 0 0 0 2 0 5 4 3 2 1 0
(End)

Alois P. Heinz, Rows n = 1..141, flattened
Michael Z. Spivey, The Humble Sum of Remainders Function, Mathematics Magazine, Vol. 78, No. 4 (Oct., 2005), pp. 300-305.
Eric Weisstein's World of Mathematics, Mod.

Row sums are given by A004125.
Cf. A002260.
Cf. A000007, A010766, A051126, A051127, A051731, A051777.

a048158 = mod
a048158_row n = a048158_tabl !! (n-1)
a048158_tabl = zipWith (map . mod) [1..] a002260_tabl
-- Reinhard Zumkeller, Apr 29 2015, Jan 20 2014 (fixed), Aug 13 2013

T:= (n, k)-> modp(n, k):
seq(seq(T(n, k), k=1..n), n=1..20); # Alois P. Heinz, Apr 04 2012

Flatten[Table[Mod[n, Range[n]], {n, 15}]]

def A048158_T(n,k): return n%k # Chai Wah Wu, May 13 2024

A051731(n,k) = A000007(T(n,k)). - Reinhard Zumkeller, Nov 01 2009
T(n,k) = n - k*A010766(n,k). - Mats Granvik, Gary W. Adamson, Feb 20 2010
G.f. for the k-th column: x^(k+1)*Sum_{i=0..k-2} (i + 1)*x^i/(1 - x^k). - Stefano Spezia, May 08 2024

More terms from David Wasserman, Oct 01 2008

A051127 Table T(n,k) = k mod n read by antidiagonals (n >= 1, k >= 1).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 1, 2, 1, 0, 0, 0, 2, 1, 0, 1, 1, 3, 2, 1, 0, 0, 2, 0, 3, 2, 1, 0, 1, 0, 1, 4, 3, 2, 1, 0, 0, 1, 2, 0, 4, 3, 2, 1, 0, 1, 2, 3, 1, 5, 4, 3, 2, 1, 0, 0, 0, 0, 2, 0, 5, 4, 3, 2, 1, 0, 1, 1, 1, 3, 1, 6, 5, 4, 3, 2, 1, 0, 0, 2, 2, 4, 2, 0, 6, 5, 4, 3, 2, 1, 0, 1, 0, 3, 0, 3, 1, 7, 6, 5, 4, 3, 2, 1

Offset: 1

N. J. A. Sloane

Note that the upper right half of this sequence when formatted as a square array is essentially the same as this whole sequence when formatted as an upper right triangle. Sums of antidiagonals are A004125. - Henry Bottomley, Jun 22 2001

			0  0  0  0  0  0  0  0  0  0 ...
1  0  1  0  1  0  1  0  1  0 ...
1  2  0  1  2  0  1  2  0  1 ...
1  2  3  0  1  2  3  0  1  2 ...
1  2  3  4  0  1  2  3  4  0 ...
1  2  3  4  5  0  1  2  3  4 ...
1  2  3  4  5  6  0  1  2  3 ...
1  2  3  4  5  6  7  0  1  2 ...
1  2  3  4  5  6  7  8  0  1 ...
1  2  3  4  5  6  7  8  9  0 ...
1  2  3  4  5  6  7  8  9 10 ...
1  2  3  4  5  6  7  8  9 10 ...
1  2  3  4  5  6  7  8  9 10 ...

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Transpose of A051126.
Cf. A048158, A051777, A122750.
Cf. A002260, A004125, A004736.

T[n_, m_] = Mod[n - m + 1, m + 1]; Table[Table[T[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%] (* Roger L. Bagula, Sep 04 2008 *)

T(n, k)=k%n \\ Charles R Greathouse IV, Feb 09 2017

As a linear array, the sequence is a(n) = A004736(n) mod A002260(n) or a(n) = ((t*t+3*t+4)/2-n) mod (n-(t*(t+1)/2)), where t = floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 17 2012

G.f. for the n-th row: y*Sum_{i=0..n-2} (i + 1)*y^i/(1 - y^n). - Stefano Spezia, May 08 2024

More terms from James Sellers, Dec 11 1999

A062347 a(n) = (product of first n primes) modulo prime(n+1).

Original entry on oeis.org

1, 2, 1, 2, 1, 9, 8, 18, 15, 17, 19, 11, 6, 26, 35, 27, 15, 55, 30, 24, 28, 4, 47, 56, 28, 28, 75, 25, 105, 30, 9, 52, 111, 89, 46, 132, 53, 97, 104, 106, 42, 94, 158, 184, 155, 5, 6, 45, 87, 78, 86, 26, 120, 33, 242, 66, 86, 2, 214, 2, 198, 127, 14, 112, 7, 141, 77, 309

Offset: 0

Henry Bottomley, Jul 06 2001

nonn

			a(4)=1 since 2*3*5*7 = 210 = 19*11 + 1.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Harry J. Smith)

Cf. A000040, A002110, A051126, A102647.

Mod[ #[ [ 1 ] ], #[ [ 2 ] ] ]&/@ Transpose[ {FoldList[ Times, 1, Prime[ Range[ 70 ] ] ], Prime[ Range[ 71 ] ]} ]
Join[{1},Module[{nn=70,prs},prs=Prime[Range[nn]];Table[Mod[Fold[Times,Take[prs,n-1]],prs[[n]]],{n,2,nn}]]] (* Harvey P. Dale, Jun 30 2024 *)

{ n=-1; f=1; forprime (p=2, prime(1001), write("b062347.txt", n++, " ", f%p); f*=p ) } \\ Harry J. Smith, Aug 05 2009

from sympy import sieve, primorial
print([1] + [primorial(k) % sieve[k+1] for k in range(1, 71)])
# Karl-Heinz Hofmann, Jan 26 2022

a(n) = A051126(A002110(n), A000040(n+1)).

A051778 Triangle read by rows, where row (n) = n mod (n-1), n mod (n-2), n mod (n-3), ...n mod 2.

Original entry on oeis.org

1, 1, 0, 1, 2, 1, 1, 2, 0, 0, 1, 2, 3, 1, 1, 1, 2, 3, 0, 2, 0, 1, 2, 3, 4, 1, 0, 1, 1, 2, 3, 4, 0, 2, 1, 0, 1, 2, 3, 4, 5, 1, 3, 2, 1, 1, 2, 3, 4, 5, 0, 2, 0, 0, 0, 1, 2, 3, 4, 5, 6, 1, 3, 1, 1, 1, 1, 2, 3, 4, 5, 6, 0, 2, 4, 2, 2, 0, 1, 2, 3, 4, 5, 6, 7, 1, 3, 0, 3, 0, 1, 1, 2, 3, 4, 5, 6, 7, 0, 2, 4, 1, 0, 1, 0

Offset: 3

Asher Auel, Dec 09 1999

Central terms: a(2*n+1,n) = n for n > 0. - Reinhard Zumkeller, Dec 03 2014

Deleting column 1 of the array at A051126 gives the array A051778 in square format (see Example). - Clark Kimberling, Feb 04 2016

			row (7) = 7 mod 6, 7 mod 5, 7 mod 4, 7 mod 3, 7 mod 2 = 1, 2, 3, 1, 1.
1;
1  0 ;
1  2  1 ;
1  2  0  0 ;
1  2  3  1  1 ;
1  2  3  0  2  0 ;
1  2  3  4  1  0  1 ;
1  2  3  4  0  2  1  0 ;
1  2  3  4  5  1  3  2  1 ;
1  2  3  4  5  0  2  0  0  0 ;
1  2  3  4  5  6  1  3  1  1  1 ;
Northwest corner of square array:
1 1 1 1 1 1 1 1 1 1 1
0 2 2 2 2 2 2 2 2 2 2
1 0 3 3 3 3 3 3 3 3 3
0 1 0 4 4 4 4 4 4 4 4
1 2 1 0 5 5 5 5 5 5 5
0 0 2 1 0 6 6 6 6 6 6
1 1 3 2 1 0 7 7 7 7 7
- _Clark Kimberling_, Feb 04 2016

Reinhard Zumkeller, Rows n=3..125 of triangle, flattened

Cf. A051777, A051127, A051126.

Cf. A004125 (row sums), A000027 (central terms), A049820 (number of nonzeros per row), A032741 (number of ones per row), A070824 (number of zeros per row).

a051778 n k = a051778_tabl !! (n-3) !! (k-1)
a051778_row n = a051778_tabl !! (n-3)
a051778_tabl = map (\xs -> map (mod (head xs + 1)) xs) $
                   iterate (\xs -> (head xs + 1) : xs) [2]
-- Reinhard Zumkeller, Dec 03 2014

Flatten[Table[Mod[n,i],{n,3,20},{i,n-1,2,-1}]] (* Harvey P. Dale, Sep 09 2012 *)
TableForm[Table[Mod[n, k], {n, 1, 12}, {k, 2, 12}]] (* square *)
(* Clark Kimberling, Feb 04 2016 *)

A372523 Triangle read by rows: T(n, k) is equal to n/k if k | n, else to the concatenation of A003988(n, k) = floor(n/k) and A051127(k, n) = n mod k.

Original entry on oeis.org

1, 2, 1, 3, 11, 1, 4, 2, 11, 1, 5, 21, 12, 11, 1, 6, 3, 2, 12, 11, 1, 7, 31, 21, 13, 12, 11, 1, 8, 4, 22, 2, 13, 12, 11, 1, 9, 41, 3, 21, 14, 13, 12, 11, 1, 10, 5, 31, 22, 2, 14, 13, 12, 11, 1, 11, 51, 32, 23, 21, 15, 14, 13, 12, 11, 1, 12, 6, 4, 3, 22, 2, 15, 14, 13, 12, 11, 1

Offset: 1

Stefano Spezia, May 04 2024

			The triangle begins:
  1;
  2,  1;
  3, 11,  1;
  4,  2, 11,  1;
  5, 21, 12, 11,  1;
  6,  3,  2, 12, 11,  1;
  7, 31, 21, 13, 12, 11, 1;
  ...

Stefano Spezia, First 150 rows of the triangle, flattened

Cf. A003988, A004216, A051127, A055642.
Cf. A000012 (right diagonal), A000027 (1st column).
Cf. A048158, A051126, A051127, A234575.

T[n_,k_]:=If[Divisible[n,k],n/k,FromDigits[Join[IntegerDigits[Floor[n/k]],IntegerDigits[Mod[n,k]]]]]; Table[T[n,k],{n,12},{k,n}]//Flatten (* or *)
T[n_,k_]:=Floor[n/k]10^IntegerLength[Mod[n,k]]+Mod[n,k]; Table[T[n,k],{n,12},{k,n}]//Flatten (* or *)
T[n_, k_]:=SeriesCoefficient[x^k(1+Sum[(i + 10^(1+Floor[Log10[Mod[n,k]]]))*x^i, {i, k-1}] - Sum[i*x^(k+i), {i, k-1}])/(1-x^k)^2, {x, 0, n}]; Table[T[n, k], {n, 12}, {k, n}]//Flatten

T(n, k) = floor(n/k)*10^(1+floor(log10(n mod k))) + (n mod k) if n is not divisible by k.
T(n, n) = 1.
T(n, 1) = n.
T(n, k) = 2*T(n-k, k) - T(n-2*k, k) for n >= 3*k.
T(n, k) = [x^n] x^k*(1 + (Sum_{i=1..k-1} (i + 10^(1+floor(log10(n mod k))))*x^i) - (Sum_{i=1..k-1} i*x^(k+i)))/(1 - x^k)^2.

A100990 a(n) = n^21 mod 100.

Original entry on oeis.org

0, 1, 52, 3, 4, 25, 56, 7, 8, 9, 0, 11, 12, 13, 64, 75, 16, 17, 68, 19, 0, 21, 72, 23, 24, 25, 76, 27, 28, 29, 0, 31, 32, 33, 84, 75, 36, 37, 88, 39, 0, 41, 92, 43, 44, 25, 96, 47, 48, 49, 0, 51, 52, 53, 4, 75, 56, 57, 8, 59, 0, 61, 12, 63, 64, 25, 16, 67, 68, 69, 0, 71, 72, 73, 24

Offset: 0

Henry Bottomley, Nov 25 2004

Also n^(20k+1) mod 100 for any positive integer k.
There are 63 numbers (A075821) where the final two digits of n^21, n^41, n^61, etc. are equal to n.
Period 100.

			a(11) = 11 since 11^21 = 7400249944258160101211 and the final two digits are 11.

Jianing Song, Table of n, a(n) for n = 0..99 (a period)

Cf. A010809, A051126, A075821.

[n^21 mod 100: n in [0..1000]]; // Vincenzo Librandi, Apr 21 2011

Table[Mod[n^21,100],{n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2011 *)
PowerMod[Range[0,80],21,100] (* Harvey P. Dale, Mar 15 2015 *)

a(n)=n^21%100 \\ Charles R Greathouse IV, Apr 06 2016

a(n) = A051126(A010809(n), 100) = a(n-100).

A309176 a(n) = n^2 * (n + 1)/2 - Sum_{k=1..n} sigma_2(k).

Original entry on oeis.org

0, 0, 2, 3, 12, 13, 33, 40, 66, 81, 135, 135, 212, 249, 319, 354, 489, 511, 681, 725, 876, 981, 1233, 1235, 1509, 1660, 1920, 2032, 2437, 2472, 2936, 3091, 3488, 3755, 4275, 4290, 4955, 5292, 5854, 6024, 6843, 6968, 7870, 8190, 8839, 9340, 10420, 10442, 11568, 12038, 13014, 13474, 14851, 15098, 16436

Offset: 1

Ilya Gutkovskiy, Jul 15 2019

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000326, A001157, A004125, A048158, A051126, A154585, A256532.

Cf. A002411, A064602.

Table[n^2 (n + 1)/2 - Sum[DivisorSigma[2, k], {k, 1, n}], {n, 1, 55}]
nmax = 55; CoefficientList[Series[x (1 + 2 x)/(1 - x)^4 - 1/(1 - x) Sum[k^2 x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[Mod[n, k] k, {k, 1, n}], {n, 1, 55}]

a(n) = n^2*(n+1)/2 - sum(k=1, n, sigma(k, 2)); \\ Michel Marcus, Sep 18 2021

from math import isqrt
def A309176(n): return (n**2*(n+1)>>1)+((s:=isqrt(n))**2*(s+1)*(2*s+1)-sum((q:=n//k)*(6*k**2+q*(2*q+3)+1) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 21 2023

G.f.: x * (1 + 2*x)/(1 - x)^4 - (1/(1 - x)) * Sum_{k>=1} k^2 * x^k/(1 - x^k).
a(n) = Sum_{k=1..n} (n mod k) * k.
a(n) = A002411(n) - A064602(n).

A329970 a(n) = (-1)^(n + 1) * n * ceiling(n/2) + Sum_{k=1..n} (-1)^k * k^2 * floor(n/k).

Original entry on oeis.org

0, 0, -2, 3, 0, -3, -7, 16, 2, -15, -21, 31, 24, -15, -57, 34, 25, -17, -27, 77, 8, -99, -111, 155, 117, -36, -140, 40, 25, -80, -96, 259, 112, -157, -249, 202, 183, -156, -354, 224, 203, -40, -62, 342, -21, -524, -548, 562, 488, -34, -358, 194, 167, -262

Offset: 1

Ilya Gutkovskiy, Nov 26 2019

sign

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A004125, A051126, A093005, A154585, A309176.

Table[(-1)^(n + 1) n Ceiling[n/2] + Sum[(-1)^k k^2 Floor[n/k], {k, 1, n}], {n, 1, 54}]
nmax = 54; CoefficientList[Series[x (1 - x + 2 x^2)/((1 - x)^2 (1 + x)^3) + 1/(1 - x) Sum[(-1)^k k^2 x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[(-1)^(k + 1) Mod[n, k] k, {k, 1, n}], {n, 1, 54}]

a(n) = (-1)^(n + 1)*n*ceil(n/2) + sum(k=1, n, (-1)^k * k^2 * (n\k)); \\ Michel Marcus, Sep 20 2021

G.f.: x * (1 - x + 2*x^2) / ((1 - x)^2 * (1 + x)^3) + (1/(1 - x)) * Sum_{k>=1} (-1)^k * k^2 * x^k / (1 - x^k).

a(n) = Sum_{k=1..n} (-1)^(k + 1) * (n mod k) * k.

A070705 LCM of first n prime powers modulo next prime power.

Original entry on oeis.org

2, 2, 2, 4, 4, 3, 1, 4, 8, 5, 14, 15, 5, 18, 1, 20, 16, 2, 15, 15, 8, 21, 29, 21, 16, 32, 29, 23, 22, 30, 54, 71, 37, 7, 37, 43, 45, 30, 36, 77, 100, 72, 64, 7, 56, 33, 42, 54, 132, 18, 90, 156, 91, 29, 86, 149, 139, 111, 112, 96, 62, 5, 204, 103, 41, 197, 81, 218, 128, 238, 58

Offset: 1

Lekraj Beedassy, May 15 2002

nonn

			The case n=7 implies a(7) = A051451(7) (mod A000961(8)) = lcm(2,3,4,5,7,8,9) (mod 11) = 2520 (mod 11) = 1.

Cf. A051451, A000961.

a(n) = A051451(n) (mod A000961(n+1)) = A051126(A051451(n), A000961(n-1)).

More terms from Don Reble, May 16 2002

cp's OEIS Frontend

A048158 Triangular array T read by rows: T(n,k) = n mod k, for k=1,2,...,n, n=1,2,...

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A051127 Table T(n,k) = k mod n read by antidiagonals (n >= 1, k >= 1).

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A062347 a(n) = (product of first n primes) modulo prime(n+1).

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A051778 Triangle read by rows, where row (n) = n mod (n-1), n mod (n-2), n mod (n-3), ...n mod 2.

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A372523 Triangle read by rows: T(n, k) is equal to n/k if k | n, else to the concatenation of A003988(n, k) = floor(n/k) and A051127(k, n) = n mod k.

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A100990 a(n) = n^21 mod 100.

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A309176 a(n) = n^2 * (n + 1)/2 - Sum_{k=1..n} sigma_2(k).

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