A050873 -id:A050873 - cp's OEIS frontend

A384655 a(n) = Sum_{k=1..n} A051903(gcd(n,k)).

0, 1, 1, 3, 1, 4, 1, 7, 4, 6, 1, 11, 1, 8, 7, 15, 1, 14, 1, 17, 9, 12, 1, 25, 6, 14, 13, 23, 1, 22, 1, 31, 13, 18, 11, 36, 1, 20, 15, 39, 1, 30, 1, 35, 26, 24, 1, 53, 8, 32, 19, 41, 1, 44, 15, 53, 21, 30, 1, 59, 1, 32, 34, 63, 17, 46, 1, 53, 25, 46, 1, 81, 1, 38

Offset: 1

Amiram Eldar, Jun 06 2025

nonn

The terms of this sequence can be calculated efficiently using the 1st formula. The value the of function f(n, k) is equal to the number of integers i from 1 to n such that gcd(i, n) is 1 if k = 1, or k-free if k >= 2 (k-free numbers are numbers that are not divisible by a k-th power other than 1). E.g., f(n, 1) = A000010(n), f(n, 2) = A063659(n), and f(n, 3) = A254926(n).

			a(4) = A051903(gcd(4,1)) + A051903(gcd(4,2)) + A051903(gcd(4,3)) + A051903(gcd(4,4)) = A051903(1) + A051903(2) + A051903(1) + A051903(4) = 0 + 1 + 0 + 2 = 3.

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

Cf. A000010, A033150, A005117, A050873, A051903, A051953, A063659, A067259, A254926, A357310, A383156, A384656 (unitary analog).

e[n_] := If[n == 1, 0, Max[FactorInteger[n][[;;, 2]]]]; a[n_] := Sum[e[GCD[n, k]], {k, 1, n}]; Array[a, 100]
(* or *)
f[p_, e_, k_] := p^e - If[e < k, 0, p^(e - k)]; a[n_] := Module[{fct = FactorInteger[n], emax, s}, emax = Max[fct[[;; , 2]]]; s = emax * n; Do[s -= Times @@ (f[#1, #2, k] & @@@ fct), {k, 1, emax}]; s]; a[1] = 0; Array[a, 100]

e(n) = if(n == 1, 0, vecmax(factor(n)[,2]));
a(n) = sum(k = 1, n, e(gcd(n, k)));

a(n) = if(n == 1, 0, my(f = factor(n), p = f[,1], e = f[,2], emax = vecmax(e), s = emax*n); for(k = 1, emax, s -= prod(i = 1, #p, p[i]^e[i] - if(e[i] < k, 0, p[i]^(e[i]-k)))); s);

a(n) = Sum_{k=1..A051903(n)} (n - f(n, k)) = A051903(n) * n  - Sum_{k=1..A051903(n)} f(n, k), where f(n, k) is multiplicative for a given k, with f(p^e, k) = p^e - p^(e-k) if e >= k and f(p^e, k) = p^e if e < k.
a(n) = 1 if and only if n is prime.
a(n) >= 2 if and only if n is composite.
a(n) >= A051953(n) with equality if and only if n is squarefree.
a(n) >= 2*n - A000010(n) - A063659(n) with equality if and only if n is cubefree that is not squarefree (i.e., n in A067259, or equivalently, A051903(n) = 2).
a(p^e) = (p^e-1)/(p-1) for a prime p and e >= 1.
a(n) < c*n and lim sun_{n->oo} a(n)/n = c, where c is Niven's constant (A033150).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Sum{k>=1} (1-1/zeta(2*k)) = 0.49056393035179738598... .

A167192 Triangle read by rows: T(n,k) = (n-k)/gcd(n,k), 1 <= k <= n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Oct 30 2009

			The triangle T(n,k) begins:
n\k   1   2   3   4  5  6  7  8  9 10 11 12 13  14  15 ...
1:    0
2:    1   0
3:    2   1   0
4:    3   1   1   0
5:    4   3   2   1  0
6:    5   2   1   1  1  0
7:    6   5   4   3  2  1  0
8:    7   3   5   1  3  1  1  0
9:    8   7   2   5  4  1  2  1  0
10:   9   4   7   3  1  2  3  1  1  0
11:  10   9   8   7  6  5  4  3  2  1  0
12:  11   5   3   2  7  1  5  1  1  1  1  0
13:  12  11  10   9  8  7  6  5  4  3  2  1  0
14:  13   6  11   5  9  4  1  3  5  2  3  1  1   0
15:  14  13   4  11  2  3  8  7  2  1  4  1  2   1   0
- _Wolfdieter Lang_, Feb 20 2013

Indranil Ghosh, Rows 1..120 of triangle, flattened

Cf. A054531, A112543, A164306.

Flatten[Table[(n-k)/GCD[n,k],{n,20},{k,n}]] (* Harvey P. Dale, Nov 27 2015 *)

for(n=1,10, for(k=1,n, print1((n-k)/gcd(n,k), ", "))) \\ G. C. Greubel, Sep 13 2017

T(n,k) = (n-k)/gcd(n,k), 1 <= k <= n.
T(n,k) = A025581(n,k)/A050873(n,k);
T(n,1) = A001477(n-1);
T(n,2) = A026741(n-2) for n > 1;
T(n,3) = A051176(n-3) for n > 2;
T(n,4) = A060819(n-4) for n > 4;
T(n,n-3) = A144437(n) for n > 3;
T(n,n-2) = A000034(n) for n > 2;
T(n,n-1) = A000012(n);
T(n,n) = A000004(n).

A186711 Greatest common divisor of the n-th and (n+1)st 3-smooth numbers.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 4, 2, 6, 3, 1, 4, 12, 6, 2, 8, 9, 3, 12, 4, 16, 18, 6, 24, 27, 1, 32, 36, 12, 48, 54, 2, 64, 72, 81, 3, 96, 108, 4, 128, 144, 162, 6, 192, 216, 8, 1, 9, 288, 324, 12, 384, 432, 16, 2, 18, 576, 648, 24, 3, 27, 864, 32, 4, 36, 1152, 1296, 48, 6, 54, 1728, 64, 8, 72, 9, 81, 2592, 96, 12

Offset: 1

Reinhard Zumkeller, Feb 26 2011

nonn

A186712 shows where this function and the 3-smooth numbers A003586 are in the same range: a(A186712(n)) = A003586(n) and a(m) != A003586(n) for m < A186712(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A003586, A050873, A061987, A186771.

a186711 n = a186711_list !! (n-1)
a186711_list = zipWith gcd a003586_list $ tail a003586_list

A186711 := proc(n) igcd(A003586(n),A003586(n+1)) ; end proc: # R. J. Mathar, Feb 28 2011

S3 = Select[Range[3*10^4], FactorInteger[#][[-1, 1]] <= 3&]; Table[GCD[ S3[[n]], S3[[n+1]] ], {n, 1, Length[S3]-1}] (* Jean-François Alcover, Feb 02 2018 *)

a(n) = A050873(A003586(n+1), A003586(n)).

a(A186771(n)) = 1.

A384039 The number of integers k from 1 to n such that gcd(n,k) is a powerful number.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 6, 6, 7, 4, 10, 6, 12, 6, 8, 12, 16, 7, 18, 12, 12, 10, 22, 12, 21, 12, 21, 18, 28, 8, 30, 24, 20, 16, 24, 21, 36, 18, 24, 24, 40, 12, 42, 30, 28, 22, 46, 24, 43, 21, 32, 36, 52, 21, 40, 36, 36, 28, 58, 24, 60, 30, 42, 48, 48, 20, 66, 48, 44

Offset: 1

Amiram Eldar, May 18 2025

The number of integers k from 1 to n such that the powerfree part (A055231) of gcd(n,k) is 1.

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

Cf. A000010, A001694, A005117, A050873, A055231, A063658, A206369.

The number of integers k from 1 to n such that gcd(n,k) is: A026741 (odd), A062570 (power of 2), A063659 (squarefree), A078429 (cube), A116512 (power of a prime), A117494 (prime), A126246 (1 or 2), A206369 (square), A254926 (cubefree), A372671 (3-smooth), this sequence (powerful), A384040 (cubefull), A384041 (exponentially odd), A384042 (5-rough).

f[p_, e_] := If[e == 1, p-1, (p^2-p+1)*p^(e-2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, f[i,1]-1, (f[i,1]^2-f[i,1]+1)*f[i,1]^(f[i,2]-2)));}

Multiplicative with a(p^e) = (p^2-p+1)*p^(e-2) if e >= 2, and p-1 otherwise.
a(n) >= A000010(n), with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^s + 1/p^(2*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 1/p^4) = 0.66922021803510257394... .

A074712 Number of (interiors of) cells touched by a diagonal in a regular n X k grid (enumerated antidiagonally).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 4, 4, 4, 5, 4, 3, 4, 5, 6, 6, 6, 6, 6, 6, 7, 6, 7, 4, 7, 6, 7, 8, 8, 6, 8, 8, 6, 8, 8, 9, 8, 9, 8, 5, 8, 9, 8, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 9, 8, 11, 6, 11, 8, 9, 10, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Offset: 1

Jens Voß, Sep 04 2002

From Yifan Xie, Nov 17 2024: (Start)
A(n, k) is the minimum sum of side lengths of squares that exactly cover a n X k rectangle.
A(n, k) is the minimum number of nonzero elements of a n X k matrix such that the sum of each row is n, and the sum of each column is k.
(End)

			The square array A(n,k) (n >= 1, k >= 1) begins:
  1 2  3  4  5  6  7  8
  2 2  4  4  6  6  8  8
  3 4  3  6  7  6  9 10
  4 4  6  4  8  8 10  8
  5 6  7  8  5 10 11 12
  6 6  6  8 10  6 12 12
  7 8  9 10 11 12  7 14
  8 8 10  8 12 12 14  8
  ...
From _Seiichi Manyama_, Apr 05 2025: (Start)
The triangle T(n,k) (1 <= k <= n) begins:
   1;
   2,  2;
   3,  2,  3;
   4,  4,  4,  4;
   5,  4,  3,  4,  5;
   6,  6,  6,  6,  6,  6;
   7,  6,  7,  4,  7,  6,  7;
   8,  8,  6,  8,  8,  6,  8,  8;
   9,  8,  9,  8,  5,  8,  9,  8,  9;
  10, 10, 10, 10, 10, 10, 10, 10, 10, 10;
  ... (End)

Nathaniel Johnston, First 150 antidiagonals, flattened
Micky Bullock, The Diagonal Problem (2 dimensions).
Alberto L. Delgado's Problem of the Week No. 145.

Cf. A050873, A199408.

A074712 := proc(m,n) local d: d:=gcd(m,n): if(d=1)then return m+n-1: else return d*procname(m/d,n/d): fi: end: seq(seq(A074712(n-d+1,d),d=1..n),n=1..8); # Nathaniel Johnston, May 09 2011

A[m_,n_]=m+n-GCD[m,n];Table[A[m,s-m],{s,2,10},{m,1,s-1}]//Flatten (* Luc Rousseau, Sep 16 2017 *)

(A(n,k)=n+k-gcd(n,k));for(s=2,10,for(n=1,s-1,k=s-n;print1(A(n,k),", "))) \\ Luc Rousseau, Sep 16 2017

A(n, k) = n + k - 1 if n and k are coprime; A(n, k) = d * A(n/d, k/d) where d is the greatest common divisor of n and k, otherwise.
A(n, k) = n + k - gcd(n, k). - Luc Rousseau, Sep 15 2017
T(n,k) = A(k,n-k+1) = n+1 - A050873(n+1,k). - Seiichi Manyama, Apr 05 2025

A127185 Triangle of distances between n>=1 and n>=m>=1 measured by the number of non-common prime factors.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Mar 25 2007

Consider the non-directed graph where each integer n >= 1 is a unique node labeled by n and where nodes n and m are connected if their list of exponents in their prime number decompositions n=p_1^n_1*p_2^n_2*... and m=p_1^m_1*p_2^m_2*... differs at one place p_i by 1. [So connectedness means n/m or m/n is a prime.] The distance between two nodes is defined by the number of hops on the shortest path between them. [Actually, the shortest path is not unique if the graph is not pruned to a tree by an additional convention like connecting only numbers that differ in the exponent of the largest prime factors; this does not change the distance here.] The formula says this can be computed by passing by the node of the greatest common divisor.

			T(8,10)=T(2^3,2*5)=3 as one must lower the power of p_1=2 two times and rise the power of p_3=5 once to move from 8 to 10. A shortest path is 8<->4<->2<->10 obtained by division through 2, division through 2 and multiplication by 5.
Triangle is read by rows and starts
   n\m 1 2 3 4 5 6 7 8 9 10
   ------------------------
    1| 0
    2| 1 0
    3| 1 2 0
    4| 2 1 3 0
    5| 1 2 2 3 0
    6| 2 1 1 2 3 0
    7| 1 2 2 3 2 3 0
    8| 3 2 4 1 4 3 4 0
    9| 2 3 1 4 3 2 3 5 0
   10| 2 1 3 2 1 2 3 3 4 0

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened)
Diego Dominici, An Arithmetic Metric, arXiv:0906.0632 [math.NT], 2009.

Cf. A130836.

t[n_, n_] = 0; t[n_, 1] := PrimeOmega[n]; t[n_, m_] := With[{g = GCD[n, m]}, PrimeOmega[n/g] + PrimeOmega[m/g]]; Table[t[n, m], {n, 1, 14}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jan 08 2014 *)

T(n, k) = my(g=gcd(n,k)); bigomega(n/g) + bigomega(k/g);
tabl(nn) = for(n=1, nn, for (k=1, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Dec 26 2018

A127185(m,n)=vecsum(abs(factor(m/n)[, 2])) \\ M. F. Hasler, Dec 07 2019

T(n,m) = A001222(n/g)+A001222(m/g) where g=gcd(n,m)=A050873(n,m).
Special cases: T(n,n)=0. T(n,1)=A001222(n).
T(m,n) = A130836(m,n) = Sum |e_k| if m/n = Product p_k^e_k. - M. F. Hasler, Dec 08 2019

A143051 Smallest number not occurring earlier and smaller than the largest square so far, the next square if no such number exists.

Original entry on oeis.org

0, 1, 4, 2, 3, 9, 5, 6, 7, 8, 16, 10, 11, 12, 13, 14, 15, 25, 17, 18, 19, 20, 21, 22, 23, 24, 36, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 49, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 64, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 81, 65, 66, 67, 68, 69, 70, 71

Offset: 0

Reinhard Zumkeller, Jul 20 2008

nonn

Permutation of the natural numbers, inverse: A143052;
A143053(n) = a(a(n));
a(A000290(n)) = A005563(n-1);
a(A002522(n)) = A000290(n+1);
for n>1: GCD(a(n^2),a(n^2+1)) = A050873(A000290(n),A002522(n)) =
A022998(n+1).

a(n) = if n=0 then 0 else if n=1+k^2 then n+2*k else n-1.

A130335 Smallest k > 0 such that gcd(n(n+1)/2, (n+k)(n+k+1)/2) = 1.

Original entry on oeis.org

1, 2, 7, 2, 2, 4, 2, 2, 4, 2, 2, 10, 2, 2, 7, 2, 2, 4, 2, 2, 4, 2, 2, 13, 2, 2, 10, 2, 2, 7, 2, 2, 4, 2, 2, 10, 2, 2, 7, 2, 2, 4, 2, 2, 7, 2, 2, 10, 2, 2, 7, 2, 2, 4, 2, 2, 4, 2, 2, 13, 2, 2, 10, 2, 2, 4, 2, 2, 4, 2, 2, 10, 2, 2, 7, 2, 2, 4, 2, 2, 4, 2, 2, 22, 2, 2, 7, 2, 2, 16, 2, 2, 4, 2, 2, 10, 2, 2, 7, 2

Offset: 1

Reinhard Zumkeller, May 28 2007

nonn

First occurrence of 3k+1, k=0.. or 0 if unknown, limit = 2^31: 1, 6, 3, 12, 24, 90, 231, 84, 792, 0, 195, 3432, 780, 0, 3255, 6075, 73644, 51482970, 0, 924, 183540, 0, 45219, 0, 509124, 3842375445, 29259, 71484, 0, 0, 0, 2311539, 238547880, 0, 55380135, 893907420, 23303784, 0, 0, 208260975, 0, 0, 1744264599, 0, 0, 0, 1487657079, 665710275, 0, 0, 1963994955, 0, 319589424, 0, 0, 0, 4181294964, 0, 0, 383229924, ..., . - Robert G. Wilson v, Jun 03 2007

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000217, A050873.
Cf. A001651, A008585.
Cf. A130334.
See A130336 and A130337 for record values and where they occur.

f[n_] := Block[{k = If[ n == 1 || Mod[n, 3] == 0, 1, 2]}, While[ GCD[n(n + 1)/2, (n + k)(n + k + 1)/2] != 1, k += 3 ]; k]; Array[f, 100] (* Robert G. Wilson v, Jun 03 2007 *)

a(n) = my(k=1); while (gcd(n*(n+1)/2, (n+k)*(n+k+1)/2) != 1, k++); k;

from math import gcd
def A130335(n):
    k, Tn, Tm = 1, n*(n+1)//2, (n+1)*(n+2)//2
    while gcd(Tn,Tm) != 1:
        k += 1
        Tm += k+n
    return k # Chai Wah Wu, Sep 16 2021

a(n) = Min{k>0: A050873(A000217(n+k),A000217(n))=1};
a(n) = A130334(n) - n;
a(n) > 1 for n>1; a(n) > 2 iff n mod 3 = 0: a(A001651(n))=2, a(A008585(n)) > 2 for n > 1.
a(n) == 1 (mod 3) if a(n) != 2. - Robert G. Wilson v, Jun 03 2007

A132442 Triangle whose n-th row consists of the first n terms of the n-th row of A134866.

Original entry on oeis.org

1, 1, 3, 1, 1, 4, 1, 3, 1, 7, 1, 1, 1, 1, 6, 1, 3, 4, 3, 1, 12, 1, 1, 1, 1, 1, 1, 8, 1, 3, 1, 7, 1, 3, 1, 15, 1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 3, 1, 3, 6, 3, 1, 3, 1, 18, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 3, 4, 7, 1, 12, 1, 7, 4, 3, 1, 28, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 14, 1, 3, 1, 3, 1, 3, 8, 3, 1, 3, 1, 3, 1, 24

Offset: 1

Gary W. Adamson, Nov 14 2007

Previous name: Triangle, n-th row = first n terms of n-th row of an array formed by A051731 * A127093 (transform).
Right border = sigma(n), A000203.
Row sums = A038040.
The function T(n,k) = T(k,n) is defined for k > n, but only the values of k in 1..n as a triangular array are listed here.

			First few rows of the A134866 array:
  1,  1,  1,  1,  1,  1,  1, ...
  1,  3,  1,  3,  1,  3,  1, ...
  1,  1,  4,  1,  1,  4,  1, ...
  1,  3,  1,  7,  1,  3,  1, ...
  1,  1,  1,  1,  6,  1,  1, ...
  1,  3,  4,  3,  1, 12,  1, ...
  ...
First few rows of the triangle:
  1;
  1,  3;
  1,  1,  4;
  1,  3,  1,  7;
  1,  1,  1,  1,  6;
  1,  3,  4,  3,  1, 12;
  1,  1,  1,  1,  1,  1,  8;
  1,  3,  1,  7,  1,  3,  1, 15;
  ...

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A051731, A127093.
Cf. A038040 (row sums), A000203 (right border), A050873 (gcd(n,k)).
Cf. A000142 (determinant).
Cf. A134866.

a132442 n k = a132442_tabl !! (n-1) !! (k-1)
a132442_row n = a132442_tabl !! (n-1)
a132442_tabl = map (map a000203) a050873_tabl
-- Reinhard Zumkeller, Dec 12 2015

T[ n_, k_] := If[ n < 1 || k < 1, 0, If[ k > n, T[ k, n], If[ k == 1, 1, If[ n > k, T[ k, Mod[ n, k, 1]],  DivisorSigma [1, n]]]]] (* Michael Somos, Jul 18 2011 *)

{T(n, k) = if( n<1 || k<1, 0, if( k>n, T(k, n), if( k==1, 1, if( n>k, T(k, (n-1)%k+1), sigma( n)))))} /* Michael Somos, Jul 18 2011 */

T(n,k) = A000203(gcd(n,k)). - Reinhard Zumkeller, Dec 12 2015

Missing T(10,9) = 1 inserted by Reinhard Zumkeller, Dec 12 2015

Name edited by Michel Marcus, Dec 21 2022

A134866 Table read by antidiagonals: T(n,k) = sigma(gcd(n,k)).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 7, 1, 3, 1, 1, 1, 4, 1, 1, 4, 1, 1, 1, 3, 1, 3, 6, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 7, 1, 12, 1, 7, 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 8, 3, 1, 3, 1, 3, 1

Offset: 1

Gary W. Adamson, Nov 14 2007

Previous name was: Triangle, antidiagonals of an array formed by A051731 * A127093 (transform).

Row sums give A094471.

			First few rows of the array:
  1, 1, 1, 1, 1, 1, 1, ...
  1, 3, 1, 3, 1, 3, 1, ...
  1, 1, 4, 1, 1, 4, 1, ...
  1, 3, 1, 7, 1, 3, 1, ...
  1, 1, 1, 1, 6, 1, 1, ...
  ...
First antidiagonals:
  1;
  1, 1;
  1, 3, 1;
  1, 1, 1, 1;
  1, 3, 4, 3, 1;
  1, 1, 1, 1, 1, 1;
  1, 3, 1, 7, 1, 3, 1;
  1, 1, 4, 1, 1, 4, 1, 1;
  1, 3, 1, 3, 6, 3, 1, 3, 1;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened)

Cf. A051731, A127093, A094471, A132442.

Table[DivisorSigma[1, GCD[#, k]] &[n - k + 1], {n, 13}, {k, n}] // Flatten (* Michael De Vlieger, Dec 19 2022 *)

T(n, k) = sigma(gcd(n, k)); \\ Michel Marcus, Dec 19 2022

T(n,k) = A000203(A050873(n,k)). - Michel Marcus, Dec 19 2022

New name and data corrected by Michel Marcus, Dec 19 2022

cp's OEIS Frontend

A384655 a(n) = Sum_{k=1..n} A051903(gcd(n,k)).

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A167192 Triangle read by rows: T(n,k) = (n-k)/gcd(n,k), 1 <= k <= n.

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A186711 Greatest common divisor of the n-th and (n+1)st 3-smooth numbers.

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A384039 The number of integers k from 1 to n such that gcd(n,k) is a powerful number.

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A074712 Number of (interiors of) cells touched by a diagonal in a regular n X k grid (enumerated antidiagonally).

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A127185 Triangle of distances between n>=1 and n>=m>=1 measured by the number of non-common prime factors.

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A143051 Smallest number not occurring earlier and smaller than the largest square so far, the next square if no such number exists.

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A130335 Smallest k > 0 such that gcd(n(n+1)/2, (n+k)(n+k+1)/2) = 1.