Nathaniel J. Strout - cp's OEIS frontend

A327439 a(0)=1. If a(n-1) and n are relatively prime and a(n-1)!=1, a(n) = a(n-1) - 1. Otherwise (i.e., if a(n-1) and n share a common factor or a(n-1)=1), a(n) = a(n-1) + gcd(a(n-1),n) + 1.

1, 3, 2, 1, 3, 2, 5, 4, 9, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 3, 2, 5, 4, 9, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 3, 2, 5, 4, 7, 6, 9, 8, 11, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11

Offset: 0

Nathaniel J. Strout, Feb 24 2020

Graphically at large scales this sequence is vaguely self-similar, though in certain sections it acts in a rough manner, in particular in regions surrounding apparent cusps. See the program for a graph to zoom in on these sections.

See Python program for zoomable graph.

Nathaniel J. Strout, Table of n, a(n) for n = 0..116000

a[0] = 1; a[n_] := a[n] = If[a[n - 1] != 1 && CoprimeQ[n, a[n - 1]], a[n - 1] - 1, a[n - 1] + GCD[a[n - 1], n] + 1]; Array[a, 101, 0] (* Amiram Eldar, Feb 24 2020 *)
nxt[{n_,a_}]:={n+1,If[CoprimeQ[n+1,a]&&a!=1,a-1,a+GCD[a,n+1]+1]}; NestList[nxt,{0,1},70][[;;,2]] (* Harvey P. Dale, Jun 08 2024 *)

import math
import matplotlib.pyplot as plt
num = 10000
x = []
y = []
# y is the main sequence
def sequence():
    a = 1
    y.append(a)
    for i in range(num):
        if (a != 1) and (math.gcd(a,i+1) == 1):
            a -= 1
        else:
            a += math.gcd(a,i+1)+1
        x.append(i)
        y.append(a)
    x.append(num)
sequence()
# code only regarding the plot.
plt.xlim(0,num)
plt.ylim(0,num)
plt.plot(x, y)
plt.xlabel('x - axis')
plt.ylabel('y - axis')
plt.title('Plot of Sequence')
plt.show()

A330190 Symmetric matrix read by antidiagonals: f(i,j) = 1 + gcd(f(i-1,j), f(i,j-1)), where f(1,j) and f(i,1) are 1.

Original entry on oeis.org

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

Offset: 1

Nathaniel J. Strout, Dec 04 2019

This matrix when displayed in a gray scale, from least to greatest, forms spikes of increasing numbers because large sections of the antidiagonals are the same number. See examples section.

			An example of a triangle described in the comment:
  ...........
  ...........
  ..........2
  ........2 3
  ......2 3 4
  ....2 3 4 5
Array begins:
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...
  1, 2, 3, 2, 3, 2, 3, 2, 3, 2, ...
  1, 2, 2, 3, 4, 3, 4, 3, 4, 3, ...
  1, 2, 3, 4, 5, 2, 3, 4, 5, 2, ...
  1, 2, 2, 3, 2, 3, 4, 5, 6, 3, ...
  1, 2, 3, 4, 3, 4, 5, 6, 7, 2, ...
  1, 2, 2, 3, 4, 5, 6, 7, 8, 3, ...
  1, 2, 3, 4, 5, 6, 7, 8, 9, 4, ...
  1, 2, 2, 3, 2, 3, 2, 3, 4, 5, ...
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (Rows n = 1..150, flattened)
Nathaniel J. Strout, 1000 X 1000 grid
Michael De Vlieger, 2048 X 2048 grid with color function where black = 1, red = 2 and magenta represents the maximum value in the grid (i.e., f(312,768) = f(768,312) = 41).

f[1, j_] := f[1, j] = 1; f[i_, 1] := f[i, 1] = 1; f[i_, j_] := f[i, j] = 1 + GCD[f[i - 1, j], f[i, j - 1]]; Table[f[m - k + 1, k], {m, 13}, {k, m, 1, -1}] // Flatten (* Michael De Vlieger, Aug 03 2022 *)

T(n)={my(M=matrix(n,n,i,j,1)); for(i=2, n, for(j=2, n, M[i,j] = 1 + gcd(M[i-1,j], M[i,j-1]))); M}
{ my(A=T(10)); for(i=1, #A, print(A[i,])) } \\ Andrew Howroyd, Jan 25 2020

A321297 a(n) is the sum of proper divisors of n that are >= sqrt(n) minus the sum of the divisors that are greater than 1 and less than sqrt(n).

Original entry on oeis.org

0, 0, 0, 2, 0, 1, 0, 2, 3, 3, 0, 5, 0, 5, 2, 10, 0, 10, 0, 9, 4, 9, 0, 17, 5, 11, 6, 15, 0, 21, 0, 18, 8, 15, 2, 36, 0, 17, 10, 27, 0, 31, 0, 27, 16, 21, 0, 45, 7, 28, 14, 33, 0, 43, 6, 37, 16, 27, 0, 67, 0, 29, 20, 50, 8, 55, 0, 45, 20, 45, 0, 76, 0, 35, 32

Offset: 1

Nathaniel J. Strout, Nov 02 2018

nonn

Cf. A000203 (sigma), A070038, A079667.

Array[Function[{d, s}, 1 + Subtract @@ (Total /@ Reverse@ TakeDrop[d, LengthWhile[d, # < s &]])] @@ {Most@ Divisors[#], Sqrt@ #} &, 74, 2] (* Michael De Vlieger, Nov 25 2018 *)

a(n)=sumdiv(n, d, if(d>1&&d=n/d, d, -d)))

a(p) = 0, for primes p.

a(n) = 2*A070038(n) - sigma(n) - n + 1 for n > 1. - Andrew Howroyd, Nov 25 2018

A304780 Consider a triangle whose first row is {1,2} and, for n > 1, has as its n-th row the integers k through 2k where k is the sum of the numbers in the (n-1)th row. Then a(n) is the first number in the n-th row.

Original entry on oeis.org

1, 3, 18, 513, 395523, 234658258578, 82596747478641253260993, 10233334041075645341729789249315281196742910563, 157081688394356396673208173772909833928515988895188885472258972148661958252271815996039831298

Offset: 1

Nathaniel J. Strout, May 18 2018

nonn

			Triangle begins:
  1, 2; (row sum = 3)
  3, 4, 5, 6; (row sum = 18)
  18, 19, 20, 21, ... 33, 34, 35, 36; (row sum = 513)
  513, 514, 515, 516, ..., 1023, 1024, 1025, 1026;
  ...

Cf. A000217.

RecurrenceTable[{a[1] == 1, a[n] == (3*a[n-1]*(a[n-1] + 1))/2}, a, {n, 1, 10}] (* Vaclav Kotesovec, Jul 23 2018 *)
Nest[Append[#, Range[#, 2 #] &@ Total@ Last@ #] &, {{1, 2}}, 3] // Flatten (* Michael De Vlieger, Jul 26 2018 *)

a(n) = if (n==1, 1, (3*a(n - 1)*(a(n - 1) + 1))/2); \\ Michel Marcus, May 24 2018

a(1) = 1, a(n) = (3*a(n-1)*(a(n-1) + 1))/2 for n > 1.

a(n) ~ (2/3) * c^(2^n), where c = 1.515006464529590220430714781603262955960312205695360166833... - Vaclav Kotesovec, Jul 23 2018

A296422 Primes that can be represented in the form b^n+1 or b^n-1 where b >= 2 and n >= 2.

Original entry on oeis.org

3, 5, 7, 17, 31, 37, 101, 127, 197, 257, 401, 577, 677, 1297, 1601, 2917, 3137, 4357, 5477, 7057, 8101, 8191, 8837, 12101, 13457, 14401, 15377, 15877, 16901, 17957, 21317, 22501, 24337, 25601, 28901, 30977, 32401, 33857, 41617, 42437, 44101, 50177, 52901

Offset: 1

Nathaniel J. Strout, Dec 12 2017

nonn

Union of A000668 and A121326. - Andrey Zabolotskiy, Dec 21 2017

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040 (primes), A001597 (perfect powers).

Cf. A000668 (Mersenne primes), A121326.

N:= 10^5: # to get terms <= N
R:= 3:
for b from 2 while b^2+1 <= N do
  p:= 2:
  do
    p:= nextprime(p);
    if b^p-1 > N then break fi;
    if isprime(b^p-1) then R:= R, b^p-1 fi;
  od:
  p:= 1:
  do
    p:= 2*p;
    if b^p+1 > N then break fi;
    if isprime(b^p+1) then R:= R, b^p+1 fi;
  od;
od:
sort(convert({R},list)); # Robert Israel, Jan 08 2018

Select[Prime@ Range[2, 10^4], AnyTrue[# + {-1, 1}, Or[# == 1, GCD @@ FactorInteger[#][[All, -1]] > 1] &] &] (* Michael De Vlieger, Dec 13 2017 *)

lista(nn) = {forprime(p=2, nn, if ((p==2) || ispower(p+1) || ispower(p-1), print1(p, ", ")); ); } \\ Michel Marcus, Dec 13 2017

cp's OEIS Frontend

User: Nathaniel J. Strout

A327439 a(0)=1. If a(n-1) and n are relatively prime and a(n-1)!=1, a(n) = a(n-1) - 1. Otherwise (i.e., if a(n-1) and n share a common factor or a(n-1)=1), a(n) = a(n-1) + gcd(a(n-1),n) + 1.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

Python

A330190 Symmetric matrix read by antidiagonals: f(i,j) = 1 + gcd(f(i-1,j), f(i,j-1)), where f(1,j) and f(i,1) are 1.

Views

Author

Keywords

Comments

Examples

Links

Programs

Mathematica

PARI

A321297 a(n) is the sum of proper divisors of n that are >= sqrt(n) minus the sum of the divisors that are greater than 1 and less than sqrt(n).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A304780 Consider a triangle whose first row is {1,2} and, for n > 1, has as its n-th row the integers k through 2k where k is the sum of the numbers in the (n-1)th row. Then a(n) is the first number in the n-th row.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A296422 Primes that can be represented in the form b^n+1 or b^n-1 where b >= 2 and n >= 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI