This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
Here are the first terms of the sequence: 100,1000,101,103,104,135,171,128,119,120,... and here are the cumulative sums: 100,1100,1201,1304,1408,1543,1714,1842,1961,2081,... If we align a(n) and its cumulative sum, we see that at least three digits are shared: 100,1000, 101, 103, 104 ,135, 171, 128, 119, 120,... 100,1100,1201,1304,1408,1543,1714,1842,1961,2081,...
Here are the first terms of the sequence: 1000,10000,1001,1003,1004,1035,1061,1127,1184,... and here are the cumulative sums: 1000,11000,12001,13004,14008,15043,16104,17231,18415,... If we align a(n) and its cumulative sum, we see that at least four digits are shared: 1000,10000, 1001, 1003, 1004, 1035 ,1061, 1127, 1184,... 1000,11000,12001,13004,14008,15043,16104,17231,18415,...
Here are the first terms of the sequence: 10000,100000,10001,10003,10004,10035,10061,10127,... and here are the cumulative sums: 10000,110000,120001,130004,140008,150043,160104,170231,... If we align a(n) and its cumulative sum, we see that at least five digits are shared: 10000,100000, 10001, 10003, 10004, 10035, 10061, 10127,... 10000,110000,120001,130004,140008,150043,160104,170231,...
Block[{a = {1, 10}, m = {1, 0}, k}, Do[k = 2; While[Nand[FreeQ[a, k], GCD[k, a[[-1]]] > 1, IntersectingQ[m, IntegerDigits[k]]], k++]; AppendTo[a, k]; Set[m, IntegerDigits[k]], {i, 73}]; a] (* Michael De Vlieger, Mar 11 2021 *)
from sympy import factorint
def aupton(terms):
alst, aset = [1, 10], {1, 10}
for n in range(3, terms+1):
an = 1
anm1_digs, anm1_factors = set(str(alst[-1])), set(factorint(alst[-1]))
while True:
while an in aset: an += 1
if set(str(an)) & anm1_digs != set():
if set(factorint(an)) & anm1_factors != set():
alst.append(an); aset.add(an); break
an += 1
return alst
print(aupton(75)) # Michael S. Branicky, Mar 09 2021
Block[{a = {1, 2}, m = {2}, k}, Do[k = 2; While[Nand[FreeQ[a, k], GCD[k, a[[-1]]] > 1, ! IntersectingQ[m, IntegerDigits[k]]], k++]; AppendTo[a, k]; Set[m, IntegerDigits[k]], {i, 74}]; a] (* Michael De Vlieger, Mar 11 2021 *)
from sympy import factorint
def aupton(terms):
alst, aset = [1, 2], {1, 2}
for n in range(3, terms+1):
an = 1
anm1_digs, anm1_factors = set(str(alst[-1])), set(factorint(alst[-1]))
while True:
while an in aset: an += 1
if set(str(an)) & anm1_digs == set():
if set(factorint(an)) & anm1_factors != set():
alst.append(an); aset.add(an); break
an += 1
return alst
print(aupton(76)) # Michael S. Branicky, Mar 09 2021
Block[{a = {1}, m = {1}, k}, Do[k = 2; While[Nand[FreeQ[a, k], GCD[k, a[[-1]]] == 1, IntersectingQ[m, IntegerDigits[k]]], k++]; AppendTo[a, k]; Set[m, IntegerDigits[k]], {i, 74}]; a] (* Michael De Vlieger, Mar 11 2021 *)
from sympy import factorint
def aupton(terms):
alst, aset = [1], {1}
for n in range(2, terms+1):
an = 1
anm1_digs, anm1_factors = set(str(alst[-1])), set(factorint(alst[-1]))
while True:
while an in aset: an += 1
if set(str(an)) & anm1_digs != set():
if set(factorint(an)) & anm1_factors == set():
alst.append(an); aset.add(an); break
an += 1
return alst
print(aupton(74)) # Michael S. Branicky, Mar 09 2021
I:=[1,6,4]; [n le 3 select I[n] else - Self(n-1) + 10*Self(n-2) + 10*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 20 2019
1, seq(op([6*(10^i-1)/9, 3*(10^i-1)/9+1]), i=1..30); # Robert Israel, Jul 15 2019
CoefficientList[Series[(1 + 7 x)/((1 + x) (1 - 10 x^2)), {x, 0, 26}], x] (* Michael De Vlieger, Jul 18 2019 *)
LinearRecurrence[{-1,10,10},{1,6,4},30] (* Harvey P. Dale, Jan 02 2022 *)
Vec((1+7*x)/((1+x)*(1-10*x^2)) + O(x^20)) \\ Felix Fröhlich, Jul 15 2019
a(n) = if(n==1, 1, if(n%2==0, 6*(10^(n/2)-1)/9, 3*(10^((n-1)/2)-1)/9+1)) \\ Felix Fröhlich, Jul 15 2019
a(1) = 91 a(1) + a(2) = 101 a(1) + a(2) + a(3) = 102 a(1) + a(2) + a(3) + a(4) = 204 a(1) + a(2) + a(3) + a(4) + a(5) = 224 a(1) + a(2) + a(3) + a(4) + a(5) + a(6) = 228; etc. The succession of the above results is: 91, 101, 102, 204, 224, 228, ... The first terms of the sequence are: 91, 10, 1, 102, 20, 4, 2, 24, 22, 8, ... We see that the successive digits are the same in the two sequences.
Nest[(a=#;AppendTo[a,(new=Flatten[IntegerDigits/@Accumulate@#][[Length@Flatten[IntegerDigits/@a]+1;;]];
k=1;While[MemberQ[a,FromDigits@new[[;;k]]]||new[[k+1]]==0,k++];FromDigits@new[[;;k]])])&,{91,10},77] (* Giorgos Kalogeropoulos, Aug 05 2023 *)
a(1) = 1 a(1) * a(2) = 11 a(1) * a(2) * a(3) = 22 a(1) * a(2) * a(3) * a(4) = 550 a(1) * a(2) * a(3) * a(4) * a(5) = 27500 a(1) * a(2) * a(3) * a(4) * a(5) * a(6) = 742500; etc. The succession of the above results is: 1, 11, 22, 550, 27500, 742500, ... The first terms of the sequence are: 1, 11, 2, 25, 50, 27, 500, 7, 4, 2500,, ... We see that the successive digits are the same in the two sequences.
Nest[(a=#;AppendTo[a,(new=Flatten[IntegerDigits/@Table[Times@@a[[;;i]],{i,Length@a}]][[Length@Flatten[IntegerDigits/@a]+1;;]];
k=1;While[MemberQ[a,FromDigits@new[[;;k]]]||new[[k+1]]==0,k++];FromDigits@new[[;;k]])])&,{1,11,2,25},45] (* Giorgos Kalogeropoulos, Aug 05 2023 *)
Comments