cp's OEIS Frontend

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.

Showing 1-5 of 5 results.

A269307 Consider the sum of the divisors of a number x>1. Take the sum of its digits. Repeat the process deleting the first addendum and adding the previous sum. The sequence lists the numbers that after some iterations reach x.

Original entry on oeis.org

17, 28, 31, 44, 51, 132, 133, 198, 208, 2528, 9241, 13570, 16577, 177568, 228742, 780889, 878078, 1854920, 2775787, 3663541, 8204010, 66326143, 73734437, 164211532, 670396359, 803230921, 832581731, 1036125551, 1572413223
Offset: 1

Views

Author

Paolo P. Lava, Feb 24 2016

Keywords

Comments

44 works in both directions: sigma(n) -> n and n -> sigma(n). See A269308.

Examples

			Sigma(17) = 18 :   1 + 8 = 9;  8 + 9 = 17.
Sigma(133) = 160 :  1 + 6 + 0 = 7;  6 + 0 + 7 = 13;  0 + 7 + 13 = 20; 7 + 13 + 20 = 40;  13 + 20 + 40 = 73;  20 + 40 + 73 = 133.

Crossrefs

Cf. A007629, A269308, A269309, A269310, A269311, A269312.

Programs

  • Maple
    with(numtheory): P:=proc(q,h) local a,b,k,n,t,v; v:=array(1..h);
for n from 2 to q do a:=sigma(n); b:=ilog10(a)+1; if b>1 then
for k from 1 to b do v[b-k+1]:=(a mod 10); a:=trunc(a/10); od; t:=b+1; v[t]:=add(v[k], k=1..b);
while v[t]
  • Mathematica
    Select[Range[2,10^5], (t = #; d = IntegerDigits[DivisorSigma[1, #]]; While[Total[d] < t, d = Join[Rest[d], {Total[d]}]]; Total[d] == t) &] (* Robert Price, May 21 2019 *)

Extensions

a(20)-a(29) from Lars Blomberg, Jan 18 2018

A269312 Consider a number x. Take the sum of its digits. Repeat the process deleting the first addendum and adding the previous sum. The sequence lists the numbers that after some iterations reach the arithmetic derivative of x.

Original entry on oeis.org

14, 51, 145, 285, 629, 708, 807, 1318, 2362, 2548, 2869, 3789, 4087, 4811, 6031, 6355, 10201, 15563, 17143, 17287, 17561, 19883, 20567, 21731, 22429, 23461, 26269, 27301, 30967, 33389, 69529, 73211, 85927, 86087, 90133, 96781, 110159, 116011, 159767, 161701, 162055, 190079
Offset: 1

Views

Author

Paolo P. Lava, Feb 24 2016

Keywords

Examples

			14’ = 9 : 1 + 4 = 5; 4 + 5 = 9.
51’ = 20 : 5 + 1 = 6; 1 + 6 = 7; 6 + 7  = 13; 7 + 13 = 20.

Links

Crossrefs

Cf. A003415, A007629, A269307, A269308, A269309, A269310, A269311.

Programs

  • Maple
    with(numtheory): P:=proc(q,h) local a,b,c,k,n,p,t,v; v:=array(1..h);
for n from 1 to q do a:=n; b:=ilog10(a)+1; if b>1 then
for k from 1 to b do v[b-k+1]:=(a mod 10); a:=trunc(a/10); od; t:=b+1; v[t]:=add(v[k], k=1..b);c:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
while v[t]
  • Mathematica
    dn[n_] := If[Abs@n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[Abs@n]]]; (* after Michael Somos,Apr 12 2011 *)
Select[Range[10^5], # >= 10 && (s = dn[#]; d = IntegerDigits[#]; While[Total[d] < s, d = Join[Rest[d], {Total[d]}]]; Total[d] == s) &] (* Robert Price, May 22 2019 *)

A269309 Consider the Euler totient function of a number x. Take the sum of its digits. Repeat the process deleting the first addendum and adding the previous sum. The sequence lists the numbers that after some iterations reach x.

Original entry on oeis.org

13, 21, 63, 177, 206, 412, 655, 824, 1990, 2637, 11438, 12485, 23846, 34132, 40528, 56202, 87539, 94597, 1288999, 3075239, 3106991, 20689503, 31173397, 46230492, 65889985, 79476719, 170859904, 266368503, 295845211, 420471958, 445169688, 797687940, 962257660
Offset: 1

Views

Author

Paolo P. Lava, Feb 24 2016

Keywords

Examples

			phi(13) = 12 : 1 + 2 = 3; 2 + 3 = 5; 3 + 5 = 8; 5 + 8 = 13.

Crossrefs

Cf. A000010, A007629, A269307, A269308, A269310, A269311, A269312.

Programs

  • Maple
    with(numtheory): P:=proc(q,h) local a,b,k,n,t,v; v:=array(1..h);
for n from 2 to q do a:=phi(n); b:=ilog10(a)+1; if b>1 then
for k from 1 to b do v[b-k+1]:=(a mod 10); a:=trunc(a/10); od; t:=b+1; v[t]:=add(v[k], k=1..b);
while v[t]
  • Mathematica
    Select[Range[10^5], EulerPhi[#] >= 10 && (d = IntegerDigits[EulerPhi[#]]; While[Total[d] < #, d = Join[Rest[d], {Total[d]}];]; Total[d] == #) &] (* Robert Price, May 21 2019 *)

Extensions

a(19)-a(33) from Lars Blomberg, Jan 18 2018

A269311 Consider the arithmetic derivative of a number x. Take the sum of its digits. Repeat the process deleting the first addendum and adding the previous sum. The sequence lists the numbers that after some iterations reach x.

Original entry on oeis.org

8, 18, 21, 160, 1590, 2420, 18620, 69229, 83790, 279964, 494520, 598810, 676450, 1183147, 4233720, 5600348, 14217074, 20025836, 64278677, 425208387, 604048830, 750851470, 1981942354
Offset: 1

Views

Author

Paolo P. Lava, Feb 24 2016

Keywords

Examples

			8’ = 12 : 1 + 2 = 3; 2 + 3 = 5; 3 + 5 = 8.
18’ = 21 : 2 + 1 = 3; 1 + 3 = 4; 3 + 4 = 7; 4 + 7 = 11; 7 + 11 = 18.

Crossrefs

Cf. A003415, A007629, A269307, A269308, A269309, A269310, A269312.

Programs

  • Maple
    with(numtheory): P:=proc(q,h) local a,b,k,p,t,v; global n; v:=array(1..h);
for n from 1 to q do a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]); b:=ilog10(a)+1; if b>1 then
for k from 1 to b do v[b-k+1]:=(a mod 10); a:=trunc(a/10); od; t:=b+1; v[t]:=add(v[k], k=1..b);
while v[t]
  • Mathematica
    dn[n_] := If[Abs@n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[Abs@n]]]; (* from Michael Somos, Apr 12 2011 *)
Select[Range[10^5], dn[#] >= 10 && (d = IntegerDigits[dn[#]]; While[Total[d] < #, d = Join[Rest[d], {Total[d]}]]; Total[d] == #) &] (* Robert Price, May 22 2019 *)

Extensions

a(19)-a(23) from Lars Blomberg, Jan 18 2018

A269308 Consider a number x. Take the sum of its digits. Repeat the process deleting the first addendum and adding the previous sum. The sequence lists the numbers that after some iterations reach the sum of the divisors of x.

Original entry on oeis.org

20, 25, 43, 44, 49, 59, 122, 206, 2485, 11899, 17608, 24141, 56207, 195236, 2424613, 2842925, 6241233, 59087970, 111205290, 124735931, 224269761, 1086241193
Offset: 1

Views

Author

Paolo P. Lava, Feb 24 2016

Keywords

Comments

44 works in both directions: n -> sigma(n) and sigma(n) -> n. See A269307.

Examples

			sigma(20) = 42 :  2 + 0 = 2; 0 + 2 = 2; 2 + 2 = 4; 2 + 4 = 6; 4 + 6 = 10; 6 + 10 = 16; 10 + 16 = 26; 16 +26 = 42.

Crossrefs

Cf. A000203, A007629, A269307, A269309, A269310, A269311, A269312.

Programs

  • Maple
    with(numtheory): P:=proc(q,h) local a,b,k,n,t,v; v:=array(1..h);
for n from 2 to q do a:=n; b:=ilog10(a)+1; if b>1 then
for k from 1 to b do v[b-k+1]:=(a mod 10); a:=trunc(a/10); od; t:=b+1; v[t]:=add(v[k], k=1..b);
while v[t]
  • Mathematica
    Select[Range[10,10^5], (s = DivisorSigma[1, #]; d = IntegerDigits[#]; While[Total[d] < s, d = Join[Rest[d], {Total[d]}]]; Total[d] == s) &] (* Robert Price, May 21 2019 *)

Extensions

a(16)-a(22) from Lars Blomberg, Jan 18 2018
Showing 1-5 of 5 results.

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