A269309 -id:A269309 - cp's OEIS frontend

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.

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

Paolo P. Lava, Feb 24 2016

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

			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.

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

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]

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 *)

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

Paolo P. Lava, Feb 24 2016

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

Lars Blomberg, Table of n, a(n) for n = 1..663

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

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]

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 *)

A269310 Consider 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 the Euler totient function of x.

Original entry on oeis.org

22, 44, 55, 88, 98, 136, 162, 166, 241, 462, 1020, 2040, 2416, 2899, 3060, 4080, 5110, 7942, 10738, 10996, 15006, 24822, 57040, 67054, 70625, 75588, 96888, 261524, 301834, 507471, 735840, 816584, 2893877, 6081064, 8155616, 16513570, 18772258, 40833543

Offset: 1

Paolo P. Lava, Feb 24 2016

			phi(22) = 10: 2 + 2 = 4; 2 + 4 = 6; 4 + 6 = 10.

Lars Blomberg, Table of n, a(n) for n = 1..47

Cf. A000010, A007629, A269307, A269308, A269309, A269311, A269312.

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]

Select[Range[2,10^5], (t = EulerPhi[#]; d = IntegerDigits[#]; While[Total[d] < t, d = Join[Rest[d], {Total[d]}]]; Total[d] == t) &] (* Robert Price, May 17 2019 *)

a(38) 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

Paolo P. Lava, Feb 24 2016

			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.

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

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]

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 *)

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

Paolo P. Lava, Feb 24 2016

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

			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.

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

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]

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 *)

a(16)-a(22) from Lars Blomberg, Jan 18 2018

A289868 Consider the digit reverse of a number x. Take the sum of these digits and repeat the process deleting the first addend and adding the previous sum. The sequence lists the numbers that after some iterations reach a sum equal to x.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 17, 21, 25, 42, 63, 84, 143, 286, 2355, 5821, 6618, 11709, 12482, 33747, 39571, 129109, 466957, 1162248, 1565166, 1968084, 3636638, 3853951, 4898376, 13443745, 13933175, 17118698, 22421197, 24153462, 147440984, 209989875, 245742153

Offset: 0

Paolo P. Lava, Jul 14 2017

Numbers of iterations for the listed terms are  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 5, 4, 5, 5, 5, 6, 6, 9, 10, 10, 11, 11, 12, 12, 14, 15, 17, 17, 17, 18, 18, 18, 20, 20, 20, 21, 21, 23, 23, 24.
From David A. Corneth, Jul 20 2017: (Start)
If n is in the sequence and its highest digit is m then n * (10\m) is in the sequence for m * (10\m) < 10.
Let T(q, k, n) = b(n) from the following recursion: for 0 <= i <= q-1, b(i) = 1 if i = k, else, b(i) = 0. Then b(n) = Sum(j=1..n, b(n-j)). If some m has q digits d1,..,dk,..,dq with d1 nonzero then after n iterations, we have Sum(j=1..q, T(q, k, n)*d(q-k+1)). This enables an iterative approach to finding solutions with q digits. (End)

			Digit reverse of 17 is 71 and 7 + 1 = 8, 1 + 8 = 9, 8 + 9 = 17;
Digit reverse of 286 is 682 and 6 + 8 + 2 = 16, 8 + 2 + 16 = 26, 2 + 16 + 26 = 44, 16 + 26 + 44 = 86, 26 + 44 + 86 = 156, 44 + 86 + 156 = 286.

Cf. A000045, A000073, A001590, A004086, A248134, A269309, A269307, A269311.

P:=proc(q) local a,b,k,n; for n from 0 to q do a:=convert(n,base,10); b:=convert(a,`+`); while b

Select[Range[10^6], Function[n, Total@ NestWhile[Append[Drop[#, 1], Total@ #] &, Reverse@ IntegerDigits@ n, Total@ # < n &] == n]] (* Michael De Vlieger, Jul 20 2017 *)

is(n) = {my(d = Vecrev(digits(n))); while(vecsum(d)David A. Corneth, Jul 20 2017

cp's OEIS Frontend

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.

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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.

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A269310 Consider 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 the Euler totient function of x.

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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.

Views

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Maple

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Extensions

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions

A289868 Consider the digit reverse of a number x. Take the sum of these digits and repeat the process deleting the first addend and adding the previous sum. The sequence lists the numbers that after some iterations reach a sum equal to x.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

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