A007629 -id:A007629 - cp's OEIS frontend

A281918 7th power analog of Keith numbers.

1, 18, 27, 31, 34, 43, 53, 58, 68, 145, 187, 314, 826, 2975, 37164, 40853, 58530, 72795, 77058, 160703, 187617, 1926759, 6291322, 6628695, 25285305, 31292514, 33968486, 54954185, 71593237, 125921697, 555963577, 575307142, 2393596216, 2444508547, 42544333760, 97812197525

Offset: 1

Paolo P. Lava, Feb 02 2017

Like Keith numbers but starting from n^7 digits to reach n.
Consider the digits of n^7. Take their sum and repeat the process deleting the first addend and adding the previous sum. The sequence lists the numbers that after some number of iterations reach a sum equal to n.
a(30) > 10^8. - Jinyuan Wang, Jan 30 2020

			145^7 = 1347646586640625:
1 + 3 + 4 + 7 + 6 + 4 + 6 + 5 + 8 + 6 + 6 + 4 + 0 + 6 + 2 + 5 = 73;
3 + 4 + 7 + 6 + 4 + 6 + 5 + 8 + 6 + 6 + 4 + 0 + 6 + 2 + 5 + 73 = 145.

Cf. A226971, A007629, A246544, A263534.

Cf. A274769, A274770, A281915, A281916, A281917, A281919, A281920, A281921.

with(numtheory): P:=proc(q, h,w) local a, b, k, t, v; global n; v:=array(1..h);
for n from 1 to q do b:=n^w; a:=[];
for k from 1 to ilog10(b)+1 do a:=[(b mod 10), op(a)]; b:=trunc(b/10); od;
for k from 1 to nops(a) do v[k]:=a[k]; od; b:=ilog10(n^w)+1;
t:=nops(a)+1; v[t]:=add(v[k], k=1..b); while v[t]

(* function keithQ[ ] is defined in A007629 *)
a281918[n_] := Join[{1}, Select[Range[10, n], keithQ[#, 7]&]]
a281918[10^6] (* Hartmut F. W. Hoft, Jun 03 2021 *)

a(28)-a(29) from Jinyuan Wang, Jan 30 2020

a(30)-a(36) from Giovanni Resta, Feb 03 2020

A281919 8th-power analog of Keith numbers.

Original entry on oeis.org

1, 30, 46, 54, 63, 207, 394, 693, 694, 718, 20196, 42664, 80051, 90135, 91447, 93136, 207846, 324121, 361401, 421609, 797607, 802702, 882227, 1531788, 1788757, 1789643, 4028916, 4176711, 6692664, 15643794, 31794346, 65335545, 140005632, 144311385, 153364253

Offset: 1

Paolo P. Lava, Feb 02 2017

Like Keith numbers but starting from n^8 digits to reach n.

Consider the digits of n^8. Take their sum and repeat the process deleting the first addend and adding the previous sum. The sequence lists the numbers that after some number of iterations reach a sum equal to n.

Giovanni Resta, Table of n, a(n) for n = 1..40

Cf. A007629, A246544, A263534.

Cf. A274769, A274770, A281915, A281916, A281917, A281918, A281920, A281921.

with(numtheory): P:=proc(q, h,w) local a, b, k, t, v; global n; v:=array(1..h);
for n from 1 to q do b:=n^w; a:=[];
for k from 1 to ilog10(b)+1 do a:=[(b mod 10), op(a)]; b:=trunc(b/10); od;
for k from 1 to nops(a) do v[k]:=a[k]; od; b:=ilog10(n^w)+1;
t:=nops(a)+1; v[t]:=add(v[k], k=1..b); while v[t]

(* function keithQ[ ] is defined in A007629 *)
a281919[n_] := Join[{1}, Select[Range[10, n], keithQ[#, 8]&]]
a281919[10^6] (* Hartmut F. W. Hoft, Jun 03 2021 *)

207^8 = 3371031134626313601:
+ 3 + 7 + 1 + 0 + 3 + 1 + 1 + 3 + 4 + 6 + 2 + 6 + 3 + 1 + 3 + 6 + 0 + 1 = 54;
+ 7 + 1 + 0 + 3 + 1 + 1 + 3 + 4 + 6 + 2 + 6 + 3 + 1 + 3 + 6 + 0 + 1 + 54 = 105;
+ 1 + 0 + 3 + 1 + 1 + 3 + 4 + 6 + 2 + 6 + 3 + 1 + 3 + 6 + 0 + 1 + 54 + 105 = 207.

a(32) from Jinyuan Wang, Feb 01 2020

Terms a(33) and beyond from Giovanni Resta, Feb 03 2020

A281920 9th-power analog of Keith numbers.

Original entry on oeis.org

1, 54, 71, 81, 196, 424, 451, 2394, 9057, 51737, 52141, 104439, 227914, 228088, 1019555, 1096369, 1202713, 1687563, 1954556, 3332130, 3652731, 4177592, 31669012, 79937731, 81478913, 148341053, 168763202, 182573136, 342393476, 773367191, 1450679282, 2914657310, 3282344153

Offset: 1

Paolo P. Lava, Feb 02 2017

Like Keith numbers but starting from n^9 digits to reach n.

Consider the digits of n^9. Take their sum and repeat the process deleting the first addend and adding the previous sum. The sequence lists the numbers that after some number of iterations reach a sum equal to n.

			196^9 = 426878854210636742656:
4 + 2 + 6 + 8 + 7 + 8 + 8 + 5 + 4 + 2 + 1 + 0 + 6 + 3 + 6 + 7 + 4 + 2 + 6 + 5 + 6 = 100;
2 + 6 + 8 + 7 + 8 + 8 + 5 + 4 + 2 + 1 + 0 + 6 + 3 + 6 + 7 + 4 + 2 + 6 + 5 + 6 + 100 = 196.

Cf. A007629, A246544, A263534.

Cf. A274769, A274770, A281915, A281916, A281917, A281918, A281919, A281921.

with(numtheory): P:=proc(q, h,w) local a, b, k, t, v; global n; v:=array(1..h);
for n from 1 to q do b:=n^w; a:=[];
for k from 1 to ilog10(b)+1 do a:=[(b mod 10), op(a)]; b:=trunc(b/10); od;
for k from 1 to nops(a) do v[k]:=a[k]; od; b:=ilog10(n^w)+1;
t:=nops(a)+1; v[t]:=add(v[k], k=1..b); while v[t]

(* function keithQ[ ] is defined in A007629 *)
a281920[n_] := Join[{1}, Select[Range[10, n], keithQ[#, 9]&]]
a281920[10^6] (* Hartmut F. W. Hoft, Jun 03 2021 *)

a(24) from Jinyuan Wang, Feb 02 2020

a(25)-a(33) from Giovanni Resta, Feb 03 2020

A281921 10th-power analog of Keith numbers.

Original entry on oeis.org

1, 82, 85, 94, 97, 106, 117, 459, 1197, 24615, 24657, 26184, 87664, 117099, 538168, 1049708, 1229174, 2210323, 4587773, 11019224, 96167938, 104719358, 202511251, 226456915, 821871524, 1811437987, 1832881095, 3530066559, 7414362499, 7906250753, 15607432165, 15631766564

Offset: 1

Paolo P. Lava, Feb 02 2017

Like Keith numbers but starting from n^10 digits to reach n.

Consider the digits of n^10. Take their sum and repeat the process deleting the first addend and adding the previous sum. The sequence lists the numbers that after some number of iterations reach a sum equal to n.

			106^10 = 179084769654285362176: 1 + 7 + 9 + 0 + 8 + 4 + 7 + 6 + 9 + 6 + 5 + 4 + 2 + 8 + 5 + 3 + 6 + 2 + 1 + 7 + 6 = 106.

Cf. A007629, A246544, A263534.

Cf. A274769, A274770, A281915, A281916, A281917, A281918, A281919, A281920.

with(numtheory): P:=proc(q, h,w) local a, b, k, t, v; global n; v:=array(1..h);
for n from 1 to q do b:=n^w; a:=[];
for k from 1 to ilog10(b)+1 do a:=[(b mod 10), op(a)]; b:=trunc(b/10); od;
for k from 1 to nops(a) do v[k]:=a[k]; od; b:=ilog10(n^w)+1;
t:=nops(a)+1; v[t]:=add(v[k], k=1..b); while v[t]

(* function keithQ[ ] is defined in A007629 *)
a281921[n_] := Join[{1}, Select[Range[10, n], keithQ[#, 10]&]]
a281921[10^6] (* Hartmut F. W. Hoft, Jun 03 2021 *)

a(21) from Jinyuan Wang, Feb 02 2020

a(22)-a(32) from Giovanni Resta, Feb 03 2020

A187713 Base-5 Keith numbers.

Original entry on oeis.org

5, 9, 10, 11, 13, 15, 20, 22, 31, 40, 43, 53, 62, 71, 84, 93, 124, 154, 221, 483, 3044, 18748, 125973, 232085, 1705260, 2091605, 5616236, 8067806, 8849508, 58944155, 84572166, 164487062, 421825427, 469435978, 744740232

Offset: 1

Alonso del Arte, Mar 17 2011

Among bases b = 2 to 36, in b = 5 there is the third highest percentage of Keith numbers between b and b^3 (that is, numbers with two or three digits); only binary and ternary have more Keith numbers in that range.

			a(2) = 9. In base 5, the number 9 is written 14, and the second order linear recurrence is then 1, 4, 5, 9, ... therefore 9 is a Keith number in base 5.
The number 14 is a Keith number in base 10 but not base 5, as we have: 2, 4, 6, 10, 16, ...

Cf. A007629, base-10 Keith numbers; A162724, binary Keith numbers.

(* First run the program for A186830 to define keithSeq *) Select[Range[5, 10^6], Last[keithSeq[#, 5]] == # &]

Extended by T. D. Noe, Mar 23 2011

A188195 Base-3 Keith numbers.

Original entry on oeis.org

3, 5, 6, 7, 57, 102, 127, 206, 217, 677, 805, 840, 1486, 1680, 1887, 2090, 2834, 8329, 10145, 12866, 21127, 23002, 50782, 67925, 82685, 96841, 153861, 178852, 357896, 3826652, 17985694, 38610616, 38610808, 70587766, 160804168, 341014432, 632582224

Offset: 1

T. D. Noe, Mar 24 2011

Keith numbers are described in A007629.

			57 is here because, in base 3, 57 is 2010 and applying the Keith iteration to this number produces the numbers 2, 0, 1, 0, 3, 4, 8, 15, 30, 57.

Cf. A007629 (base 10), A162724 (base 2), A187713 (base 5), A188196-A188200.

IsKeith[n_,b_] := Module[{d, s, k}, d = IntegerDigits[n, b]; s = Total[d]; k = 1; While[AppendTo[d, s]; s = 2 s - d[[k]]; s < n, k++]; s == n]; Select[Range[3,10^5], IsKeith[#,3]&]

A188200 Base-9 Keith numbers.

Original entry on oeis.org

17, 21, 25, 42, 67, 81, 96, 101, 149, 162, 173, 202, 243, 303, 324, 346, 404, 405, 486, 519, 567, 648, 692, 732, 857, 1189, 1464, 2199, 4398, 11644, 18325, 33774, 34453, 37999, 70348, 92664, 141557, 256820, 263412, 326778, 349484, 526824, 535754, 579708, 1461987, 1519308, 1621052, 2688905, 4650964, 8027458, 8198651, 8374956, 13504910, 17858551, 20002383, 55640285, 154513633, 170801638

Offset: 1

T. D. Noe, Mar 24 2011

Keith numbers are described in A007629. Base 9 appears to be unusually rich in Keith numbers. Why?

			101 is here because, in base 9, 101 is 122 and applying the Keith iteration to this number produces the numbers 1, 2, 2, 5, 9, 16, 30, 55, 101. Note that the multiples 202, 303, and 404 are here also.

Cf. A007629 (base 10), A162724 (base 2), A187713 (base 5), A188195-A188199.

IsKeith[n_,b_] := Module[{d, s, k}, d = IntegerDigits[n, b]; s = Total[d]; k = 1; While[AppendTo[d, s]; s = 2 s - d[[k]]; s < n, k++]; s == n]; Select[Range[3,10^5], IsKeith[#,9]&]

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

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 *)

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

Paolo P. Lava, Feb 24 2016

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

Cf. A000010, A007629, A269307, A269308, 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:=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]

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 *)

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

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A281918 7th power analog of Keith numbers.

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A281919 8th-power analog of Keith numbers.

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A281920 9th-power analog of Keith numbers.

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A281921 10th-power analog of Keith numbers.

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A187713 Base-5 Keith numbers.

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A188195 Base-3 Keith numbers.

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A188200 Base-9 Keith numbers.

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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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