A007629 -id:A007629 - cp's OEIS frontend

A050235 a(n) is the number of n-digit Keith numbers A007629.

0, 6, 2, 9, 7, 10, 2, 3, 2, 0, 2, 4, 2, 3, 3, 3, 5, 3, 5, 3, 1, 1, 3, 1, 1, 3, 7, 1, 2, 5, 2, 4, 6, 3

Offset: 1

Mike Keith, Keith Numbers
Eric Weisstein's World of Mathematics, Keith Number

More terms from Eric W. Weisstein using results from D. Lichtblau, Aug 24 2004
a(27)-a(31) from Eric W. Weisstein from computations by Daniel Lichtblau Eric W. Weisstein, Jun 23 2009
a(32)-a(34) from computations by Daniel Lichtblau -- Eric W. Weisstein, Aug 26 2009

A246544 Consider the aliquot parts, in ascending order, of a composite number. Take their sum and repeat the process deleting the minimum number and adding the previous sum. The sequence lists the numbers that after some iterations reach a sum equal to themselves.

Original entry on oeis.org

6, 21, 28, 85, 496, 2133, 8128, 19521, 77125, 97273, 176661, 615281, 4948133, 33550336, 68353213, 129127041, 8589869056

Offset: 1

Paolo P. Lava, Aug 29 2014

Similar to Keith numbers and Primonacci numbers, using proper divisors instead of digits or prime factors.
The perfect numbers A000396 are a subset.
The numbers of iterations are: 1, 2, 1, 3, 1, 2, 1, 2, 3, 4, 2, 5, ...; with 1 when the term is perfect. - Michel Marcus, Aug 30 2014
The 2-hyperperfect numbers A007593 are a subset, with number of iterations 2. - Michel Marcus, Sep 22 2014

			Aliquot parts of 85 are 1, 5 and 17:
  1 + 5 + 17 = 23;
  5 + 17 + 23 = 45;
  17 + 23 + 45 = 85.
Aliquot parts of 19521 are 1, 3, 9, 27, 81, 241, 723, 2169 and 6507:
  1 + 3 + 9 + 27 + 81 + 241 + 723 + 2169 + 6507 = 9761;
  3 + 9 + 27 + 81 + 241 + 723 + 2169 + 6507 + 9761 = 19521.

Cf. A000396, A007629, A212875.

with(numtheory): P:=proc(q,h)
local a,b,k,n,t,v; v:=array(1..h);
for n from 2 to q do if not isprime(n) then
a:=sort([op(divisors(n))]); b:=nops(a)-1;
for k from 1 to b do v[k]:=a[k]; od;
t:=b+1; v[t]:=add(v[k],k=1..b);
while v[t]

A246544 = {};
For[n = 4, n <= 1000000, n++,
 If[PrimeQ[n], Continue[]];
 a = Most[Divisors[n]];
 sum = Total[a];
 While[sum < n, sum = Total[a = Join[Rest[a], {sum}]]];
 If[sum == n, AppendTo[A246544, n]];
]; A246544 (* Robert Price, Sep 08 2019 *)

lista(nn) = {forcomposite(n=1, nn, d = divisors(n); v = vector(#d-1, i, d[i]); vs = sum(i=1, #v, v[i]); ind = 1; while (vs < n, v = concat(v, vs); vs += vs - v[ind]; ind++;); if (vs == n, print1(n, ", ")););} \\ Michel Marcus, Aug 29 2014

import math
def divs(n):
    large_divisors = []
    for i in range(1, int(math.sqrt(n) + 1)):
        if n % i == 0:
            yield i
            if i !=  n // i:
                large_divisors.insert(0, n / i)
    for divisor in large_divisors:
        yield divisor
a = 2
while a < 1000000000:
    q = list(divs(a))[:-1]
    r = sum(q)
    if r > a or len(q) == 1:
        pass
    elif r == a:
        print(a)
    else:
        c = 1
        while r < a:
            q.append(r)
            r = sum(q[c:])
            c += 1
        if r == a:
            print(a)
    a += 1
# David Consiglio, Jr., Sep 09 2014

from sympy import divisors, isprime
A246544_list = []
for n in range(2,10**5):
    if not isprime(n):
        x = divisors(n)
        x.pop()
        y = sum(x)
        while y < n:
            x, y = x[1:]+[y], 2*y-x[0]
        if y == n:
            A246544_list.append(n)
# Chai Wah Wu, Nov 03 2014

a(13)-a(15) from Michel Marcus, Aug 29 2014
a(16) from David Consiglio, Jr., Sep 06 2014
a(17) from Lars Blomberg, Oct 27 2014

A282757 2*n analog to Keith numbers.

Original entry on oeis.org

5, 9, 10, 15, 19, 20, 25, 28, 30, 35, 40, 45, 47, 66, 88, 132, 198, 2006, 2740, 4012, 4419, 13635, 56357, 338540, 354164, 419966, 441972, 685704, 803678, 1528803, 1844810, 9127005, 12305952, 14315686, 14650155, 15828353, 17838087, 22618003, 37826729, 71644613

Offset: 1

Paolo P. Lava, Feb 22 2017

Like Keith numbers but starting from 2*n digits to reach n.

Consider the digits of 2*n. Take their sum 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 themselves.

			2*28 = 56 :
5 + 6 = 11;
6 + 11 = 17;
11 + 17 = 28.

Cf. A282758 - A282765.

with(numtheory): P:=proc(q, h,w) local a, b, k, n, t, v; v:=array(1..h);
for n from 1 to q do a:=w*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[10^6], Function[n, Module[{d = IntegerDigits[2 n], s, k = 0}, s = Total@ d; While[s < n, AppendTo[d, s]; k++; s = 2 s - d[[k]]]; s == n]]] (* Michael De Vlieger, Feb 22 2017, after T. D. Noe at A007629 *)

A282765 10*n analog to Keith numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 19, 28, 56, 176, 904, 3347, 4795, 5301, 9775, 10028, 16165, 16715, 35103, 49693, 111039, 191103, 370287, 439385, 845772, 1727706, 1836482, 3631676, 3767812, 4363796, 4499932, 5351605, 6940437, 20090073, 28246243, 38221997, 60220332

Offset: 1

Paolo P. Lava, Feb 22 2017

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

Consider the digits of 10*n. Take their sum 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 themselves.

			10*14 = 140:
1 + 4 + 0 = 5;
4 + 0 + 5 = 9;
0 + 5 + 9 = 14.

Cf. A282757 - A282764.

with(numtheory): P:=proc(q, h,w) local a, b, k, n, t, v; v:=array(1..h);
for n from 1 to q do a:=w*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[10^6], Function[n, Module[{d = IntegerDigits[10 n], s, k = 0}, s = Total@ d; While[s < n, AppendTo[d, s]; k++; s = 2 s - d[[k]]]; s == n]]] (* Michael De Vlieger, Feb 22 2017, after T. D. Noe at A007629 *)

A274769 Square analog to Keith numbers.

Original entry on oeis.org

1, 9, 37, 40, 43, 62, 70, 74, 160, 1264, 1952, 2847, 12799, 16368, 16584, 42696, 83793, 97415, 182011, 352401, 889871, 925356, 1868971, 1881643, 3661621, 7645852, 15033350, 21655382, 63288912, 88192007, 158924174, 381693521, 792090500, 2025078249, 2539401141

Offset: 1

Paolo P. Lava, Jul 06 2016

Like Keith numbers but starting from n^2 digits to reach n.

Consider the digits of the square of a number n. Take their sum 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 themselves.

			1264^2 = 1597696 :
1 + 5 + 9 + 7 + 6 + 9 + 6 = 43;
5 + 9 + 7 + 6 + 9 + 6 + 43 = 85;
9 + 7 + 6 + 9 + 6 + 43 + 85 = 165;
7 + 6 + 9 + 6 + 43 + 85 + 165 = 321;
6 + 9 + 6 + 43 + 85 + 165 + 321 = 635;
9 + 6 + 43 + 85 + 165 + 321 + 635 = 1264.

Cf. A007629, A246544, A263534, A274770.

with(numtheory): P:=proc(q, h) local a,b,k,n,t,v; v:=array(1..h);
for n from 1 to q do b:=n^2; 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^2)+1;
t:=nops(a)+1; v[t]:=add(v[k], k=1..b); while v[t]

Select[Range[10^6], Function[n, Module[{d = IntegerDigits[n^2], s, k = 0}, s = Total@ d; While[s < n, AppendTo[d, s]; k++; s = 2 s - d[[k]]]; s == n]]] (* Michael De Vlieger, Feb 22 2017, after T. D. Noe at A007629 *)
(* function keithQ[ ] is defined in A007629 *)
a274769[n_] := Join[{1, 9}, Select[Range[10, n], keithQ[#, 2]&]]
a274769[10^6] (* Hartmut F. W. Hoft, Jun 02 2021 *)

a(32)-a(35) from Giovanni Resta, Jul 08 2016

A274770 Cube analog to Keith numbers.

Original entry on oeis.org

1, 8, 17, 18, 26, 27, 44, 55, 63, 80, 105, 187, 326, 776, 1095, 2196, 6338, 13031, 13131, 25562, 27223, 70825, 140791, 553076, 632489, 1402680, 1404312, 3183253, 11311424, 50783292, 51231313, 182252596, 255246098, 522599548, 1180697763, 2025114819, 2137581414

Offset: 1

Paolo P. Lava, Jul 06 2016

Like Keith numbers but starting from n^3 digits to reach n.

Consider the digits of the cube of a number n . Take their sum 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 themselves.

			776^3 = 467288576 :
4 + 6 + 7 + 2 + 8 + 8 + 5 + 7 + 6 = 53;
6 + 7 + 2 + 8 + 8 + 5 + 7 + 6 + 53 = 102;
7 + 2 + 8 + 8 + 5 + 7 + 6 + 53 + 102 = 198;
2 + 8 + 8 + 5 + 7 + 6 + 53 + 102 + 198 = 389;
8 + 8 + 5 + 7 + 6 + 53 + 102 + 198 + 389 = 776.

Cf. A007629, A246544, A263534, A274769.

with(numtheory): P:=proc(q, h) local a,b,k,n,t,v; v:=array(1..h);
for n from 1 to q do b:=n^3; 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^3)+1;
t:=nops(a)+1; v[t]:=add(v[k], k=1..b); while v[t]

(* function keithQ[ ] is defined in A007629 *)
a274770[n_] := Join[{1, 8}, Select[Range[10, n], keithQ[#, 3]&]]
a274770[10^6] (* Hartmut F. W. Hoft, Jun 02 2021 *)

a(34)-a(37) from Giovanni Resta, Jul 08 2016

A263534 Consider the 10's complements mod 10 of the digits of a number k. Take their sum 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 k.

Original entry on oeis.org

29, 76, 157, 174, 191, 475, 713, 1129, 1961, 3286, 4424, 7812, 8973, 19978, 24317, 35845, 37041, 51712, 68022, 166838, 443275, 444247, 445219, 509439, 706317, 1189312, 1933197, 2686010, 10809303, 55558901, 58338037, 257990335, 504050156, 839186880

Offset: 1

Paolo P. Lava, Oct 20 2015

Like Keith numbers but using the ten's complements of their digits.

a(35) > 10^9. - Robert Price, Apr 08 2019

			For 29, the 10's complements of its digits are 8, 1. Then:
  8 + 1 = 9;
  1 + 9 = 10;
  9 + 10 = 19;
  10 + 19 = 29.
For 475, the 10's complements of its digits are 6, 3, 5. Then:
  6 + 3 + 5 = 14;
  3 + 5 + 14 = 22;
  5 + 14 + 22 = 41;
  14 + 22 + 41 = 77;
  22 + 41 + 77 = 140;
  41 + 77 + 140 = 258;
  77 + 140 + 258 = 475.

Cf. A007629.

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

Select[Range[10^5], Function[{m, n}, Last@ NestWhile[Append[#, Total@ Take[#, -m]] &, Flatten[{#, Total@ #}] &[IntegerDigits[n] /. d_?Positive :> 10 - d], Last@ # < n &, 1, 10^2] == n] @@ {IntegerLength@#, #} &] (* Michael De Vlieger, Mar 09 2018 *)

Name clarified, some terms and Maple code corrected by Paolo P. Lava, Mar 08 2018
a(30)-a(32) from Robert Price, Apr 05 2019
a(33)-a(34) from Robert Price, Apr 08 2019

A281915 4th power analog of Keith numbers.

Original entry on oeis.org

1, 7, 19, 20, 22, 25, 28, 36, 77, 107, 110, 175, 789, 1528, 1932, 3778, 5200, 7043, 8077, 38855, 41234, 44884, 49468, 204386, 763283, 9423515, 73628992, 87146144, 146124072, 146293356, 326194628, 1262293219, 1321594778, 2767787511, 11511913540, 12481298961, 13639550655

Offset: 1

Paolo P. Lava, Feb 02 2017

Like Keith numbers but starting from n^4 digits to reach n.

Consider the digits of n^4. 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.

			175^4 = 937890625:
9 + 3 + 7 + 8 + 9 + 0 + 6 + 2 + 5 = 49;
3 + 7 + 8 + 9 + 0 + 6 + 2 + 5 + 49 = 89;
7 + 8 + 9 + 0 + 6 + 2 + 5 + 49 + 89 = 175.

Cf. A055575, A007629, A246544, A263534.

Cf. A274769, A274770, A281916, A281917, A281918, 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 *)
a281915[n_] := Join[{1, 7}, Select[Range[10, n], keithQ[#, 4]&]]
a281915[10^6] (* Hartmut F. W. Hoft, Jun 02 2021 *)

a(27)-a(28) from Jinyuan Wang, Jan 30 2020

Missing a(25) and a(29)-a(37) from Giovanni Resta, Jan 31 2020

A281916 5th power analog of Keith numbers.

Original entry on oeis.org

1, 28, 35, 36, 46, 51, 99, 109, 191, 239, 476, 491, 1022, 1126, 1358, 1362, 15156, 21581, 44270, 63377, 100164, 375830, 388148, 2749998, 5215505, 10158487, 81082532, 87643314, 410989134, 1485204944, 3496111364, 3829840893, 15889549579, 16107462404, 16766005098, 17608009898

Offset: 1

Paolo P. Lava, Feb 02 2017

Like Keith numbers but starting from n^5 digits to reach n.

Consider the digits of n^5. 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.

			109^5 = 15386239549:
1 + 5 + 3 + 8 + 6 + 2 + 3 + 9 + 5 + 4 + 9 = 55;
5 + 3 + 8 + 6 + 2 + 3 + 9 + 5 + 4 + 9 + 55 = 109.

Cf. A055576, A007629, A246544, A263534.

Cf. A274769, A274770, A281915, A281917, A281918, 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 *)
a281916[n_] := Join[{1}, Select[Range[10, n], keithQ[#, 5]&]]
a281916[5*10^5] (* Hartmut F. W. Hoft, Jun 03 2021 *)

a(27)-a(28) from Jinyuan Wang, Jan 31 2020

a(29)-a(36) from Giovanni Resta, Jan 31 2020

A281917 6th power analog of Keith numbers.

Original entry on oeis.org

1, 18, 45, 54, 64, 125, 218, 246, 935, 1125, 6021, 6866, 7887, 40210, 89330, 457625, 577655, 613385, 640118, 5200210, 6809148, 7293243, 10013591, 50980917, 216864574, 885859983, 4556794863, 4939169289, 8580755055, 8672110451, 18562634876, 18992278338, 36013476739

Offset: 1

Paolo P. Lava, Feb 02 2017

Like Keith numbers but starting from n^6 digits to reach n.

Consider the digits of n^6. 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.

			125^6 = 3814697265625:
3 + 8 + 1 + 4 + 6 + 9 + 7 + 2 + 6 + 5 + 6 + 2 + 5 = 64;
8 + 1 + 4 + 6 + 9 + 7 + 2 + 6 + 5 + 6 + 2 + 5 + 64 = 125.

Cf. A055577, A007629, A246544, A263534.

Cf. A274769, A274770, A281915, A281916, A281918, 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[n_, e_] is defined in A007629 *)
a281917[n_] := Join[{1}, Select[Range[10, n], keithQ[#, 6]&]]
a281917[10^4] (* Hartmut F. W. Hoft, Jun 03 2021 *)

a(24) from Jinyuan Wang, Jan 31 2020

a(25)-a(33) from Giovanni Resta, Jan 31 2020

cp's OEIS Frontend

A050235 a(n) is the number of n-digit Keith numbers A007629.

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A246544 Consider the aliquot parts, in ascending order, of a composite number. Take their sum and repeat the process deleting the minimum number and adding the previous sum. The sequence lists the numbers that after some iterations reach a sum equal to themselves.

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A282757 2*n analog to Keith numbers.

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A282765 10*n analog to Keith numbers.

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A274769 Square analog to Keith numbers.

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A274770 Cube analog to Keith numbers.

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A263534 Consider the 10's complements mod 10 of the digits of a number k. Take their sum 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 k.

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A281915 4th power analog of Keith numbers.

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