A246544 -id:A246544 - cp's OEIS frontend

A274769 Square analog to Keith numbers.

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

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

A281918 7th power analog of Keith numbers.

Original entry on oeis.org

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

A247012 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 the reverse of themselves.

Original entry on oeis.org

6, 133, 172, 841, 1005, 1603, 4258, 5299, 192901, 498906, 1633303, 5307589, 16333303, 20671542, 41673714, 42999958, 73687923

Offset: 1

Paolo P. Lava, Sep 09 2014

A072234 is a subset of this sequence.

a(18) > 2*10^8. - Tyler Busby, Mar 19 2023

			Aliquot parts of 1005 are 1, 3, 5, 15, 67, 201 and 335:
  1 + 3 + 5 + 15 + 67 + 201 + 335 = 627;
  3 + 5 + 15 + 67 + 201 + 335 + 627 = 1253;
  5 + 15 + 67 + 201 + 335 + 627 + 1253 = 2503;
  15 + 67 + 201 + 335 + 627 + 1253 + 2503 = 5001 that is the reverse of 1005.
Aliquot parts of 1603 are 1, 7 and 229:
  1 + 7 + 229 = 237;
  7 + 229 + 237 = 473;
  229 + 237 + 473 = 939;
  237 + 473 + 939 = 1649;
  473 + 939 + 1649 = 3061 that is the reverse of 1603;

Cf. A072234, A097090, A246544, A247013.

with(numtheory): R:=proc(w) local x,y; x:=w; y:=0;
while x>0 do y:=10*y+(x mod 10); x:=trunc(x/10); od: y; end:
P:=proc(q,h) local a,b,c,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; c:=ilog10(n)+1;
for k from 1 to b do v[k]:=a[k]; od;
t:=b+1; v[t]:=add(v[k], k=1..b);
if R(v[t])=n then print(n); else
while ilog10(v[t])+1<=c do t:=t+1; v[t]:=add(v[k], k=t-b..t-1);
if R(v[t])=n then print(n); break; fi; od; fi; fi; od;
end: P(10^9, 1000);

A247012 = {};
For[n = 4, n <= 1000000, n++,
 If[PrimeQ[n], Continue[]];
 r = IntegerReverse[n];
 a = Most[Divisors[n]];
 sum = Total[a];
 While[sum < r, sum = Total[a = Join[Rest[a], {sum}]]];
 If[sum == r, AppendTo[A247012, n]];
]; A247012 (* Robert Price, Sep 08 2019 *)

from sympy import isprime, divisors
A247012_list = []
for n in range(2,10**9):
    m = int(str(n)[::-1])
    if not isprime(n):
        x = divisors(n)
        x.pop()
        y = sum(x)
        while y < m:
            x, y = x[1:]+[y], 2*y-x[0]
        if y == m:
            A247012_list.append(n) # Chai Wah Wu, Sep 12 2014

a(9), a(11)-a(17) from Chai Wah Wu, Sep 13 2014

cp's OEIS Frontend

A274769 Square analog to Keith numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions

A274770 Cube analog to Keith numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions

A281915 4th power analog of Keith numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions

A281916 5th power analog of Keith numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions

A281917 6th power analog of Keith numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions

A281918 7th power analog of Keith numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions

A281919 8th-power analog of Keith numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A281920 9th-power analog of Keith numbers.