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 1-9 of 9 results.

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

Views

Author

Paolo P. Lava, Jul 06 2016

Keywords

Comments

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.

Examples

			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.

Crossrefs

Cf. A007629, A246544, A263534, A274770.

Programs

  • Maple
    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]
  • Mathematica
    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 *)

Extensions

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

Views

Author

Paolo P. Lava, Jul 06 2016

Keywords

Comments

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.

Examples

			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.

Crossrefs

Cf. A007629, A246544, A263534, A274769.

Programs

  • Maple
    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]
  • Mathematica
    (* 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 *)

Extensions

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

Views

Author

Paolo P. Lava, Feb 02 2017

Keywords

Comments

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.

Examples

			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.

Crossrefs

Cf. A055575, A007629, A246544, A263534.
Cf. A274769, A274770, A281916, A281917, A281918, A281919, A281920, A281921.

Programs

  • Maple
    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]
  • Mathematica
    (* 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 *)

Extensions

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

Views

Author

Paolo P. Lava, Feb 02 2017

Keywords

Comments

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.

Examples

			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.

Crossrefs

Cf. A055576, A007629, A246544, A263534.
Cf. A274769, A274770, A281915, A281917, A281918, A281919, A281920, A281921.

Programs

  • Maple
    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]
  • Mathematica
    (* 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 *)

Extensions

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

Views

Author

Paolo P. Lava, Feb 02 2017

Keywords

Comments

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.

Examples

			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.

Crossrefs

Cf. A055577, A007629, A246544, A263534.
Cf. A274769, A274770, A281915, A281916, A281918, A281919, A281920, A281921.

Programs

  • Maple
    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]
  • Mathematica
    (* 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 *)

Extensions

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

Views

Author

Paolo P. Lava, Feb 02 2017

Keywords

Comments

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

Examples

			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.

Crossrefs

Cf. A226971, A007629, A246544, A263534.
Cf. A274769, A274770, A281915, A281916, A281917, A281919, A281920, A281921.

Programs

  • Maple
    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]
  • Mathematica
    (* 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 *)

Extensions

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

Views

Author

Paolo P. Lava, Feb 02 2017

Keywords

Comments

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.

Links

Crossrefs

Cf. A007629, A246544, A263534.
Cf. A274769, A274770, A281915, A281916, A281917, A281918, A281920, A281921.

Programs

  • Maple
    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]
  • Mathematica
    (* 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 *)

Formula

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

Extensions

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

Views

Author

Paolo P. Lava, Feb 02 2017

Keywords

Comments

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.

Examples

			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.

Crossrefs

Cf. A007629, A246544, A263534.
Cf. A274769, A274770, A281915, A281916, A281917, A281918, A281919, A281921.

Programs

  • Maple
    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]
  • Mathematica
    (* 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 *)

Extensions

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

Views

Author

Paolo P. Lava, Feb 02 2017

Keywords

Comments

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.

Examples

			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.

Crossrefs

Cf. A007629, A246544, A263534.
Cf. A274769, A274770, A281915, A281916, A281917, A281918, A281919, A281920.

Programs

  • Maple
    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]
  • Mathematica
    (* 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 *)

Extensions

a(21) from Jinyuan Wang, Feb 02 2020
a(22)-a(32) from Giovanni Resta, Feb 03 2020
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