A282757 -id:A282757 - cp's OEIS frontend

A282765 10*n analog to Keith numbers.

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 *)

A282766 n/2 analog of Keith numbers.

Original entry on oeis.org

50, 642, 1284, 1926, 2292, 5088, 29828, 42922, 53046, 95968, 512050, 1043160, 1723714, 14819056, 154860206, 159251186, 752516578, 946218018, 54728972948

Offset: 1

Paolo P. Lava, Feb 27 2017

Like Keith numbers but starting from n/2 digits to reach n.
Consider the digits of n/2. 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.
If it exists, a(20) > 10^12. - Lars Blomberg Mar 13 2017

			642/2 = 321:
3 + 2 + 1 = 6;
2 + 1 + 6 = 9;
1 + 6 + 9 = 16;
6 + 9 + 16 = 31;
9 + 16 + 31 = 56;
16 + 31 + 56 = 103;
31 + 56 + 103 = 190;
56 + 103 + 190 = 349;
103 + 190 + 349 = 642.

Cf. A282757-A282765, A282767, A282768, A282769.

with(numtheory): P:=proc(q,h,w) local a, b, k, n, t, v; v:=array(1..h);
for n from 1/w by 1/w 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]

With[{n = 2}, Select[Range[10 n, 10^6, n], Function[k, Last@ NestWhile[Append[Rest@ #, Total@ #] &, IntegerDigits[k/n], Total@ # <= k &] == k]]] (* Michael De Vlieger, Feb 27 2017 *)

a(15)-a(19) from Lars Blomberg, Mar 13 2017

A282769 n/7 analog of Keith numbers.

Original entry on oeis.org

301, 602, 1113, 4942, 478205, 23942940, 47885880, 178114489749

Offset: 1

Paolo P. Lava, Feb 27 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 iterations reach a sum equal to themselves.
If it exists, a(9) > 10^12. - Lars Blomberg Mar 07 2017

			1113/7 = 159:
    1 +   5 +   9 =   15;
    5 +   9 +  15 =   29;
    9 +  15 +  29 =   53;
   15 +  29 +  53 =   97;
   29 +  53 +  97 =  179;
   53 +  97 + 179 =  329;
   97 + 179 + 329 =  605;
  179 + 329 + 605 = 1113.

Cf. A282757 - A282765, A282766 - A282768.

with(numtheory): P:=proc(q,h,w) local a, b, k, n, t, v; v:=array(1..h);
for n from 1/w by 1/w 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]

With[{n = 7}, Select[Range[10 n, 10^6, n], Function[k, Last@ NestWhile[Append[Rest@ #, Total@ #] &, IntegerDigits[k/n], Total@ # <= k &] == k]]] (* Michael De Vlieger, Feb 27 2017 *)

a(8) from Lars Blomberg, Mar 07 2017

A282768 n/5 analog of Keith numbers.

Original entry on oeis.org

55, 110, 165, 220, 275, 330, 385, 440, 495, 530, 47270, 119710, 685385, 21526000, 6055017240

Offset: 1

Paolo P. Lava, Feb 27 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 iterations reach a sum equal to themselves.
If it exists, a(16) > 10^12. - Lars Blomberg Mar 07 2017

			530/5 = 106:
   1 +   0 +   6 =   7;
   0 +   6 +   7 =  13;
   6 +   7 +  13 =  26;
   7 +  13 +  26 =  46;
  13 +  26 +  46 =  85;
  26 +  46 +  85 = 157;
  46 +  85 + 157 = 288;
  85 + 157 + 288 = 530.

Cf. A282757 - A282765, A282766, A282767, A282769.

with(numtheory): P:=proc(q,h,w) local a, b, k, n, t, v; v:=array(1..h);
for n from 1/w by 1/w 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]

With[{n = 5}, Select[Range[10 n, 10^6, n], Function[k, Last@ NestWhile[Append[Rest@ #, Total@ #] &, IntegerDigits[k/n], Total@ # <= k &] == k]]] (* Michael De Vlieger, Feb 27 2017 *)

a(15) from Lars Blomberg, Mar 07 2017

A282758 3*n analog to Keith numbers.

Original entry on oeis.org

7, 9, 14, 19, 21, 28, 38, 53, 54, 76, 92, 124, 1299, 18185, 20468, 31871, 32054, 37903, 128200, 152057, 175539, 193399, 214631, 303677, 1806425, 3250457, 3616693, 7870170, 10793441, 12047403, 13781464, 15035426, 18663611, 19917573, 22905596, 46531972, 101743590

Offset: 1

Paolo P. Lava, Feb 22 2017

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

Consider the digits of 3*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.

			3*28 = 84:
8 + 4 = 12;
4 + 12 = 16;
12 + 16 = 28.

Cf. A282757, A282759 - 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[3 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 *)

A282759 4*n analog to Keith numbers.

Original entry on oeis.org

3, 6, 9, 12, 19, 29, 40, 787, 1679, 2137, 2508, 2728, 5016, 7524, 12773, 36183, 46116, 192952, 246916, 681538, 1316065, 4826672, 7571204, 8709926, 9716827, 24922317

Offset: 1

Paolo P. Lava, Feb 22 2017

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

Consider the digits of 4*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.

			4*29 = 116:
1 + 1 + 6 = 8;
1 + 6 + 8 = 15;
6 + 8 + 15 = 29.

Cf. A282757, A282758, A282760 - 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[4 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 *)

A282760 5*n analog to Keith numbers.

Original entry on oeis.org

2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 19, 28, 56, 147, 566, 1301, 4288, 8576, 13088, 119396, 518800, 634825, 654780, 993476, 2109420, 3034105, 6466772, 17838948, 80148824

Offset: 1

Paolo P. Lava, Feb 22 2017

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

Consider the digits of 5*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.

			5*14 = 70:
7 + 0 = 7;
0 + 7 = 7;
7 + 7 = 14.

Cf. A282757 - A282759, A282761 - 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[5 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 *)

A282761 6*n analog to Keith numbers.

Original entry on oeis.org

9, 23, 85, 88, 208, 953, 12339, 122420, 251925, 286400, 467608, 1207360, 1308519, 1537214, 1638373, 1844108, 2314739, 2736742, 9331102, 11692851, 28349534, 85403569

Offset: 1

Paolo P. Lava, Feb 22 2017

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

Consider the digits of 6*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.

			6*23 = 138:
1 + 3 + 8 = 12;
3 + 8 + 12 = 23.

Cf. A282757 - A282760, A282762 - 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[6 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 *)

A282762 7*n analog to Keith numbers.

Original entry on oeis.org

3, 6, 9, 12, 25, 29, 33, 58, 62, 66, 70, 87, 91, 95, 99, 124, 128, 150, 152, 165, 178, 191, 204, 217, 592, 801, 1184, 3860, 15728, 59800, 117711, 157701, 230720, 270737, 496085, 795918, 869366, 954639, 1549319, 1826773, 3169440, 3170466, 3973793, 3974819, 3975845, 4012718, 4013744, 5120160, 5653357, 5978943

Offset: 1

Paolo P. Lava, Feb 22 2017

Like Keith numbers but starting from 7*n digits to reach n.

Consider the digits of 7*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.

			7*25 = 175:
1 + 7 + 5 = 13;
7 + 5 + 13 =25.

Cf. A282757 - A282761, A282763 - 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[7 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 *)

A282763 8*n analog to Keith numbers.

Original entry on oeis.org

9, 20, 176, 184, 277, 2669, 15705, 233202, 241202, 445657, 742714, 2095479, 4697536, 10508788, 20308656, 55683878, 86603874

Offset: 1

Paolo P. Lava, Feb 22 2017

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

Consider the digits of 8*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.

			8*20 = 160:
1 + 6 + 0 = 7;
6 + 0 + 7 = 13;
0 + 7 + 13 = 20.

Cf. A282757 - A282762, A282764, 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[8 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 *)

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A282765 10*n analog to Keith numbers.

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A282766 n/2 analog of Keith numbers.

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A282769 n/7 analog of Keith numbers.

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A282768 n/5 analog of Keith numbers.

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A282758 3*n analog to Keith numbers.

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A282759 4*n analog to Keith numbers.

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A282760 5*n analog to Keith numbers.

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A282761 6*n analog to Keith numbers.

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