A004216 -id:A004216 - cp's OEIS frontend

A077430 a(n) = floor(log_10(2*n^2)) + 1.

1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Reinhard Zumkeller, Nov 05 2002

nonn

Cf. A077431, A077432, A004216, A077429, A077433.

A114206 Length of the preperiodic part of the decimal expansion of 1/n, and also leading zeros from the period part if the preperiodic digits are all 0s (if any).

Original entry on oeis.org

1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 1, 2

Offset: 2

N. J. A. Sloane, Oct 17 2006

Differs from A386406 at n = 92, 208, 248...

			For n = 11, 1/n = 0.(09), so the preperiodic part is empty and the initial 0 of the periodic part is included for length a(11) = 1.
For n = 92, 1/n = 0.01(0869565217391304347826) , so the preperiodic part is "01" and has length a(92) = 2 and the initial 0 in the periodic part is not included since that preperiodic part is not all 0s (unlike the way A386406(92) = 3 does include that inat periodic initial 0).

Hans Havermann, Table of n, a(n) for n = 2..10000
Index entries for sequences related to decimal expansion of 1/n.

Cf. A004216, A051628, A114205, A386406.

fb[n_] := Block[{rd, o, p},rd = RealDigits[1/n];o = Last[rd];p = First[rd];If[ ! IntegerQ[Last[p]], p = Most[p]];Length[p] - o];Table[fb[n], {n, 120}] (* Ray Chandler, Oct 18 2006 *)
r[x_]:=RealDigits[1/x]; w[x_]:=First[r[x]]; f[x_]:=First[w[x]]; l[x_]:=Last[w[x]]; z[x_]:=Last[r[x]]; b[x_]:=Which[IntegerQ[l[x]], Length[w[x]]-1*z[x], IntegerQ[f[x]]==False, -1*z[x], True, Length[Drop[w[x],-1]]-1*z[x]]; (* Hans Havermann, Oct 18 2006 *)
Table[b[i], {i,2,128}]

a(n) = max(logint(n,10), max(valuation(n,2), valuation(n,5))); \\ Kevin Ryde, Jul 22 2025

a(n) = max(A004216(n), A051628(n)). - Kevin Ryde, Jul 22 2025

More terms from Ray Chandler and Hans Havermann, Oct 18 2006

Edited by Andrei Zabolotskii and Kevin Ryde, Jul 20 2025

A004228 a(n) = 10000*log_10(n) rounded down.

Original entry on oeis.org

0, 3010, 4771, 6020, 6989, 7781, 8450, 9030, 9542, 10000, 10413, 10791, 11139, 11461, 11760, 12041, 12304, 12552, 12787, 13010, 13222, 13424, 13617, 13802, 13979, 14149, 14313, 14471, 14623

Offset: 1

N. J. A. Sloane

nonn

Cf. A004229, A004230.

Cf. A004216, A004219, A004222, A004225.

Floor[10^4 Log[10,Range[30]]] (* Harvey P. Dale, Feb 07 2015 *)

A077431 n repeated in decimal representation, but separated by enough zeros that the square has the pattern (n^2)(2n^2)(n^2).

Original entry on oeis.org

11, 22, 303, 404, 505, 606, 707, 8008, 9009, 10010, 11011, 12012, 13013, 14014, 15015, 16016, 17017, 18018, 19019, 20020, 21021, 22022, 230023, 240024, 250025, 260026, 270027, 280028, 290029, 300030, 310031, 320032, 330033, 340034, 350035

Offset: 1

Reinhard Zumkeller, Nov 05 2002

A077433(n) = number of separating zeros.

			a(17) = 17017, as 17017^2 = 289578289 = A077432(17) = 289'578'289 and 289=17^2 and 578=2*289.

Cf. A020338, A004216, A077430.

a[n_] := For[idn = IntegerDigits[n]; k = 0, True, k++, an = FromDigits[ Join[idn, Table[0, k], idn]]; If[MatchQ[IntegerDigits[an^2], {b__ /; IntegerQ[Sqrt[FromDigits[{b}]]], c___, 0..., b__} /; FromDigits[{c}] == 2*FromDigits[{b}]], Return[an]]];
Array[a, 35] (* Jean-François Alcover, Nov 13 2017 *)

a(n) = n*(1+10^(1+floor(log_10(2*n^2)))).

A077432 Squares of the form u'v'w, where in decimal representation u=n^2, v=2*n^2 and w=n^2 possibly with a leading zero.

Original entry on oeis.org

121, 484, 91809, 163216, 255025, 367236, 499849, 64128064, 81162081, 100200100, 121242121, 144288144, 169338169, 196392196, 225450225, 256512256, 289578289, 324648324, 361722361, 400800400, 441882441, 484968484, 52910580529

Offset: 1

Reinhard Zumkeller, Nov 05 2002

a(n) = A077431(n)^2.

			n=42, n^2=1764 and 2*n^2=3528: a(42) = 1764'3528'1764 = 420042^2 = A077431(42)^2.

Cf. A004216, A077430, A077431.

a(n) = (n*(1 + 10^(1 + floor(log_10(2*n^2)))))^2.

A290148 a(n) is the integer resulting from the concatenation of the unit digit of n-1 to the digits of n without its own unit digit.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 91, 1, 11, 21, 31, 41, 51, 61, 71, 81, 92, 2, 12, 22, 32, 42, 52, 62, 72, 82, 93, 3, 13, 23, 33, 43, 53, 63, 73, 83, 94, 4, 14, 24, 34, 44, 54, 64, 74, 84, 95, 5, 15, 25, 35, 45, 55, 65, 75, 85, 96, 6, 16, 26, 36, 46, 56, 66, 76, 86, 97, 7

Offset: 1

Michel Marcus, Jul 21 2017

Take list of integers n >= 1, move the right-most digit of each term to the start of the next term.

Every number appears, see A381225. - N. J. A. Sloane, Feb 23 2025

			For n=46, n-1 is 45, so a(46) is the concatenation of 5 (the unit digit of 45) and 4 (46 without 6), giving 54.
For n=123, n-1 is 122, so a(123) is the concatenation of 2 (the unit digit of 122) and 12 (123 without 3), giving 212.

Michael S. Branicky, Table of n, a(n) for n = 1..20000

Cf. A004216, A032762, A381225.

f:= n -> (n-1 mod 10) * 10^ilog10(n) + floor(n/10);

a(n) = my(precd = (n-1)%10); if (n < 10, precd, eval(concat(Str(precd), Str(n\10))));

def a(n): return 0 if n == 1 else int(str((n-1)%10)+ str(n)[:-1])
print([a(n) for n in range(1, 72)]) # Michael S. Branicky, Feb 22 2025

a(n) = (n-1 mod 10)*10^A004216(n) + floor(n/10). # Robert Israel, Jul 21 2017

A348674 Number of distinct values that can be produced by splitting n and adding the parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4

Offset: 0

Alois P. Heinz, Oct 29 2021

Differs from A055642 first at n=120: a(120) = 4 != 3 = A055642(120).

The number of split positions can vary from 0 to length(n)-1.

			a(0) = 1: 0.
a(10) = 2: 1 = 1+0, 10.
a(100) = 3: 1 = 1+0+0, 10 = 10+0, 100.
a(120) = 4: 3 = 1+2+0, 12 = 12+0, 21 = 1+20, 120.
a(2493690) = 62 = |{33, 51, 60, 69, 78, 87, 96, 105, 114, 123, 132, 141, 150, 159, 168, 177, 186, 195, 213, 267, 294, 321, 348, 375, 384, 402, 420, 510, 564, 591, 618, 708, 726, 744, 789, 807, 942, 951, 969, 1032, 1050, 1185, 2508, 2562, 2589, 3183, 3705, 3723, 3741, 3939, 4947, 5028, 9375, 9393, 24945, 25026, 49371, 93696, 93714, 249369, 493692, 2493690}|.

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Ordinal transform gives A349315.
Where records occur: A349316.
Cf. A000041, A002275, A004216, A007953, A011557, A055642, A089695.

b:= proc(s) option remember; (n-> {parse(s), seq(seq(seq(x+y,
      y=b(s[i+1..n])), x=b(s[1..i])), i=1..n-1)})(length(s))
    end:
a:= n-> nops(b(""||n)):
seq(a(n), n=0..120);

a(n) <= 2^floor(log_10(n)) = 2^A004216(n) for n>0.
a((10^n-1)/9) = a(A002275(n)) <= A000041(n) with equality only for n <= 23.
a(10^n) = a(A011557(n)) = n+1.

A372523 Triangle read by rows: T(n, k) is equal to n/k if k | n, else to the concatenation of A003988(n, k) = floor(n/k) and A051127(k, n) = n mod k.

Original entry on oeis.org

1, 2, 1, 3, 11, 1, 4, 2, 11, 1, 5, 21, 12, 11, 1, 6, 3, 2, 12, 11, 1, 7, 31, 21, 13, 12, 11, 1, 8, 4, 22, 2, 13, 12, 11, 1, 9, 41, 3, 21, 14, 13, 12, 11, 1, 10, 5, 31, 22, 2, 14, 13, 12, 11, 1, 11, 51, 32, 23, 21, 15, 14, 13, 12, 11, 1, 12, 6, 4, 3, 22, 2, 15, 14, 13, 12, 11, 1

Offset: 1

Stefano Spezia, May 04 2024

			The triangle begins:
  1;
  2,  1;
  3, 11,  1;
  4,  2, 11,  1;
  5, 21, 12, 11,  1;
  6,  3,  2, 12, 11,  1;
  7, 31, 21, 13, 12, 11, 1;
  ...

Stefano Spezia, First 150 rows of the triangle, flattened

Cf. A003988, A004216, A051127, A055642.
Cf. A000012 (right diagonal), A000027 (1st column).
Cf. A048158, A051126, A051127, A234575.

T[n_,k_]:=If[Divisible[n,k],n/k,FromDigits[Join[IntegerDigits[Floor[n/k]],IntegerDigits[Mod[n,k]]]]]; Table[T[n,k],{n,12},{k,n}]//Flatten (* or *)
T[n_,k_]:=Floor[n/k]10^IntegerLength[Mod[n,k]]+Mod[n,k]; Table[T[n,k],{n,12},{k,n}]//Flatten (* or *)
T[n_, k_]:=SeriesCoefficient[x^k(1+Sum[(i + 10^(1+Floor[Log10[Mod[n,k]]]))*x^i, {i, k-1}] - Sum[i*x^(k+i), {i, k-1}])/(1-x^k)^2, {x, 0, n}]; Table[T[n, k], {n, 12}, {k, n}]//Flatten

T(n, k) = floor(n/k)*10^(1+floor(log10(n mod k))) + (n mod k) if n is not divisible by k.
T(n, n) = 1.
T(n, 1) = n.
T(n, k) = 2*T(n-k, k) - T(n-2*k, k) for n >= 3*k.
T(n, k) = [x^n] x^k*(1 + (Sum_{i=1..k-1} (i + 10^(1+floor(log10(n mod k))))*x^i) - (Sum_{i=1..k-1} i*x^(k+i)))/(1 - x^k)^2.

A004225 a(n) = 1000*log_10(n) rounded down.

Original entry on oeis.org

0, 301, 477, 602, 698, 778, 845, 903, 954, 1000, 1041, 1079, 1113, 1146, 1176, 1204, 1230, 1255, 1278, 1301, 1322, 1342, 1361, 1380, 1397, 1414, 1431, 1447, 1462, 1477, 1491, 1505, 1518, 1531, 1544

Offset: 1

N. J. A. Sloane

nonn

Cf. A004226, A004227.

Cf. A004216, A004219, A004222, A004228.

Floor[1000 Log10[Range[40]]] (* Harvey P. Dale, May 27 2024 *)

A253445 a(n) = concatenation of n^2 with itself.

Original entry on oeis.org

11, 44, 99, 1616, 2525, 3636, 4949, 6464, 8181, 100100, 121121, 144144, 169169, 196196, 225225, 256256, 289289, 324324, 361361, 400400, 441441, 484484, 529529, 576576, 625625, 676676, 729729, 784784, 841841, 900900, 961961, 10241024, 10891089, 11561156

Offset: 1

Vincenzo Librandi, Jan 02 2015

Aaron Toponce, Table of n, a(n) for n = 1..10000

Cf. A000290, A020338, A175605.

[Seqint(Intseq(n^2) cat Intseq(n^2)): n in [1..40]];

seq(n^2*(1+10^(1+ilog10(n^2))),n=1..100); # Robert Israel, Jan 02 2015

Table[FromDigits[Join[IntegerDigits[n^2], IntegerDigits[n^2]]], {n, 40}]

vector(100,n,eval(concat(Str(n^2),Str(n^2)))) \\ Derek Orr, Jan 02 2015

a(n) = A020338(n^2) = n^2 * (1 + 10^(A004216(n^2)+1)).

cp's OEIS Frontend

A077430 a(n) = floor(log_10(2*n^2)) + 1.

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A114206 Length of the preperiodic part of the decimal expansion of 1/n, and also leading zeros from the period part if the preperiodic digits are all 0s (if any).

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A004228 a(n) = 10000*log_10(n) rounded down.

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A077431 n repeated in decimal representation, but separated by enough zeros that the square has the pattern (n^2)(2n^2)(n^2).

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A077432 Squares of the form u'v'w, where in decimal representation u=n^2, v=2*n^2 and w=n^2 possibly with a leading zero.

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A290148 a(n) is the integer resulting from the concatenation of the unit digit of n-1 to the digits of n without its own unit digit.

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Maple

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A348674 Number of distinct values that can be produced by splitting n and adding the parts.

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Maple

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A372523 Triangle read by rows: T(n, k) is equal to n/k if k | n, else to the concatenation of A003988(n, k) = floor(n/k) and A051127(k, n) = n mod k.

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A004225 a(n) = 1000*log_10(n) rounded down.

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