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.

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A269313 Consider a prime with k>1 digits. Take the sum of its k digits. Repeat the process deleting the first addendum and adding the previous sum. The sequence lists the least prime that is reached after some iterations.

Original entry on oeis.org

2, 7, 23, 19, 5, 11, 5, 17, 5, 7, 11, 11, 23, 7, 13, 17, 13, 41, 11, 17, 23, 2, 7, 53, 19, 5, 19, 5, 11, 13, 11959, 7, 13, 19, 89, 11, 17, 19, 11, 13, 17, 19, 11, 7, 11, 13, 47, 89, 7, 23, 47, 11, 17, 29, 53, 11, 13, 43, 17, 5, 7, 11, 7, 13, 971, 29, 11, 17, 29
Offset: 1

Views

Author

Paolo P. Lava, Feb 24 2016

Keywords

Comments

First values with more than five hundred digits for 2689 (563 digits), 8761 (537 digits), 24251 (690) and 40787 (609).
First values with more than one thousand digits for 44129 (1021 digits) and 82361 (1502 digits).
First value with more than two thousand digits for 40819 (2909 digits).
Within the first 10^4 primes the record is for 62659 with 4526 digits.
Does any prime reach another one?

Examples

			11: 1 + 1 = 2.
13: 1 + 3 = 4; 3 + 4 = 7;
17: 1 + 7 = 8; 7 + 8 = 15; 8 + 15 = 23.

Links

Crossrefs

Cf. A000040, A007629.

Programs

  • Maple
    with(numtheory): P:=proc(q,h) local a,b,c,d,k,n,t,v; v:=array(1..h);
for n from 10 to q do if isprime(n) then a:=n; b:=ilog10(n)+1;
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 not isprime(v[t]) do t:=t+1; v[t]:=add(v[k], k=t-b..t-1); od;
print(v[t]); fi; od; end: P(10^4,10000);
  • Mathematica
    Table[d = IntegerDigits[Prime[n]]; While[! PrimeQ[Total[d]], d = Join[Rest[d], {Total[d]}]]; Total[d], {n, 5, 100}] (* Robert Price, May 22 2019 *)

A377439 Least integer k which, by a process analogous to the Keith numbers, reach k - n.

Original entry on oeis.org

14, 18, 10, 11, 12, 10, 11, 10, 10, 10, 20, 27, 22, 25, 20, 23, 20, 21, 20, 38, 32, 30, 31, 34, 30, 32, 31, 30, 40, 47, 41, 45, 40, 43, 42, 41, 40, 58, 51, 56, 50, 54, 53, 52, 51, 50, 61, 67, 60, 65, 64, 63, 62, 61, 60, 78, 70, 76, 75, 74, 73, 72, 71, 70, 80, 87
Offset: 0

Views

Author

Paolo P. Lava, Oct 28 2024

Keywords

Examples

			a(6) = 11 because 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5 that is 11 - 6.

Links

Crossrefs

Cf. A007629, A377416.

Programs

  • Maple
    with(numtheory): P:=proc(q, h) local a, b, c, j, k, n, t, v; v:=array(1..h); c:=[];
for j from 0 to 65 do for n from 10 to q do a:=n; b:=length(a);
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]

A383230 Numbers k whose decimal representation can be split in three parts which can be used as seeds for a tribonacci-like sequence containing k itself.

Original entry on oeis.org

197, 742, 1007, 1257, 1484, 1749, 1789, 3241, 4349, 4515, 4851, 5709, 6482, 6925, 7756, 8196, 8449, 8698, 10232, 10997, 11627, 16898, 17206, 18353, 19789, 20464, 27315, 30696, 31385, 35537, 40928, 43367, 44111, 48310, 48591, 49228, 50574, 58506, 62770, 79976, 88222
Offset: 1

Views

Author

Paolo P. Lava, Apr 20 2025

Keywords

Comments

There are 306 terms < 10^8.
If the number k is rewritten as the concatenation of a, b and c, the problem is to find an integer x such that k = a*A000073(x) + b*A001590(x+1) + c*A000073(x+1).
Is there any term that results from more than one concatenation?

Examples

			1007 can be split into 10, 0, 7 and the tribonacci-like sequence contains 1007 itself: 10, 0, 7, 17, 24, 48, 89, 161, 298, 548, 1007 ... (x = 9, as per second comment);
1257 can be split into 1, 25, 7 and the tribonacci-like sequence contains 1257 itself: 1, 25, 7, 33, 65, 105, 203, 373, 681, 1257 ... (x = 8, as per second comment);
16898 can be split into 16, 8, 98 and the tribonacci-like sequence contains 16898 itself: 16, 8, 98, 122, 228, 448, 798, 1474, 2720, 4992, 9186, 16898 ... (x = 10, as per second comment).

Links

Crossrefs

Cf. A000073, A000213, A001590, A007629, A130792.

Programs

  • Maple
    P:=proc(q) local b,c,d,f1,f2,f3,i,j,m,n,t,v,y,x,w; i:=[]; for n from 100 to q do b:=length(n);
for t from 1 to b-2 do c:=n mod 10^t; m:=trunc(n/10^t); d:=length(m);
for j from 1 to d-1 do x:=trunc(m/10^j); y:=m mod 10^j; f1:=2; f2:=3; f3:=4; v:=x*f1+y*f2+c*f3;
while v

A072533 Least Keith number (or repfigit) with n digits, or 0 if there is none.

Original entry on oeis.org

14, 197, 1104, 31331, 120284, 1084051, 11436171, 129572008, 0, 24769286411, 171570159070, 1934197506555, 43520999798747, 120984833091531, 3621344088074041, 11812665388886672, 119115440241433462, 1362353777290081176, 12763314479461384279, 855191324330802397989, 7657230882259548723593, 26842994422637112523337, 229146413136585558461227, 9838678687915198599200604, 18354972585225358067718266, 133118411174059688391045955, 9493976840390265868522067200, 41796205765147426974704791528
Offset: 2

Views

Author

Benoit Cloitre, Aug 04 2002

Keywords

Comments

There are no 10-digit Keith numbers.

Crossrefs

Cf. A007629.

Extensions

Extended by T. D. Noe, Mar 15 2011

A121235 Multiply-Add Recurrence Invariant (MARI) numbers.

Original entry on oeis.org

14, 19, 28, 47, 61, 75, 191, 197, 205, 242, 302, 515, 742, 1064, 1104, 1220, 1266, 1537, 1757, 1981, 2208, 2580, 3505, 3684, 4484, 4608, 4788, 6702, 7385, 7647, 7909, 8180, 13614, 14327, 15557, 22251, 24377, 29662, 31331, 34285, 34348, 35577, 39323
Offset: 1

Views

Author

R. J. Mathar, Aug 21 2006

Keywords

Comments

Contains A042983 and A007629 as subsets.

Links

Crossrefs

Cf. A042983, A007629.

A135409 Sum of all n-digit Keith numbers.

Original entry on oeis.org

0, 244, 939, 38842, 399158, 3185054, 8997888, 89003533, 380705305, 0, 120958456566, 909738914355, 10691161155707, 215413743487843, 1146359095715014, 11741086634604569, 186260344930883645, 586697640742217928, 22434377949474656108, 86030233471862779595
Offset: 1

Views

Author

Parthasarathy Nambi, Dec 10 2007

Keywords

Comments

There are no 1-digit and 10-digit Keith numbers.

Examples

			Sum of all 2-digit Keith numbers is 244.
Sum of all 3-digit Keith numbers is 939.
Sum of all 4-digit Keith numbers is 38842.

Links

  • _Jason Earls_ and Eric Weisstein's World of Mathematics, Keith Number.

Crossrefs

Cf. A007629.

Extensions

Extended by T. D. Noe, Mar 15 2011

A188197 Base-6 Keith numbers.

Original entry on oeis.org

8, 11, 16, 27, 37, 44, 74, 88, 111, 148, 185, 409, 526, 2417, 8720, 12154, 15268, 49322, 61587, 68444, 82833, 98644, 206356, 249549, 327001, 484512, 642437, 692928, 695659, 726975, 964225, 1210087, 2141228, 2282504, 5514048, 10640601, 48453362, 69572128, 74343984, 171550728, 184847569, 204545417, 232877871, 245317977, 246133682
Offset: 1

Views

Author

T. D. Noe, Mar 24 2011

Keywords

Comments

Keith numbers are described in A007629.

Examples

			44 is here because, in base 6, 44 is 112 and applying the Keith iteration to this number produces the numbers 1, 1, 2, 4, 7, 13, 24, 44.

Crossrefs

Cf. A007629 (base 10), A162724 (base 2), A187713 (base 5), A188195-A188200.

Programs

  • Mathematica
    IsKeith[n_,b_] := Module[{d, s, k}, d = IntegerDigits[n, b]; s = Total[d]; k = 1; While[AppendTo[d, s]; s = 2 s - d[[k]]; s < n, k++]; s == n]; Select[Range[3,10^5], IsKeith[#,6]&]

A188198 Base-7 Keith numbers.

Original entry on oeis.org

8, 13, 16, 19, 24, 32, 40, 48, 57, 114, 125, 145, 171, 228, 285, 329, 342, 589, 1969, 2833, 4938, 30318, 43153, 168516, 336774, 375008, 652933, 1068018, 2955098, 5658387, 11096232, 19623430, 26245925, 81805113, 112442958, 119572340, 130712398, 407198006, 494835656, 508871625, 564319261, 712864110
Offset: 1

Views

Author

T. D. Noe, Mar 24 2011

Keywords

Comments

Keith numbers are described in A007629.

Examples

			48 is here because, in base 7, 48 is 66 and applying the Keith iteration to this number produces the numbers 6, 6, 12, 18, 30, 48.

Crossrefs

Cf. A007629 (base 10), A162724 (base 2), A187713 (base 5), A188195-A188200.

Programs

  • Mathematica
    IsKeith[n_,b_] := Module[{d, s, k}, d = IntegerDigits[n, b]; s = Total[d]; k = 1; While[AppendTo[d, s]; s = 2 s - d[[k]]; s < n, k++]; s == n]; Select[Range[3,10^5], IsKeith[#,7]&]

A265426 Primes p such that p - 1 is a binary Keith number (A162724).

Original entry on oeis.org

2, 3, 5, 17, 257, 1367, 65537, 2960687
Offset: 1

Views

Author

Jaroslav Krizek, Dec 08 2015

Keywords

Comments

See A162724 (binary Keith numbers) and A007629 (Keith numbers) for definitions.
Primes of the form A162724(n)+1.
Fermat primes (A019434) are terms.
The next term, if it exists, must be greater than 17*10^9.
Union of primes p of the form A162363(n)+1 and A000079(m)+1 for a some n or m.

Crossrefs

Cf. A000079, A007629, A019434, A162363, A162724.

Programs

  • Mathematica
    fQ[n_] := Module[{b = IntegerDigits[n, 2], s}, s = Total@ b; If[s <= 1, True, k = 1; While[s = 2 s - b[[k]]; s < n, k++]; s == n]]; Select[Prime@ Range[10^6], fQ[# - 1] &] (* Michael De Vlieger, Dec 09 2015, after T. D. Noe at A162724 *)

A319746 Quasi-Repfigit numbers (or Quasi-Keith numbers).

Original entry on oeis.org

12, 18, 32, 35, 43, 59, 142, 187, 241, 265, 610, 778, 1521, 2163, 2625, 3267, 3729, 9242, 15905, 16725, 18852, 56207, 63265, 87538, 94596, 333718, 780890, 839383, 959394, 1114534, 1745662, 2198585, 2424613, 2815415, 5501438, 7371962, 9717796, 21010738, 27800086, 31173396
Offset: 1

Views

Author

Paolo P. Lava, Sep 27 2018

Keywords

Comments

Numbers n>9 with following property: form a sequence b(i) whose initial terms are the t digits of n, later terms given by rule that b(i) = sum of t previous terms; then n - 1 or n + 1 appears in the sequence.

Examples

			a(1) = 12 because 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13 = 12 + 1.
a(2) = 18 because 1 + 8 = 9, 8 + 9 = 17 = 18 - 1.

Crossrefs

Cf. A007629, A130010.

Programs

  • Maple
    P:=proc(q) local b, k,n,v; for n from 1 to q do b:=ilog10(n)+1;
if b>1 then v:=[]; for k from 1 to b do
v:=[op(v),trunc(n/10^(b-k)) mod 10]; od; v:=[op(v),add(v[k],k=1..b)];
while v[nops(v)]
Previous Showing 51-60 of 63 results. Next

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