Thomas S. Pedigo - cp's OEIS frontend

A349951 Similar to the "Look and Say" sequence, with the added requirement that every third digit describes the location of a digit in blocks of three in the previous term.

1, 111, 311, 131211, 111131111122212, 311112132112313114224125115125, 131211212122113133123214124135115135216146227147118128158219159111012101510

Offset: 1

Thomas S. Pedigo, Dec 06 2021

The sequence has a different pattern if, instead of including integers greater than ten in the description of each block of three, one uses symbols to represent those numbers. For example, in a(7), if the subsequence "111012101510" is replaced by "11A12A15A" then the interaction of digits with each other is altered.

			a(1) is read as "one one in the first block", so a(2) is 111.
a(2) is read as "three ones in the first block", so a(3) is 311.
a(3) is read as "one three in the first block, two ones in the first block", so a(4) is 131211.
a(4) is read as "one one in the first block, one three in the first block, one one in the first block, one two in the second block, two ones in the second block", so a(5) is 111131111122212.

Cf. A005150.

{1}~Join~NestList[(s=Split/@Partition[IntegerDigits[#],3];FromDigits@Flatten@Table[Join[Reverse@#,IntegerDigits[k]]&/@({First@#,Length@#}&/@#&/@s)[[k]],{k,Length@s}])&,111,5] (* Giorgos Kalogeropoulos, Dec 08 2021 *)

A267720 The smallest prime number formed by the concatenation of consecutive powers of n beginning with 1^0 = "1," and ending with a trailing "1" - or "0," if no such prime number exists.

Original entry on oeis.org

0, 131, 1416641, 151, 16361, 17491, 181, 191, 0, 1111211, 11214417282073624883229859841, 0, 11419627441, 1151, 1162564096655361048576167772162684354564294967296687194767361099511627776175921860444161, 1171, 1181, 11936168591, 1201

Offset: 2

Thomas S. Pedigo, Jan 19 2016

It is not known whether solutions exist for A(2), A(10), or A(13); if there are any, they must be greater than 10^128.
A(10) has final exponent 35; "110100100010000...1000000000000000000000000000000000001", total length 667 digits. - Dana Jacobsen, Jan 22 2016
A(2) has exponent > 1100, length > 182841 digits.  A(13) has exponent > 750, length > 314093 digits. - Dana Jacobsen, Jan 25 2016

			A(14)=11419627441; 14^0=1; 14^1=14; 14^2=196; 14^3=2744; 11419627441 is prime.
A(19)=11936168591; 19^0=1; 19^1=19; 19^2=361; 19^3=6859; 11936168591 is prime.

Cf. A068704, A088622.

lim = 60; Table[SelectFirst[Rest@ FoldList[FromDigits@ Join[Most@ IntegerDigits@ #1, IntegerDigits[n^#2], {1}] &, 11, Range@ lim], PrimeQ], {n, 2, 20}] /. k_ /; MissingQ@ k -> 0 (* Michael De Vlieger, Jan 25 2016, Version 10.2 *)

a(8) corrected by Thomas S. Pedigo, Dec 08 2021

A266368 The smallest prime formed by the concatenation of consecutive n-th powers beginning with "1," plus a trailing "1.".

Original entry on oeis.org

11, 1231, 149161, 181, 1168125662512961, 1321, 1647294096156251, 11282187163847812527993682354320971524782969100000001948717135831808627485171, 125665616553639062516796161, 15121

Offset: 0

Thomas S. Pedigo, Dec 28 2015

a(10) has 437 and a(11) has 1810 decimal digits respectively. - Michael De Vlieger, Jan 05 2016

a(13) is a bit more manageable, with 65 decimal digits - 18192159432367108864122070312513060694016968890104075497558138881. a(14) and a(15) are even shorter, with 23 and 15 decimal digits, respectively - 11638447829692684354561 and 132768143489071. - Thomas S. Pedigo, Jan 06 2016

			a(6)=1647294096156251; 1=1^6; 64=2^6; 729=3^6; 4096=4^6; 15625=5^6; 1647294096156251 is prime.

Cf. A019521, A019522.

f[k_, n_] := FromDigits@ Flatten@ Map[IntegerDigits, Append[Range[k + 1]^n, 1], 1]; Table[If[n == 0, k = 0, k = 1]; While[! PrimeQ@ f[k, n], k++]; f[k, n], {n, 0, 9}] (* Michael De Vlieger, Jan 05 2016 *)

A265181 Prime numbers resulting from the concatenation of at least two copies of a cubic number followed by a trailing "1.".

Original entry on oeis.org

881, 27271, 7297291, 133113311, 337533751, 19683196831, 42875428751, 68921689211, 1038231038231, 1574641574641, 2053792053791, 2744274427441, 4218754218751, 6859685968591, 7290007290001, 7297297297291, 106120810612081, 224809122480911, 274400027440001, 280322128032211, 317652331765231, 500021150002111, 812060181206011, 1251251251251251, 1757617576175761, 1968319683196831, 5931959319593191

Offset: 1

Thomas S. Pedigo, Dec 03 2015

Subsequence of A030430 (primes of the form 10n+1). - Michel Marcus, Dec 04 2015
If m is a term then (m-1)/10 is divisible by a cube (A000578) and the resulting quotient, different from 1, is in A076289. - Michel Marcus, Dec 05 2015
Without the "repeated at least twice" constraint, A168147 would be a subsequence. - Michel Marcus, Dec 05 2015

			8 = 2^3; 881 is prime.
27 = 3^3; 27271 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000578, A030430, A066592, A076289, A168147 , A232066.

N:= 20: # to get all terms with at most N digits
M:= floor((N-1)/2):
res:= {}:
for s from 1 to floor(10^(M/3)) do
   x:= s^3;
   m:= 1+ilog10(x);
   for k from 2 to floor((N-1)/m) do
     p:= x*add(10^(1+m*i),i=0..k-1)+1;
     if isprime(p) then res:= res union {p} fi;
   od
od:
sort(convert(res,list)); # Robert Israel, Jan 13 2016

Take[Sort@ Flatten[Select[#, PrimeQ] & /@ Table[FromDigits@ Append[Flatten@ IntegerDigits@ Table[n^3, {#}], 1] & /@ Range[2, 20], {n, 1, 300}] /. {} -> Nothing], 27] (* Michael De Vlieger, Jan 05 2016 *)

from itertools import count, islice
from sympy import isprime
def A265181_gen(): # generator of terms
    return filter(isprime,(int(str(k**3)*2)*10+1 for k in count(1)))
A265181_list = list(islice(A265181_gen(),20)) # Chai Wah Wu, Feb 20 2023

A216233 Numbers whose squares have 2R-1 digits, such that the number represented by leftmost R digits and number represented by rightmost R digits divide each other evenly.

Original entry on oeis.org

245, 249, 251, 255, 264, 1245, 1249, 2490, 2498, 2502, 2510, 10984, 12490, 12498, 15449, 18735, 18751, 18868, 22714, 24980, 24996, 27907, 28302, 31225, 31249, 31579, 101852, 124996, 139535, 152174, 187494, 187510, 218751, 238165, 249992, 279070, 281249

Offset: 1

Thomas S. Pedigo, Mar 14 2013

Trivial solutions where the rightmost R digits are all zeros have been omitted. The first indices k for which the rightmost R digits of a(k)^2 do not contain leading zeros are 5, 12, 15, 19, 26, 27, 30, 34, 39, 52, 53, 62, 67, 80.

			The square of 22714 is 515925796, and 51592 = 2*25796.

Giovanni Resta, Table of n, a(n) for n = 1..165

Cf. A115528, A116501, A145848.

cnt = 0; Do[p = 10^Floor[nd/2]; Do[x = Floor[n*n/p]; y = Mod[n*n, 10*p]; If[y>0 && Mod[x,y]*Mod[y,x] == 0, Print[++cnt, " ", n, " ", n*n]], {n, p, Floor[10^(nd/2)]}], {nd,3,11,2}] (* Giovanni Resta, Mar 15 2013 *)

Missing a(25) and a(27)-a(37) from Giovanni Resta, Mar 15 2013

Comment corrected by Giovanni Resta, Mar 15 2013

A218539 Numbers that are equal to the sum of the uniform platonic polyhedral (figurate) numbers (tetrahedral, cubic, octahedral, dodecahedral, or icosahedral) on each of their digits.

Original entry on oeis.org

0, 1, 20, 21, 24, 153, 240, 241, 289, 304, 324, 370, 371, 407, 440, 441, 593, 739, 2167, 2284, 2348, 2484, 2583, 2860, 2861, 3009, 3029, 3093, 3249, 4288, 5859, 6888, 7996, 9898

Offset: 1

Thomas S. Pedigo, Nov 01 2012

153, 370, 371, and 407 are well known with regard to the cubic numbers.

			The octahedral numbers are represented by the formula, y(x)=(2x^3+x)/3; apply this formula to each of the digits in a(18)=739, i.e., y(7)=231, y(3)=19, y(9)=489; sum=739; the dodecahedral numbers are represented by the formula, y(x)=x(3x-1)(3x-2)/2; apply this formula to each of the digits in a(34)=9898, i.e., y(9)=2725, y(8)=2024; y(9)=2725, y(8)=2024; sum=9898.

Cf. A005188, A007532, A036057.

A210691 Modified Golomb's sequence: a(n) is the number of times n occurs, starting with a(1)=2.

Original entry on oeis.org

2, 1, 1, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17

Offset: 1

Thomas S. Pedigo, Mar 30 2012

nonn

			a(5)=4 because there are four fives in the series.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001462

v=List([2,1,1]);for(n=3,50,for(i=1,v[n],listput(v,n)));Vec(v) \\ Charles R Greathouse IV, Apr 11 2012

a(21)-a(75) from Charles R Greathouse IV, Apr 11 2012

A167118 Each subsequent term in the series describes the digits of the previous term, by tallying their cumulative total within the term. This is similar to the "look and say" sequence, but deviates from it in that blocks of like digits are described individually, according to position.

Original entry on oeis.org

1, 11, 1121, 11211231, 1121123141221351, 11211231412213511461324271231581, 11211231412213511461324271231581912416

Offset: 1

Thomas S. Pedigo, Oct 27 2009

More terms can be represented beyond those listed, but only with symbols to quantify integers greater than ten. Each term contains a replicate of the previous term; the length of each term is twice that of the previous term (The last term in the series does not adhere to this rule because its omitted parts contain nonintegers.).

			Read aloud the running count of each distinct digit within each term, followed by the digit thus described, to determine each subsequent term, i. e., A(1)=1 produces "first one," or "11," thus A(2)=11; A(2)=11 produces "first one, second one," or "1121," thus A(3)=1121; A(3)=1121 produces "first one, second one, first two, third one," or "11211231," thus A(4)=11211231; A(4)=11211231 produces "first one, second one, first two, third one, fourth one, second two, first three, fifth one," or "1121123141221351," thus A(5)=1121123141221351.

Cf. A005150.

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User: Thomas S. Pedigo

A349951 Similar to the "Look and Say" sequence, with the added requirement that every third digit describes the location of a digit in blocks of three in the previous term.

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A267720 The smallest prime number formed by the concatenation of consecutive powers of n beginning with 1^0 = "1," and ending with a trailing "1" - or "0," if no such prime number exists.

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A266368 The smallest prime formed by the concatenation of consecutive n-th powers beginning with "1," plus a trailing "1.".

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Mathematica

A265181 Prime numbers resulting from the concatenation of at least two copies of a cubic number followed by a trailing "1.".

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A216233 Numbers whose squares have 2R-1 digits, such that the number represented by leftmost R digits and number represented by rightmost R digits divide each other evenly.

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A218539 Numbers that are equal to the sum of the uniform platonic polyhedral (figurate) numbers (tetrahedral, cubic, octahedral, dodecahedral, or icosahedral) on each of their digits.

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A210691 Modified Golomb's sequence: a(n) is the number of times n occurs, starting with a(1)=2.

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PARI

Extensions

A167118 Each subsequent term in the series describes the digits of the previous term, by tallying their cumulative total within the term. This is similar to the "look and say" sequence, but deviates from it in that blocks of like digits are described individually, according to position.

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Author

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