A173426 -id:A173426 - cp's OEIS frontend

A260856 Base-6 representation is the concatenation of the base-6 representations of 1, 2, ..., n, n-1, ..., 1.

0, 1, 49, 1849, 67081, 2418025, 522134761, 676678989289, 876975982612969, 1136560874204496361, 1472982892995886760425, 1908985829323636470956521, 2474045634803467686907986409, 3206363142705295375772778742249, 4155446632946062852128962559066601

Offset: 0

M. F. Hasler, Aug 01 2015

			a(0) = 0 is the result of the empty sum corresponding to 0 digits.
a(2) = 49 = (6+1)^2 = 6^2 + 2*6 + 1 = 121_6 is the concatenation of (1, 2, 1).
a(7) = 676678989289 = 1234510111054321_6 is the concatenation of (1, 2, 3, 4, 5, 10, 11, 10, 5, 4, 3, 2, 1), where the middle "10, 11, 10" are the base 6 representations of 6, 7, 6.

D. Broadhurst, Primes from concatenation: results and heuristics, NmbrThry List, August 1, 2015

Base 6 variant of A173426 (base 10) and A173427 (base 2). See A260853 - A260866 for variants in other bases b = 3, ..., 16.

a(n,b=6)=sum(i=1,#n=concat(vector(n*2-1,k,digits(min(k,n*2-k),b))),n[i]*b^(#n-i))

A260857 Base-7 representation of a(n) is the concatenation of the base-7 representations of 1, 2, ..., n, n-1, ..., 1.

Original entry on oeis.org

0, 1, 64, 3249, 160000, 7845601, 384473664, 131870760799, 316621469105950, 760208147660763999, 1825259762561514314050, 4382448689911580334132199, 10522259304477772232578647150, 25263944590051134455098854865399, 60658730960712773989601560650105250

Offset: 0

M. F. Hasler, Aug 01 2015

Base-7 variant of A173426 (base 10) and A173427 (base 2). See A260853 - A260866 for variants in other bases.

The base 7 is not listed in A260343, because a(7) = A260851(7) = 131870760799 = 12345610654321_7 is not prime and therefore not in A260852. See these sequences for more information.

			a(0) = 0 is the result of the empty sum corresponding to 0 digits.
a(2) = 64 = (7+1)^2 = 7^2 + 2*7 + 1 = 121_7 is the concatenation of (1, 2, 1).
a(8) = 316621469105950 = 123456101110654321_7 is the concatenation of (1, 2, 3, 4, 5, 6, 10, 11, 10, 6, 5, 4, 3, 2, 1), where the middle "10, 11, 10" are the base-7 representations of 7, 8, 7.

D. Broadhurst, Primes from concatenation: results and heuristics, NmbrThry List, August 1, 2015

Table[FromDigits[Flatten[Join[IntegerDigits[Range[n],7], IntegerDigits[ Range[ n-1,1,-1],7]]],7],{n,0,20}] (* Harvey P. Dale, Nov 02 2017 *)

a(n,b=7)=sum(i=1,#n=concat(vector(n*2-1,k,digits(min(k,n*2-k),b))),n[i]*b^(#n-i))

A260858 Base-8 representation of a(n) is the concatenation of the base-8 representations of 1, 2, ..., n, n-1, ..., 1.

Original entry on oeis.org

0, 1, 81, 5329, 342225, 21911761, 1402427601, 89755965649, 45954960939217, 188231512819194065, 770996276517410920657, 3158000748616424634669265, 12935171066332946781853145297, 52982460687699754593548358342865, 217016158976818195107979529799293137

Offset: 0

M. F. Hasler, Aug 01 2015

Base-8 variant of A173426 (base 10) and A173427 (base 2). See A260853 - A260866 for variants in other bases.

The base 8 is not listed in A260343, because a(8) = A260851(8) = 45954960939217 is not prime and therefore not in A260852. See these sequences for more information.

			a(0) = 0 is the result of the empty sum corresponding to 0 digits.
a(2) = 81 = (8+1)^2 = 8^2 + 2*8 + 1 = 121_8, the concatenation of (1, 2, 1).
a(9) = 12345671011107654321_8, concatenation of (1, 2, 3, 4, 5, 6, 7, 10, 11, 10, 7, 6, 5, 4, 3, 2, 1), where the middle "10, 11, 10" are the base-8 representations of 8, 9, 8.

D. Broadhurst, Primes from concatenation: results and heuristics, NmbrThry List, August 1, 2015

a(n,b=8)=sum(i=1,#n=concat(vector(n*2-1,k,digits(min(k,n*2-k),b))),n[i]*b^(#n-i))

For n < b = 8, we have a(n) = A_b(n) = R(b,n)^2, where R(b,n) = (b^n-1)/(b-1) are the base-b repunits.

A260862 Base-12 representation of a(n) is the concatenation of the base-12 representations of 1, 2, ..., n, n-1, ..., 1.

Original entry on oeis.org

0, 1, 169, 24649, 3553225, 511709641, 73686731209, 10610895808969, 1527969074670025, 220027547690625481, 31683966878707771849, 4562491230669011577289, 7883984846509322664831433, 163482309777203435651765004745, 3389969175540090458609916107975113

Offset: 0

M. F. Hasler, Aug 01 2015

The first prime in this sequence is a(16) = A260871(11). Since a(12) is not prime, the base 12 is not listed in A260343.

			a(0) = 0 is the result of the empty sum corresponding to 0 digits.
a(2) = (12+1)^2 = 12^2 + 2*12 + 1 = 121_12, concatenation of (1, 2, 1).
a(13) = 123456789ab101110ba987654321_12 is the concatenation of (1, 2, 3, ..., 9, a, b, 10, 11, 10, b, ..., 1), where "b, 10, 11" are the base-12 representations of 11, 12, 13.

D. Broadhurst, Primes from concatenation: results and heuristics, NmbrThry List, August 1, 2015

Base-12 variant of A173426 (base 10) and A173427 (base 2). See A260853 - A260866 for variants in other bases.

a(n,b=12)=sum(i=1,#n=concat(vector(n*2-1,k,digits(min(k,n*2-k),b))),n[i]*b^(#n-i))

For n < b = 12, we have a(n) = R(b,n)^2, where R(b,n) = (b^n-1)/(b-1) are the base-b repunits.

A260863 Base-13 representation of a(n) is the concatenation of the base-13 representations of 1, 2, ..., n, n-1, ..., 1.

Original entry on oeis.org

0, 1, 196, 33489, 5664400, 957345481, 161792190756, 27342890695849, 4620948663553600, 780940325907974961, 131978915101424183716, 22304436652439380447009, 3769449794266138309731600, 8281481197999449959084458465, 236527384496061684935031509169004

Offset: 0

M. F. Hasler, Aug 01 2015

See A260343 for the bases b such that A260851(b) = A_b(b) = b*c + (c - b)*(1 + b*c), is prime, where A_b is the base-b sequence, as here with b = 13, and c = R(b,b) = (b^n-1)/(b-1) is the base-b repunit of length b.

			a(0) = 0 is the result of the empty sum corresponding to 0 digits.
a(2) = (13+1)^2 = 13^2 + 2*13 + 1 = 121_13, concatenation of (1, 2, 1).
a(14) = 123456789abc101110cba987654321_13 is the concatenation of (1, 2, 3, ..., 9, a, b, c, 10, 11, 10, c, ..., 1), where "c, 10, 11" are the base-13 representations of 12, 13, 14.

D. Broadhurst, Primes from concatenation: results and heuristics, NmbrThry List, August 1, 2015

Base-13 variant of A173426 (base 10) and A173427 (base 2). See A260853 - A260866 for variants in other bases.

a(n,b=13)=sum(i=1,#n=concat(vector(n*2-1,k,digits(min(k,n*2-k),b))),n[i]*b^(#n-i))

For n < b = 13, we have a(n) = A_b(n) = R(b,n)^2, where R(b,n) = (b^n-1)/(b-1) are the base-b repunits.

A261138 The concatenation of 123456...n and the reverse of this number.

Original entry on oeis.org

11, 1221, 123321, 12344321, 1234554321, 123456654321, 12345677654321, 1234567887654321, 123456789987654321, 1234567891001987654321, 12345678910111101987654321, 123456789101112211101987654321, 1234567891011121331211101987654321, 12345678910111213144131211101987654321

Offset: 1

Umut Uludag, Aug 10 2015

Let R(n) denote the number obtained by formally reversing the digits of n, including any leading zeros that may appear; a(n) is the decimal concatenation of 1,2,...,n,R(n),R(n-1),...,R(3),R(2),R(1). - N. J. A. Sloane, Dec 01 2021
A palindromic version of A173426.
Has same start as A259937, but A259937 generates non-palindromic terms for n>9.
All terms are multiples of 11 (cf. A349805).

			For n=10 we concatenate 1,2,3,...,10,01,9,8,...3,2,1 getting 1234567891001987654321.

Cf. A002477, A173426, A259937, A261570, A349805.

with(StringTools);
myReverse := n -> Reverse(convert(n,string));
A349804:=proc(n) local i,L,R;
L:=""; R:="";
for i from n to 1 by -1 do
L:=Join( [convert(i,string), L],"");
R:=Join( [R, myReverse(convert(i,string))],"");
od:
parse(Join([L,R],""));
end proc; # N. J. A. Sloane, Dec 01 2021
# second Maple program:
a:= n-> (s-> parse(cat(s, seq(s[-i], i=1..length(s)))))(cat("", $1..n)):
seq(a(n), n=1..14); # Alois P. Heinz, Dec 01 2021

Table[d = Flatten[IntegerDigits /@ Range@ n]; FromDigits@ Flatten[{d, Reverse@ d}], {n, 13}] (* Michael De Vlieger, Aug 20 2015 *)

def A349804(n): return int((lambda x: x+x[::-1])(''.join(str(d) for d in range(1,n+1)))) # Chai Wah Wu, Dec 01 2021

a(n) = concatenate( A007908(n), A138793(n) retaining leading zeros).

More than the usual number of terms are shown in order to distinguish this from several similar sequences.

Edited by N. J. A. Sloane, Dec 11 2021

A323532 Numbers k such that the decimal concatenation of the numbers from 1 up to k followed by digit reversals of the numbers from (k-1) down to 1 is a prime.

Original entry on oeis.org

10, 586, 2219

Offset: 1

Mikk Heidemaa, Jan 17 2019

The definition is related to the sequence discussed by N. J. A. Sloane (in Notices of the AMS (2018), Vol. 65, No. 9, pp. 1070-71) for which only a(1)-a(2) are known.

a(1) corresponds to a memorable prime (12345678910987654321); a(4) > 10000 (if it exists).

			10 is a term because 12345678910987654321 is a prime.
2219 is a term because 1...22172218221981227122...1 is a 15534-digit probable prime (where 8122 following 2219 corresponds to the digit reversal of 2218, 7122 to that of 2217, etc. down to 1).

N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, Notices, Amer. Math. Soc., 65 (No. 9, Oct. 2018), 1062-1074.

Cf. A173426 (similar but different concatenation scheme).

a[n_]:=Block[{cn=Drop[FoldList[Append, {}, ToString/@Range@n], 2]}, ParallelMap[If[PrimeQ[FromDigits@@{#<>Reverse@StringReverse@Most@#}], Length@#, Nothing]&, cn]]; a[2300]

f(n) = eval(concat(vector(2*n-1, k, if(k<=n, Str(k), concat(apply(x->Str(x), Vecrev(digits(2*n-k))))))));
isok(n) = ispseudoprime(f(n)); \\ Michel Marcus, Jan 20 2019

A350153 Prime numbers created by concatenating all numbers 1 through k for some k > 1, then continuing to concatenate all numbers from k-1 towards 1. Primes are added to the sequence as they are found as k increases.

Original entry on oeis.org

12343, 1234543, 12345678910987, 12345678910987654321, 12345678910111213141516171819202122212019181716151413, 12345678910111213141516171819202122232425262728293029

Offset: 1

Patrick Quam, Dec 16 2021

A173426(n) is the concatenation of all numbers from 1 up to k and then back down to 1. The prime terms of A173426 have been called "memorable primes" (see the Numberphile video). These "unmemorable primes" are a superset created by concatenating 1..k in ascending order followed by concatenating the numbers k-1..1 in descending order. Any primes found during either concatenation process are added to the sequence (e.g., k = 5, 1234543 is included. If 12345 were prime, it would be included as well).

			For k=10, the first prime obtained by concatenating the numbers 1..10 and then concatenating the first one or more numbers from 9..1 is 12345678910987.

Patrick Quam, Table of n, a(n) for n = 1..29
Brady Haran and N. J. A. Sloane, The Most Wanted Prime Number, Numberphile video (2021).

Cf. A048847, A007908, A173426, A260343, A323532.

select(isprime, [seq(seq(parse(cat($1..n, n-i$i=1..t)),
                t=0..n-1), n=1..30)])[];  # Alois P. Heinz, Dec 19 2021

lst={};Table[s=Flatten[IntegerDigits/@Range@n];k=n-1;
While[k!=-1,If[PrimeQ[p=FromDigits@s],AppendTo[lst,p]];s=Join[s,IntegerDigits@k];k--],{n,100}];lst (* Giorgos Kalogeropoulos, Dec 17 2021 *)

from itertools import count, chain, islice, accumulate
from sympy import isprime
def A350153gen(): return filter(lambda p:isprime(p),(int(s) for n in count(1) for s in accumulate(str(d) for d in chain(range(1,n+1),range(n-1,0,-1)))))
A350153_list = list(islice(A350153gen(),20)) # Chai Wah Wu, Dec 20 2021

cp's OEIS Frontend

A260856 Base-6 representation is the concatenation of the base-6 representations of 1, 2, ..., n, n-1, ..., 1.

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A260857 Base-7 representation of a(n) is the concatenation of the base-7 representations of 1, 2, ..., n, n-1, ..., 1.

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A260858 Base-8 representation of a(n) is the concatenation of the base-8 representations of 1, 2, ..., n, n-1, ..., 1.

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A260862 Base-12 representation of a(n) is the concatenation of the base-12 representations of 1, 2, ..., n, n-1, ..., 1.

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A260863 Base-13 representation of a(n) is the concatenation of the base-13 representations of 1, 2, ..., n, n-1, ..., 1.

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A261138 The concatenation of 123456...n and the reverse of this number.

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A323532 Numbers k such that the decimal concatenation of the numbers from 1 up to k followed by digit reversals of the numbers from (k-1) down to 1 is a prime.

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A350153 Prime numbers created by concatenating all numbers 1 through k for some k > 1, then continuing to concatenate all numbers from k-1 towards 1. Primes are added to the sequence as they are found as k increases.

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