A260343 -id:A260343 - cp's OEIS frontend

A260854 Base-4 representation of a(n) is the concatenation of the base-4 representations of 1, 2, ..., n, n-1, ..., 1.

0, 1, 25, 441, 27961, 7148857, 1830131001, 468514084153, 119939614479673, 30704541449950521, 7860362613477971257, 2012252829087011018041, 515136724246861226808633, 131875001407205856562222393, 33760000360244849399916500281, 8642560092222683848298425324857

Offset: 0

M. F. Hasler, Aug 01 2015

The base 4 is listed in A260343, which means that a(4) = A260851(4) = 27961 is prime and therefore in A260852. See these sequences for more information.

			a(0) = 0 is the result of the empty sum corresponding to 0 digits.
a(2) = 25 = 16 + 2*4 + 1 = 121_4 is the concatenation of (1, 2, 1).
a(4) = 27961 = 12310321_4 is the concatenation of (1, 2, 3, 10, 3, 2, 1), where the middle "10" is the base-4 representation of 4.

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

Base-4 variant of A173426 (base 10) and A173427 (base 2). See A260853 - A260866 for variants in other bases b = 3, ..., 16.

a(n,b=4)=sum(i=1,#n=concat(vector(n*2-1,k,digits(min(k,n*2-k),b))),n[i]*b^(#n-i))

A260864 Base-14 representation of a(n) is the concatenation of the base-14 representations of 1, 2, ..., n, n-1, ..., 1.

Original entry on oeis.org

0, 1, 225, 44521, 8732025, 1711559641, 335466848025, 65751518430361, 12887297839395225, 2525910379700086681, 495078434465717705625, 97035373155903680328601, 19018933138565843484771225, 3727710895159027432980276121, 10228838696316240496325238416281

Offset: 0

M. F. Hasler, Aug 01 2015

See A260343 for the bases b such that A260851(b) = A_b(b) = b*r + (r - b)*(1 + b*r), is prime, where A_b is the base-b sequence, as here with b=14, and r = (b^b-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) = (14+1)^2 = 14^2 + 2*14 + 1 = 121_14, concatenation of (1, 2, 1).
a(15) = 123456789abcd101110dcba987654321_14 is the concatenation of (1, 2, 3, ..., 9, a, b, c, d, 10, 11, 10, d, ..., 1), where "d, 10, 11" are the base-14 representations of 13, 14, 15.

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

Base-14 variant of A173426 (base 10) and A173427 (base 2). See A260853 - A260866 for variants in other bases.

For primes see A261408.

a(n,b=14)=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 = 14, we have a(n) = R(14,n)^2, where R(b,n) = (b^n-1)/(b-1) are the base-b repunits.

A260865 Base-15 representation of a(n) is the concatenation of the base-15 representations of 1, 2, ..., n, n-1, ..., 1.

Original entry on oeis.org

0, 1, 256, 58081, 13075456, 2942086081, 661970995456, 148943498386081, 33512287502995456, 7540264693665886081, 1696559556157202995456, 381725900136606353386081, 85888327530754964702995456, 19324873694420145086040886081

Offset: 0

M. F. Hasler, Aug 01 2015

See A260343 for the bases b such that B(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=15, 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) = (15+1)^2 = 15^2 + 2*15 + 1 = 121_15, concatenation of (1, 2, 1).
a(16) = 123456789abcde101110edcba987654321_15 is the concatenation of (1, 2, 3, ..., 9, a, ..., e, 10, 11, 10, e, d, ..., 1), where "e, 10, 11" are the base-15 representations of 14, 15, 16.

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

Base-15 variant of A173426 (base 10) and A173427 (base 2). See A260853 - A260866 for variants in other bases.

a(n,b=15)=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 = 15, 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.

A261408 Numbers k such that A260864(k) is a prime or probable prime, where A260864(k) is the number whose base-14 representation is the concatenation of the base-14 representations of (1, ..., k, k-1, ..., 1).

Original entry on oeis.org

21, 34, 192, 310, 373, 536, 12000

Offset: 1

N. J. A. Sloane, Aug 22 2015

The subsequence of those A261171(n) for which A261172(n) = 14. - M. F. Hasler, Aug 23 2015

David Broadhurst, email to N. J. A. Sloane, Aug 23 2015

Cf. A260343, A260864.

See also A260871 which lists all primes of the form A[b](k), and A261171 and A261172 which list the k and b values of these primes. - M. F. Hasler, Aug 23 2015

More explicit title from M. F. Hasler, Aug 23 2015

A260855 Base-5 representation of a(n) is the concatenation of the base-5 representations of 1, 2, ..., n, n-1, ..., 1.

Original entry on oeis.org

0, 1, 36, 961, 24336, 3034961, 1896581836, 1185364159961, 740852620019336, 463032888020409961, 289395555025471581836, 180872221891237629784961, 113045138682031465901269336, 70653211676269864870442284961, 44158257297668670511080159081836

Offset: 0

M. F. Hasler, Aug 01 2015

Base-5 variant of A173426 (base 10) and A173427 (base 2). See A260853 - A260866 for variants in other bases b = 3, ..., 16.

The base 5 is not listed in A260343, because a(5) = A260851(5) = 3034961 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) = 36 = (5+1)^2 = 5^2 + 2*5 + 1 = 121_4 is the concatenation of (1, 2, 1).
a(5) = 3034961 = 1234104321_5 is the concatenation of (1, 2, 3, 4, 10, 4, 3, 2, 1), where the middle "10" is the base-5 representation of 5.

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

a(n,b=5)=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.

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

A260854 Base-4 representation of a(n) is the concatenation of the base-4 representations of 1, 2, ..., n, n-1, ..., 1.

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A260864 Base-14 representation of a(n) is the concatenation of the base-14 representations of 1, 2, ..., n, n-1, ..., 1.

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A260865 Base-15 representation of a(n) is the concatenation of the base-15 representations of 1, 2, ..., n, n-1, ..., 1.

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A261408 Numbers k such that A260864(k) is a prime or probable prime, where A260864(k) is the number whose base-14 representation is the concatenation of the base-14 representations of (1, ..., k, k-1, ..., 1).

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A260855 Base-5 representation of a(n) is the concatenation of the base-5 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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