A260853 -id:A260853 - cp's OEIS frontend

A261170 Number of decimal digits of A260871(n), where A260871 lists primes whose base-b representation is the concatenation of the base-b representations of (1, 2, ..., k, k-1, ..., 1).

2, 3, 4, 5, 7, 9, 17, 20, 31, 38, 43, 64, 64, 70, 91, 93, 102, 117, 120, 123, 127, 127, 127, 136, 160, 166, 176, 235, 321, 351, 353, 389, 403, 418, 418, 421, 422, 466, 542, 578, 579, 703, 706, 725, 731, 765, 780, 792, 795, 799, 803, 839, 840, 848, 849, 863

Offset: 1

M. F. Hasler, Sep 15 2015

Larger values based on computations by D. Broadhurst, cf. data file in A260871.

See A261171 and A261172 for the k- and b-values such that A260871(n) = A[b](k), where the sequences A[b] of numbers whose base-b representation is the concatenation of the base-b representations of (1, 2, ..., k, k-1, ..., 1) are listed in A173427, A260853 - A260859, A173426, A260861 - A260866 and A260860 for bases b=2, ..., b=16 and b=60.

M. F. Hasler, Table of n, a(n) for n = 1..434

A360504 Concatenate the ternary strings for 1,2,...,n-1, n, n-1, ..., 2,1.

Original entry on oeis.org

1, 121, 121021, 1210111021, 12101112111021, 121011122012111021, 1210111220212012111021, 12101112202122212012111021, 1210111220212210022212012111021, 1210111220212210010110022212012111021, 1210111220212210010110210110022212012111021, 1210111220212210010110211010210110022212012111021

Offset: 1

N. J. A. Sloane, Feb 17 2023

If the terms are read as ternary strings and converted to base 10, we get A260853. For example, a(3) = 121021_3 = 439_10, which is A260853(3). This is a prime. What is the next prime term?

If the terms are read as decimal numbers, which of them are primes? a(3) = 121021_10 is a decimal prime, but what is the next one? It is a surprise that 121021 is a prime in both base 3 and base 10.

			To get a(3) we concatenate 1, 2, 10, 2, and 1, getting 121021.

Winston de Greef, Table of n, a(n) for n = 1..123
Index entries for sequences related to Most Wanted Primes video

Cf. A007089, A360502, A360503.

This is the ternary analog of A173426.

t:= n-> (l-> parse(cat(seq(l[-i], i=1..nops(l)))))(convert(n, base, 3)):
a:= n-> parse(cat(map(t, [$1..n, n-i$i=1..n-1])[])):
seq(a(n), n=1..12);  # Alois P. Heinz, Feb 17 2023

Table[FromDigits[Flatten[Join[IntegerDigits[#,3]&/@Range[n],IntegerDigits[#,3]&/@ Range[ n-1,1,-1]]]],{n,20}] (* Harvey P. Dale, Oct 01 2023 *)

from sympy.ntheory import digits
def a(n): return int("".join("".join(map(str, digits(k, 3)[1:])) for k in list(range(1, n+1))+list(range(n-1, 0, -1))))
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Feb 18 2023

# faster version for initial segment of sequence
from sympy.ntheory import digits
from itertools import count, islice
def agen(): # generator of terms
    sf, sr = "", ""
    for n in count(1):
        sn = "".join(map(str, digits(n, 3)[1:]))
        sf += sn
        yield int(sf + sr)
        sr = sn + sr
print(list(islice(agen(), 15))) # Michael S. Branicky, Feb 18 2023

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

Original entry on oeis.org

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.

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))

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

Original entry on oeis.org

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.

cp's OEIS Frontend

A261170 Number of decimal digits of A260871(n), where A260871 lists primes whose base-b representation is the concatenation of the base-b representations of (1, 2, ..., k, k-1, ..., 1).

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A360504 Concatenate the ternary strings for 1,2,...,n-1, n, n-1, ..., 2,1.

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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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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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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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