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.

Showing 1-10 of 13 results. Next

A260852 Primes in A260851: numbers whose base n expansion is equal to the concatenation of the base n expansions of (1, 2, 3, ..., n-1, n, n-1, ..., 3, 2, 1).

Original entry on oeis.org

13, 439, 27961, 522134761, 21107054541321649, 12345678910987654321, 24197857203266734883076090685781525281
Offset: 1

Views

Author

M. F. Hasler, Aug 02 2015

Keywords

Comments

Primes of the form (1+r(b)*b)*(r(b)-b+1)-1 with r(b)=(b^b-1)/(b-1).
Sequences A173427, A260853 - A260859, A173426, A260861 - A260866, A260860 list the numbers whose base b expansion is the concatenation of the base b expansions of (1, 2, ..., n, n-1, ..., 1). For n < b these are the squares of the repdigits of length n in base b, so the first candidate is the b-th term. These are the numbers listed in A260851. For the bases listed in A260343, this candidate is indeed prime: these are the primes listed here.
a(8) = A260851(40) has already 127 digits and is therefore too large to be displayed here.

Links

Crossrefs

Cf. A173427, A260853 - A260859, A173426, A260861 - A260866, A260860, A260851.

Programs

  • PARI
    for(b=2,999,ispseudoprime(p=(1+b*c=(b^b-1)\(b-1))*(c-b+1)-1)&&print1(p","))

Formula

a(n) = A260851(A260343(n)).

A260853 Base-3 representation of a(n) is the concatenation of the base-3 representations of 1, 2, ..., n, n-1, ..., 1.

Original entry on oeis.org

0, 1, 16, 439, 35350, 2864599, 232046890, 18795930559, 1522471570630, 369960528437035, 269701223137448146, 196612191672080116867, 143330287729139571972130, 104487779754548024866115515, 76171591441065652665051372946, 55529090160536864641400481743827
Offset: 0

Views

Author

M. F. Hasler, Aug 01 2015

Keywords

Comments

The base 3 is listed in A260343, which means that a(3) = A260851(3) = 439 = 121021_3 is prime and therefore in A260852. See these sequences for more information.

Examples

			a(0) = 0 is the result of the empty sum corresponding to 0 digits.
a(2) = 16 = 121_3 is the concatenation of (1, 2, 1).
a(3) = 439 = 121021_3 is the concatenation of (1, 2, 10, 2, 1), where the middle "10" is the base-3 representation of 3.

Links

Crossrefs

Base-3 variant of A173426 (base 10) and A173427 (base 2). See A260854 - A260866 for variants in other bases b = 4, ..., 16.

Programs

  • Mathematica
    Join[{0},Table[FromDigits[Join[Flatten[IntegerDigits[Range[n], 3]], Flatten[ Reverse[ Most[ IntegerDigits[Range[n],3]]]]],3],{n,20}]] (* Harvey P. Dale, Mar 11 2019 *)
  • PARI
    a(n,b=3)=sum(i=1,#n=concat(vector(n*2-1,k,digits(min(k,n*2-k),b))),n[i]*b^(#n-i))

A260343 Numbers n such that the base-n number formed by concatenating the base-n numbers 1 2 ... n-1 n n-1 ... 2 1 is prime.

Original entry on oeis.org

2, 3, 4, 6, 9, 10, 16, 40, 104, 8840
Offset: 1

Views

Author

N. J. A. Sloane, Aug 01 2015

Keywords

Comments

n = 8840 only corresponds to a probable prime (with 69770 decimal digits).
The concatenation (in base n) of the base-n numbers 1 2 3 ... k-1 k k-1 ... 3 2 1 is a square for k
Sequence A260852 lists the actual primes, A260852(k) = A260851(a(k)). - M. F. Hasler, Aug 02 2015

Examples

			For n = 2 we get the binary number 1 10 1 = 1101 = 13 (in decimal).
For n = 10 we get (as David Broadhurst remarks) the "memorable" decimal prime 12345678910987654321.
For n = 16 the prime is the hexadecimal number 123456789abcdef10fedcba987654321.

Links

Crossrefs

For n=2 see A173427, for n=10 see A173426.
For n=3 through n=16 see A260853 - A260866.

Programs

  • Mathematica
    Select[Range[2, 120], PrimeQ@ Total[Times @@@ Transpose[{(Function[p, Power[#, p]] /@ Reverse@ Delete[Range[0, 2 # - 1], # + 1]), Flatten@ {Range[#], Reverse@ Range[# - 1]}}]] &] (* Michael De Vlieger, Aug 02 2015 *)
bnpQ[n_]:=PrimeQ[FromDigits[Flatten[Join[IntegerDigits[#,n]&/@Range[n], IntegerDigits[ #,n]&/@Reverse[Range[n-1]]]],n]]; Select[Range[2,110],bnpQ] (* Harvey P. Dale, Feb 26 2023 *)
  • PARI
    for(b=2,9e9,ispseudoprime(p=(1+b*c=(b^b-1)\(b-1))*(c-b+1)-1)&&print1(b", ")); \\ D. Broadhurst and M. F. Hasler, Aug 02 2015
  • Python
    from gmpy2 import is_prime
def intbase(dlist,b=10): # convert list of digits in base b to integer
    y = 0
    for d in dlist:
        y = y*b + d
    return y
A260343_list = [n for n in range(2,500) if is_prime(intbase(list(range(1,n))+[1,0]+list(range(n-1,0,-1)), n))] # Chai Wah Wu, Aug 01 2015

A260859 Base-9 representation of a(n) is the concatenation of the base-9 representations of 1, 2, ..., n, n-1, ..., 1.

Original entry on oeis.org

0, 1, 100, 8281, 672400, 54479161, 4412944900, 357449732641, 28953439105600, 21107054541321649, 138483384602892402628, 908589486379899193778809, 5961255620138564686107812272, 39111798123729126657669459066697, 256612507489786800304910707633347364
Offset: 0

Views

Author

M. F. Hasler, Aug 01 2015

Keywords

Comments

The base 9 is listed in A260343, because a(9) = A260851(9) = 21107054541321649 = 123456781087654321_9 is prime and therefore in A260852. See these sequences for more information.

Examples

			a(0) = 0 is the result of the empty sum corresponding to 0 digits.
a(2) = 100 = (9+1)^2 = 9^2 + 2*9 + 1 = 121_9, concatenation of (1, 2, 1).
a(10) = 1234567810111087654321_9 is the concatenation of (1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 10, 8, 7, 6, 5, 4, 3, 2, 1), where the middle "10, 11, 10" are the base-9 representations of 9, 10, 9.

Links

Crossrefs

Base-9 variant of A173426 (base 10) and A173427 (base 2). See A260853 - A260866 for the variants in other bases.

Programs

  • PARI
    a(n,b=9)=sum(i=1,#n=concat(vector(n*2-1,k,digits(min(k,n*2-k),b))),n[i]*b^(#n-i))

Formula

For n < b = 9, 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.

A260860 Base-60 representation of a(n) is the concatenation of the base-60 representations of 1, 2, ..., n, n-1, ..., 1.

Original entry on oeis.org

0, 1, 3721, 13402921, 48250954921, 173703464074921, 625332472251274921, 2251196900199483274921, 8104308840723833403274921, 29175511826606141868603274921, 105031842575782131223980603274921, 378114633272815673636150700603274921
Offset: 0

Views

Author

M. F. Hasler, Aug 01 2015

Keywords

Comments

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=60, and c = R(b,b) = (b^b-1)/(b-1) is the base-b repunit of length b.

Examples

			a(0) = 0 is the result of the empty sum corresponding to 0 digits.
a(2) = (60+1)^2 = 60^2 + 2*60 + 1 = 121_60, concatenation of (1, 2, 1).
a(61) = 123...101110...21_60, which is the concatenation of (1, 2, 3, ..., 10, 11, 10, ..., 2, 1), where the middle "10, 11, 10" are the base-60 representations of 60, 61, 60.

Links

Crossrefs

Base-60 variant of A173426 (base 10) and A173427 (base 2). See A260853 - A260866 for variants in other bases.

Programs

  • PARI
    a(n,b=60)=sum(i=1,#n=concat(vector(n*2-1,k,digits(min(k,n*2-k),b))),n[i]*b^(#n-i))

Formula

For n < b = 60, 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.

A261171 Value of k for which A260871(n) = A[b](k), with b = A261172(n); A[b](k) = the number whose base-b representation is the concatenation of the base-b representations of (1, ..., k, k-1, ..., 1).

Original entry on oeis.org

2, 3, 4, 4, 5, 6, 9, 10, 13, 16, 16, 21, 23, 23, 29, 28, 38, 39, 33, 34, 41, 40, 37, 37, 41, 42, 44, 64, 77, 82, 75, 83, 83, 87, 104, 104, 86, 94
Offset: 1

Views

Author

M. F. Hasler, Aug 23 2015

Keywords

Comments

For more data, see the 3rd column of D. Broadhurst's list of [n, b, k, length(A260871(n))] given in A260871.
This and the companion sequence A261172 are a compact way of recording the very large primes listed in A260871 by means of the k- and b-value such that A260871(n) = A[A261172(n)](A261171(n)). See A261170 for the number of decimal digits of these primes. - M. F. Hasler, Sep 15 2015

Examples

			A260871(1) = A[2](2), therefore a(1) = 2.
A260871(2) = A[3](3), therefore a(2) = 3.
A260871(3) = A[2](4), therefore a(3) = 4.

Crossrefs

Cf. A173427, A260853 - A260859, A173426, A260861 - A260866 and A260860 for A[b] with b=2, ..., b=16 and b=60.
See also A260852 = { primes of the form A260851(b) = A[b](b), b in A260343 }.

Programs

  • PARI
    A261171_list(LIM=1e499)={my(A=List(),p,d);for(b=2,9e9,for(n=b,9e9,if(LIMb)));ispseudoprime(p)&&listput(A,[log(p),n])));apply(t->t[2],vecsort(A))}

Formula

A260871(n) = A[A261172(n)](a(n)), where A[b](k) = Sum_{i=1..#d} d[i]*b^(#d-i), d = concatenation of (1, 2, ..., k, k-1, ..., 1) all written in base b.

A261172 Value of b for which A260871(n) = A[b](k), with k = A261171(n); A[b](k) = the number whose base-b representation is the concatenation of the base-b representations of (1, ..., k, k-1, ..., 1).

Original entry on oeis.org

2, 3, 2, 4, 3, 6, 9, 10, 11, 16, 12, 14, 22, 18, 25, 20, 2, 6, 18, 14, 7, 40, 31, 25, 23, 20, 22, 62, 65, 68, 29, 23, 38, 26, 104, 6, 34, 52
Offset: 1

Views

Author

M. F. Hasler, Aug 23 2015

Keywords

Comments

For more data, see the 2nd column of D. Broadhurst's list of [n, b, k, length(A260871(n))] given in A260871.

Examples

			A260871(1) = A[2](2), therefore a(1) = 2.
A260871(2) = A[3](3), therefore a(2) = 3.
A260871(3) = A[2](4), therefore a(3) = 2.

Crossrefs

Cf. A173427, A260853 - A260859, A173426, A260861 - A260866 and A260860 for A[b] with b=2, ..., b=16 and b=60.
See also A260852 = { primes of the form A260851(b) = A[b](b), b in A260343 }.

Programs

  • PARI
    A261172_list(LIM=1e499)={my(A=List(),p,d);for(b=2,9e9,for(n=b,9e9,if(LIMb)));ispseudoprime(p)&&listput(A,[log(p),n])));apply(t->t[2],vecsort(A))}

Formula

A260871(n) = A[a(n)](A261171(n)), where A[b](k) = Sum_{i=1..#d} d[i]*b^(#d-i), d = concatenation of (1, 2, ..., k, k-1, ..., 1) all written in base b.

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

Views

Author

M. F. Hasler, Aug 01 2015

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • PARI
    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

Views

Author

M. F. Hasler, Aug 01 2015

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Base-14 variant of A173426 (base 10) and A173427 (base 2). See A260853 - A260866 for variants in other bases.
For primes see A261408.

Programs

  • PARI
    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))

Formula

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.

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

Views

Author

M. F. Hasler, Aug 01 2015

Keywords

Comments

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.

Examples

			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.

Links

Programs

  • PARI
    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))
Showing 1-10 of 13 results. Next

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