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 1-6 of 6 results.

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.

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).

Original entry on oeis.org

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

Views

Author

M. F. Hasler, Sep 15 2015

Keywords

Comments

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.

Links

A260861 Base-11 representation of a(n) is the concatenation of the base-11 representations of 1, 2, ..., n, n-1, ..., 1.

Original entry on oeis.org

0, 1, 144, 17689, 2143296, 259371025, 31384248336, 3797497946089, 459497294348544, 55599173087763361, 6727499948806851600, 8954302429379707945271, 131099941868210323821706774, 1919434248892467772593071038679, 28102436838034620750856132266604106
Offset: 0

Views

Author

M. F. Hasler, Aug 01 2015

Keywords

Comments

The first prime in this sequence is a(13) = A260871(9). Since a(11) is not prime, the base 11 is not listed in A260343.

Examples

			a(0) = 0 is the result of the empty sum corresponding to 0 digits.
a(2) = (11+1)^2 = 11^2 + 2*11 + 1 = 121_11, concatenation of (1, 2, 1).
a(12) = 123456789a101110a987654321_11 is the concatenation of (1, 2, 3, ..., 9, a, 10, 11, 10, a, 9, ..., 1), where "a, 10, 11" are the base-11 representations of 10, 11, 12.

Links

Crossrefs

Base-11 variant of A173426 (base 10) and A173427 (base 2). See A260853 - A260866 for variants in other bases.

Programs

  • PARI
    a(n,b=11)=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 = 11, we have a(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

Views

Author

N. J. A. Sloane, Aug 22 2015

Keywords

Comments

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

References

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

Crossrefs

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

Extensions

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

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

Views

Author

M. F. Hasler, Aug 01 2015

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • PARI
    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))

Formula

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.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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