A083754 -id:A083754 - cp's OEIS frontend

A083755 Primes arising in A083754.

13, 137, 13711, 137119, 13711927, 1371192763, 137119276331, 13711927633153, 1371192763315321, 137119276331532113, 13711927633153211383, 1371192763315321138333, 137119276331532113833339, 13711927633153211383333949, 1371192763315321138333394951, 137119276331532113833339495177, 13711927633153211383333949517787

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 06 2003

Also all terms above after a(4) have been certified prime with Primo. See A083754 for programs to generate this sequence. - Rick L. Shepherd, May 09 2003

Cf. A083754.

More terms from Rick L. Shepherd, May 09 2003

A336893 Lexicographically earliest infinite sequence of distinct positive terms such that the sum of digits of the first n terms is coprime to their concatenation.

Original entry on oeis.org

1, 3, 7, 2, 4, 5, 9, 6, 13, 8, 19, 11, 15, 21, 10, 17, 22, 23, 12, 24, 25, 14, 27, 16, 20, 28, 26, 31, 29, 18, 33, 37, 35, 39, 40, 41, 34, 42, 44, 43, 32, 45, 30, 46, 47, 36, 49, 38, 48, 51, 55, 53, 61, 50, 57, 60, 63, 52, 59, 64, 62, 66, 67, 54, 65, 58, 68, 69

Offset: 1

David James Sycamore and Michael De Vlieger, Aug 07 2020

Conjecture: A permutation of the positive integers.
Comment from N. J. A. Sloane, Aug 15 2020: Is there a proof that this is well-defined, i.e. that the sequence exists?  If so, the condition that a(1)=1 can be omitted from the definition.
Yes, this sequence is well defined: an upper limit for a(n+1) is given by N = concatenate(M, K) with M = max{ a(k); k <= n } and K = A068695(concatenate(a(1), ..., a(n), M)). This N is distinct from (since by construction larger than) all preceding terms, it will yield a prime number for the concatenation, certainly larger than its digit sum, so satisfies all required conditions. [This proof resulted from ideas from several OEIS editors and a new proof that A068695 is always well defined, see there.] - M. F. Hasler, Nov 09 2020

			Since a(1)=1, a(2) cannot be 2 because 1+2=3 and 3|12. However, 1+3=4 and GCD(13,4)=1, so a(2)=3.

G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers, Oxford University Press,1945,Chapter II.
G.A. Jones and J. Mary Jones, Elementary Number Theory, London: Springer-Verlag, 2005, Chapter 2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Plot of 1024 terms and 3000 terms.
Michael De Vlieger, Plot of a(n) - n for 1 <= n <= 3000.

Cf. A068695, A083754, A045572

#Code by Carl Love; (Mapleprimes)
Seq1 := proc(N::posint)
local
  S:=Array(1 .. 1, [1]),
SD:=1,
C:=1,
  Used := table([1= ()]),
  k, j, C1, SD1;
  for k from 2 to N do
      for j from 2 do
          if not assigned(Used[j]) then
             C1 := Scale10(C, length(j))+j;
             SD1 := SD+`+`(convert(j, base, 10)[]);
             if igcd(C1, SD1) = 1 then
                 C := C1; SD := SD1; Used[j] :=() ; S(k) := j;
                 break
             end if
         end if
       end do
     end do;
    seq(x,x=S)
  end proc:
  Seq1(200);

Nest[Append[#, Block[{k = 2, d = Map[IntegerDigits, #]}, While[Nand[FreeQ[#, k], GCD[FromDigits[#], Total[#]] &@ Flatten@ Append[d, IntegerDigits[k]] == 1], k++]; k]] &, {1}, 100]

A113578 a(1) = 1, then the rearrangement of odd palindromes such that every concatenation is a prime for n > 1.

Original entry on oeis.org

1, 3, 7, 11, 9, 111, 33, 99, 717, 151, 383, 969, 3003, 3663, 141, 121, 10101, 11711, 393, 11811, 363, 979, 77, 34443, 171, 14941, 989, 919, 707, 34243, 929, 7557, 18781, 18681, 131, 11511, 30303, 10701, 12421, 12321, 747, 7667, 1441, 14841, 13431, 797, 16861

Offset: 1

Amarnath Murthy, Nov 06 2005

Since the first 5 terms of A083754 are odd palindromes (A029950), they are also the first 5 terms of this sequence. - Michel Marcus, Feb 06 2014

			13, 137, 13711, 137119, 137119111, 13711911133, ...,  are all prime.

Cf. A113577, A113579.

findnew(va, n, vp, ilast) = {s= ""; for (i=1, n-1, s = concat(s, Str(va[i]));); ok = 0; i = 2; while (!ok, if (vp[i] != 0, ns = concat(s, Str(vp[i])); if (isprime(eval(ns)), ok = 1);); if (!ok, i++); if (i > #vp, return (0));); i;}
lista(nn) = {vn = vector(nn, i, i); vp = select(n->is_A002113(n), vn); va = vector(nn); va[1] = 1; print1(va[1], ", "); vp[1] = 0; ilast = 1; for (n = 2, vecmax(vp), inew = findnew(va, n, vp, ilast); if (! inew, break); va[n] = vp[inew]; vp[inew] = 0; print1(va[n], ", "); ilast = inew;);} \\ Michel Marcus, Feb 06 2014

Corrected and extended by Michel Marcus, Feb 06 2014

A289994 a(1) = 1, a(2) = 2, a(n) = smallest number not occurring earlier such that the concatenation a(n-2), a(n-1) and a(n) is a prime.

Original entry on oeis.org

1, 2, 7, 11, 19, 13, 9, 21, 43, 3, 31, 49, 17, 37, 29, 41, 27, 39, 67, 33, 81, 53, 51, 23, 89, 91, 59, 47, 73, 61, 87, 99, 79, 63, 117, 113, 77, 127, 103, 93, 111, 71, 69, 101, 147, 57, 97, 133, 119, 121, 139, 131, 107, 157, 151, 123, 153, 83, 143, 169, 109, 137

Offset: 1

Seiichi Manyama, Sep 02 2017

			127, 2711, 71119, 111913, ... are primes.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A075607, A073676, A083754, A110772.

cp's OEIS Frontend

A083755 Primes arising in A083754.

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A336893 Lexicographically earliest infinite sequence of distinct positive terms such that the sum of digits of the first n terms is coprime to their concatenation.

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Maple

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A113578 a(1) = 1, then the rearrangement of odd palindromes such that every concatenation is a prime for n > 1.

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PARI

Extensions

A289994 a(1) = 1, a(2) = 2, a(n) = smallest number not occurring earlier such that the concatenation a(n-2), a(n-1) and a(n) is a prime.

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