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.

A086578 a(n) = 7*(10^n - 1).

Original entry on oeis.org

0, 63, 693, 6993, 69993, 699993, 6999993, 69999993, 699999993, 6999999993, 69999999993, 699999999993, 6999999999993, 69999999999993, 699999999999993, 6999999999999993, 69999999999999993, 699999999999999993, 6999999999999999993, 69999999999999999993, 699999999999999999993
Offset: 0

Views

Author

Ray Chandler, Jul 22 2003

Keywords

Comments

Original definition: a(n) = k where R(k+7) = 7.

Crossrefs

Cf. A002275, A004086 (R(n)).
One of family of sequences of form a(n) = k, where R(k+m) = m, m=1 to 9; m=1: A002283, m=2: A086573, m=3: A086574, m=4: A086575, m=5: A086576, m=6: A086577, m=7: A086578, m=8: A086579, m=9: A086580.
Sequences of the form m*10^n - 7: 3*A033175 (m=1, 10), A086943 (m=3), 3*A185127 (m=4), this sequence (m=7), A100412 (m=8).

Programs

  • Magma
    [7*(10^n -1): n in [0..20]]; // G. C. Greubel, Apr 14 2023
    
  • Mathematica
    LinearRecurrence[{11,-10}, {0,63}, 31] (* G. C. Greubel, Apr 14 2023 *)
  • SageMath
    [7*(10^n -1) for n in range(21)] # G. C. Greubel, Apr 14 2023

Formula

a(n) = 7*9*A002275(n) = 7*A002283(n).
R(a(n)) = A086575(n).
From Chai Wah Wu, Jul 08 2016: (Start)
a(n) = 11*a(n-1) - 10*a(n-2) for n > 1.
G.f.: 63*x/((1 - x)*(1 - 10*x)). (End)
E.g.f.: 7*(exp(10*x) - exp(x)). - G. C. Greubel, Apr 14 2023

Extensions

Edited by Jinyuan Wang, Aug 04 2021

A100413 Numbers k such that k is reversal(k)-th even composite number (k is A004086(k)-th even composite number).

Original entry on oeis.org

52, 592, 5992, 59992, 599992, 5999992, 59999992, 599999992, 5999999992, 59999999992, 599999999992, 5999999999992, 59999999999992, 599999999999992, 5999999999999992, 59999999999999992, 599999999999999992
Offset: 1

Views

Author

Farideh Firoozbakht, Dec 08 2004

Keywords

Examples

			592 is in the sequence because 592 is the 295th even composite number.
		

Crossrefs

Programs

Formula

a(n) = 6*10^n - 8.
a(n) = 2*(A086943(n) + 3). - Martin Ettl, Nov 08 2012
From Colin Barker, Oct 14 2014: (Start)
a(n) = 10*a(n-1) + a(n-2) - 10*a(n-3).
G.f.: 4*x*(13+5*x)/((1-x)*(1-10*x)). (End)
E.g.f.: 2 (1 - 4*exp(x) + 3*exp(10*x)). - G. C. Greubel, Apr 13 2023

A169830 Numbers k such that 2*reverse(k) - k = 1.

Original entry on oeis.org

1, 73, 793, 7993, 79993, 799993, 7999993, 79999993, 799999993, 7999999993, 79999999993, 799999999993, 7999999999993, 79999999999993, 799999999999993, 7999999999999993, 79999999999999993, 799999999999999993, 7999999999999999993, 79999999999999999993, 799999999999999999993
Offset: 1

Views

Author

N. J. A. Sloane, May 31 2010

Keywords

Comments

The sequence is infinite since it contains all numbers of the form 799...9993. (Cf. A101155, A101849.) [Ulrich Krug (leuchtfeuer37(AT)gmx.de), Jun 02 2010]
All numbers of the form 8*10^k-7 are members, but are there any others? - Robert G. Wilson v, Jun 01 2010
All solutions are of the form 8*10^k-7. - David Radcliffe, Jul 25 2015

Crossrefs

Same sequence as A100412.
Digit reversals of A083818.

Programs

  • Mathematica
    k = 1; lst = {}; fQ[n_] := 2 FromDigits@ Reverse@ IntegerDigits@n == 1 + n; While[k < 10^8, If[fQ@k, Print@k; AppendTo[lst, k]]; k++ ]; lst (* Robert G. Wilson v, Jun 01 2010 *)
    Rest@ CoefficientList[Series[x (1 + 62 x)/((1 - x) (1 - 10 x)), {x, 0, 20}], x] (* or *)
    Table[If[n == 1, 1, FromDigits@ Join[{7}, ConstantArray[9, n - 2], {3}]], {n, 20}] (* or *)
    LinearRecurrence[{11, -10}, {1, 73}, 20] (* Michael De Vlieger, Feb 12 2017 *)
  • PARI
    isok(n) = 2*fromdigits(Vecrev(digits(n))) - n == 1; \\ Michel Marcus, Feb 12 2017

Formula

a(n) = 8*10^(n-1) - 7. - David Radcliffe, Jul 25 2015
From Matthew House, Feb 12 2017: (Start)
G.f.: x*(1+62*x)/((1-x)*(1-10*x)).
a(n) = 11*a(n-1) - 10*a(n-2). (End)
E.g.f.: (31 - 35*exp(x) + 4*exp(10*x))/5. - Elmo R. Oliveira, Jun 12 2025

Extensions

a(6)-a(8) from Robert G. Wilson v, Jun 01 2010
More terms from David Radcliffe, Jul 25 2015

A100414 Numbers n such that n is R(n)-th composite number where R(n) is the digit reversal of n (A002808(A004086(n))=n).

Original entry on oeis.org

21, 48034, 69926, 180461, 214591, 409473, 563715, 41630193, 253385633342, 661494322636
Offset: 1

Views

Author

Farideh Firoozbakht, Dec 10 2004

Keywords

Comments

There is no further term < 3*10^9.
a(11) > 3*10^12. [Donovan Johnson, Jun 14 2009]

Examples

			41630193 is in the sequence because 41630193 is the 39103614th composite number.
		

Crossrefs

Programs

  • Mathematica
    Do[s=FromDigits[Reverse[IntegerDigits[n]]];If[s
    				

Extensions

a(9)-a(10) from Donovan Johnson, Jun 14 2009

A136543 Numbers n such that phi(n)+sigma(n)=4*reversal(n).

Original entry on oeis.org

793, 79993, 2152311, 79999993, 7999999993, 799999999993
Offset: 1

Views

Author

Farideh Firoozbakht, Jan 20 2008

Keywords

Comments

All semiprimes of the form 8*10^m-7 are in the sequence - the proof is easy. Next term is greater than 10^8.
a(7) > 10^12. - Giovanni Resta, Nov 03 2012
A100412(n) is in the sequence for n = 2, 4, 7, 9, 11, 16, 18, 23, 31, 32, 40,... - M. F. Hasler, Nov 03 2012

Examples

			phi(2152311)+sigma(2152311)=1217664+3312384=4*1132512=4*reversal(2152311), so 2152311 is in the sequence.
		

Programs

  • Mathematica
    Do[If[4*FromDigits@Reverse@IntegerDigits@n==EulerPhi@n+ DivisorSigma[1,n],Print[n]],{n,100000000}]

Extensions

a(5)-a(6) from Giovanni Resta, Nov 03 2012

A384597 Integers k such that k + 1 has a divisor that is an anagram of k, which must have the same number of digits as k.

Original entry on oeis.org

1, 41, 73, 631, 793, 6031, 6391, 6733, 7412, 7520, 7993, 8627, 9710, 25147, 37112, 43916, 49316, 51427, 60031, 60391, 60733, 62314, 63214, 63991, 66331, 67393, 67933, 70211, 71132, 72101, 74102, 74912, 75020, 75290, 78260, 79993, 81103, 85712, 86927, 89627
Offset: 1

Views

Author

Gonzalo Martínez, Jun 04 2025

Keywords

Comments

This sequence has infinitely many terms, since 60*10^m + 31 is a term for all positive integers m, as (60*10^m + 31) + 1 = 2*(30*10^m + 16).
A100412 is a subsequence of a(n), since if m is in A100412, then m + 1 = 2*reversal(m).

Examples

			73 is in this sequence since 73 + 1 = 37*2, where 37 is an anagram of 73.
		

Crossrefs

Cf. A100412.

Programs

  • Mathematica
    {1}~Join~Select[Range[100000],ContainsAny[IntegerDigits/@Divisors[#+1],Complement[Permutations[IntegerDigits[#]],{IntegerDigits[#]}]]&] (* James C. McMahon, Jun 10 2025 *)
  • PARI
    isok(k) = my(s=vecsort(digits(k))); fordiv(k+1, d, if (vecsort(digits(d)) == s, return(1))); \\ Michel Marcus, Jun 04 2025
  • Python
    def ok(k):
        return any((k+1)%d==0 and sorted(str(d))==sorted(str(k)) and len(str(d))==len(str(k)) for d in range(1,k+2))
    print(", ".join(map(str, [k for k in range(1, 100000) if ok(k)])))
    
Showing 1-6 of 6 results.