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.

A104907 Numbers n such that d(n)*reversal(n)=sigma(n), where d(n) is number of positive divisors of n.

Original entry on oeis.org

1, 73, 861, 7993, 8241, 799993, 7999993, 44908500, 82000041, 293884500, 6279090751, 8200000041, 62698513951, 79999999993, 82000000041, 374665576800, 597921764310, 7999999999993, 8200000000041
Offset: 1

Views

Author

Farideh Firoozbakht, Apr 16 2005

Keywords

Comments

All primes of the form 8*10^n-7 are in the sequence, so 8*10^A099190-3 is a subsequence of this sequence. A105322 is this subsequence. Also if p=(2*10^n+1)/3 is prime then 123*p is in the sequence, so 123*A093170 is a subsequence of this sequence. A105323 is this subsequence.
a(20) > 10^13. - Giovanni Resta, Jul 13 2015

Examples

			Let p=8*10^n-7 be a prime so d(p)=2; reversal(p)=4*10^n-3 and sigma(p)
=8*10^n-6 hence d(p)*reversal(p)=sigma(p) and this shows that p
is in the sequence. 73,7993,799993 and 7999993 are such terms.
Also let q=(2*10^n+1)/3 be a prime, so 123*q=82*10^n+41; reversal
(123*q)=14*10^n+28; d(123*q)=8 and sigma(123*q)=168*q+168=112*10^n
+224 hence d(123*q)*reversal(123*q)=sigma(123*q) and this shows
that 123*q is in the sequence. 861,8241 and 82000041 are such terms.

Crossrefs

Cf. A056657, A093170, A096507, A099190, A105322, A105323, A105324.

Programs

  • Mathematica
    reversal[n_]:= FromDigits[Reverse[IntegerDigits[n]]]; Do[If[DivisorSigma[0, n]*reversal[n] == DivisorSigma[1, n], Print[n]], {n, 1125000000}]
Select[Range[8*10^6],DivisorSigma[0,#]IntegerReverse[#]==DivisorSigma[1,#]&] (* The program generates the first 7 terms of the sequence. *) (* Harvey P. Dale, Jan 31 2023 *)

Extensions

a(11)-a(15) from Donovan Johnson, Feb 06 2010
a(16) from Giovanni Resta, Feb 06 2014
a(17)-a(19) from Giovanni Resta, Jul 13 2015

A105322 Primes of the form 8*10^n-7.

Original entry on oeis.org

73, 7993, 799993, 7999993, 79999999993, 7999999999993, 79999999999993, 7999999999999999999999999999999999999993, 7999999999999999999999999999999999999999999999993
Offset: 1

Views

Author

Farideh Firoozbakht, Apr 16 2005

Keywords

Comments

This sequence is a subsequence of A104907 also is a subsequence of A105324(see A104907 and A105324).

Examples

			7993 is in the sequence because 7993=8*10^3-7 and 7993 is prime.

Links

Crossrefs

Cf. A099190, A104907, A105324.

Programs

  • Magma
    [ a: n in [0..50] | IsPrime(a) where a is 8*10^n-7 ]; // Vincenzo Librandi, Jul 19 2012
  • Mathematica
    Do[If[PrimeQ[8*10^n - 7], Print[8*10^n - 7]], {n, 60}]
Select[Table[8*10^n-7,{n,0,80}],PrimeQ] (* Vincenzo Librandi, Jul 19 2012 *)

Formula

a(n) = 8*A099190(n) - 7.

A114928 Numbers n such that sigma(n)=4*reversal(n).

Original entry on oeis.org

42, 402, 492, 4000002, 57906504, 400000002, 4000000002, 6279090751, 62698513951, 400000000002
Offset: 1

Views

Author

Farideh Firoozbakht, Jan 28 2006

Keywords

Comments

If p=(2*10^n+1)/3 is prime then m=6*p is in the sequence because sigma(m)=sigma(6*p)=12*(2*10^n+4)/3=4*(2*10^n+4)=4* reversal(4*10^n+2)=4*reversal(6*(2*10^n+1)/3)=4*reversal(6*p) =4*reversal(m). Next term is greater than 5*10^8.
a(11) > 10^12. - Giovanni Resta, Oct 28 2012

Examples

			492 is in the sequence because sigma(492)=sigma(4*3*41)=7*4*42
=4*294=4*reversal(492).

Crossrefs

Cf. A069216, A105324, A114927.

Programs

  • Mathematica
    Do[If[DivisorSigma[1, n]==4*FromDigits[Reverse[IntegerDigits[n]]], Print[n]], {n, 500000000}]

Extensions

a(7)-a(9) from Donovan Johnson, Dec 21 2008
a(10) from Giovanni Resta, Oct 28 2012

A114927 Numbers n such that sigma(n)=3*reversal(n).

Original entry on oeis.org

41, 291552, 692133, 2946762, 8231796, 21732508611, 27892659612
Offset: 1

Views

Author

Farideh Firoozbakht, Jan 28 2006

Keywords

Comments

No more terms through 10^9. - Ryan Propper, Jan 08 2007
a(8) > 10^12. - Giovanni Resta, Oct 28 2012

Crossrefs

Cf. A069216, A105324, A114928.

Programs

  • Mathematica
    Do[If[DivisorSigma[1, n] == 3*FromDigits[Reverse[IntegerDigits[n]]], Print[n]], {n, 20000000}]

Extensions

a(6)-a(7) from Donovan Johnson, Dec 21 2008

A115748 Numbers n such that sigma(n)=7*reversal(n).

Original entry on oeis.org

63301, 651001, 6967932, 2158803990, 88858402692
Offset: 1

Views

Author

Farideh Firoozbakht, Feb 12 2006

Keywords

Comments

a(6) > 10^12. - Giovanni Resta, Oct 28 2012

Examples

			2158803990 is in the sequence because sigma(2158803990)
=6951619584=7*993088512=7*reversal(2158803990).

Crossrefs

Cf. A069216, A105324, A114928.

Extensions

a(5) from Donovan Johnson, Dec 21 2008

A115749 Numbers n such that sigma(n)=8*reversal(n).

Original entry on oeis.org

861, 951, 2070, 8241, 900051, 8864151, 9000051, 82000041, 8200000041, 82000000041
Offset: 1

Views

Author

Farideh Firoozbakht, Feb 12 2006

Keywords

Comments

If p=3*10^n+17 is prime then 3*p is in the sequence because sigma(3*p)=4*(3*10^n+18)=12*10^n+72=8*(15*10^(n-1)+9)=8* reversal(9*10^n+51)=8*reversal(3*p). Also if p=(2*10^n+1)/3 is prime then 123*p is in the sequence (the proof is easy). Next term is greater than 13*10^7.
a(11) > 10^12. - Giovanni Resta, Oct 28 2012

Examples

			82000041 is in the sequence because sigma(82000041)
=112000224=8*14000028=8*reversal(82000041).

Crossrefs

Cf. A069216, A105324, A114928, A115747, A115748.

Programs

  • Mathematica
    Do[If[DivisorSigma[1,n]==8*FromDigits[Reverse[IntegerDigits[n]]],Print[n]],{n,130000000}]

Extensions

a(9)-a(10) from Donovan Johnson, Dec 21 2008
Showing 1-6 of 6 results.

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