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 1-5 of 5 results.

A105324 Numbers n such that 2*reversal(n)=sigma(n).

Original entry on oeis.org

6, 73, 483, 4074, 4473, 4623, 7993, 42813, 69855, 253782, 799993, 7999993, 46000023, 426000213, 4600000023, 6718967838, 42600000213, 46000000023, 79999999993, 426000000213
Offset: 1

Views

Author

Farideh Firoozbakht, Apr 16 2005

Keywords

Comments

I. If p=8*10^n-7 is a prime then p is in the sequence because reversal(p)=4*10^n-3 & sigma(p)=8*10^n-6 so 2*reversal(p) =sigma(p). 73,7993,799993 & 7999993 are such terms.
II. If q=(2*10^n+1)/3 is a prime then (a): 69*q is in the sequence because 69*q=46*10^n+23; reversal (69*q)=32*10^n+64 & sigma(69*q)=96*q+96=64*10^n+128 so 2*reversal (69*q)=sigma(69*q). 483,4623 & 46000023 are such terms. (b): 639*q is in the sequence because 639*q=426*10^n+213; reversal (639*q)=312*10^n+624 & sigma(639*q)=936*q+936=624*10^n+1248 so 2*reversal(639*q)=sigma(639*q). 42813 & 426000213 are such terms.
a(21) > 10^12. - Giovanni Resta, Oct 28 2012

Examples

			253782 is in the sequence because reversal(253782)=287352; sigma(253782)=574704 & 2*287352=574704.

Crossrefs

Cf. A093170, A096507, A099190, A105322, A105323, A105325, A105326.

Programs

  • Mathematica
    reversal[n_]:= FromDigits[Reverse[IntegerDigits[n]]]; Do[If[2* reversal[n]== DivisorSigma[1, n], Print[n]], {n, 1000000000}]
Select[Range[8*10^6],2*IntegerReverse[#]==DivisorSigma[1,#]&] (* The program generates the first 12 terms of the sequence. *) (* Harvey P. Dale, Oct 29 2022 *)

Extensions

a(15)-a(19) from Donovan Johnson, Dec 21 2008
a(20) from Giovanni Resta, Oct 28 2012

A101155 Indices of primes in sequence defined by A(0) = 73, A(n) = 10*A(n-1) + 63 for n > 0.

Original entry on oeis.org

0, 2, 4, 5, 9, 11, 12, 38, 47, 53, 63, 81, 146, 147, 359, 398, 1637, 1875, 2145, 2193, 15788, 23073, 38465, 68399
Offset: 1

Views

Author

Klaus Brockhaus and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Dec 03 2004

Keywords

Comments

Numbers n such that (720*10^n - 63)/9 is prime.
Numbers n such that digit 7 followed by n >= 0 occurrences of digit 9 followed by digit 3 is prime.
Numbers corresponding to terms <= 398 are certified primes.
a(25) > 2*10^5. - Robert Price, Nov 11 2015

Examples

			7999993 is prime, hence 5 is a term.

References

  • Klaus Brockhaus and Walter Oberschelp, Zahlenfolgen mit homogenem Ziffernkern, MNU 59/8 (2006), pp. 462-467.

Links

Crossrefs

Cf. A000533, A002275, A101849, A169830, A099190.

Programs

  • Mathematica
    Select[Range[0, 100000], PrimeQ[(720*10^# - 63)/9] &] (* Robert Price, Nov 11 2015 *)
  • PARI
    a=73;for(n=0,1000,if(isprime(a),print1(n,","));a=10*a+63)
  • PARI
    for(n=0,1000,if(isprime((720*10^n-63)/9),print1(n,",")))

Formula

a(n) = A099190(n) - 1.

Extensions

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 01 2008
a(21)-a(24) from Kamada data by Ray Chandler, Apr 30 2015

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.

A099191 Smallest prime p such that p+n is twice its reversal, or 0 if impossible.

Original entry on oeis.org

73, 2, 3, 0, 5, 0, 7, 0, 0, 0, 11, 0, 211, 0, 0, 0, 53, 0, 613, 0, 0, 0, 0, 0, 43, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 23, 0, 241, 0, 0, 0, 0, 0, 13, 0, 0, 0, 251, 0, 0, 0, 0, 0, 653, 0, 97, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 271, 0, 0, 0, 0, 0, 673, 0, 0, 0, 281, 0, 67, 0, 0, 0, 683, 0, 0, 0, 0
Offset: 1

Views

Author

Robert G. Wilson v, Oct 01 2004

Keywords

Examples

			a(13) = 211 since 211+13 = 224 = 2*112. a(25) = 43 since 25+43 = 68 = 2*34.

Crossrefs

Cf. A099190.

Programs

  • Mathematica
    f[n_] := Block[{k = 1}, While[p = Prime[k]; p + n != 2FromDigits[ Reverse[ IntegerDigits[ p]]] && k < 10^6, k++ ]; If[k == 10^6, 0, Prime[k]]]; Table[ f[n], {n, 75}]
Showing 1-5 of 5 results.

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