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.

A096507 Numbers k such that 6*R_k + 1 is a prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

1, 2, 6, 8, 9, 11, 20, 23, 41, 63, 66, 119, 122, 149, 252, 284, 305, 592, 746, 875, 1204, 1364, 2240, 2403, 5106, 5776, 5813, 12456, 14235, 39606, 55544, 84239, 275922
Offset: 1

Views

Author

Labos Elemer, Jul 12 2004

Keywords

Comments

Also numbers k such that (2*10^k + 1)/3 is prime.
These numbers form a near-repdigit sequence (6)w7.
All the terms from k = 2403 through 14235 correspond to primes. - Joao da Silva (zxawyh66(AT)yahoo.com), Oct 03 2005

Examples

			k = 9 gives 2000000001/3 = 666666667, which is prime.
k = 20 gives 66666666666666666667, which is prime.

Links

Crossrefs

Cf. A002275, A056657, A093170, A096503, A096504, A096505, A096506, A096508.

Programs

  • Mathematica
    Select[Range@ 2500, PrimeQ[FromDigits@ Table[6, {#}] + 1] &] (* or *)
Select[Range@ 2500, PrimeQ[2 (10^# - 1)/3 + 1] &] (* Michael De Vlieger, Jul 04 2016 *)

Formula

a(n) = A056657(n) + 1.

Extensions

More terms from Julien Peter Benney (jpbenney(AT)ftml.net), Sep 14 2004
39606 and 55544 from Serge Batalov, Jun 2009
84239 from Serge Batalov, Jul 06 2009 confirmed as next term by Ray Chandler, Feb 23 2012
a(33) from Kamada data by Tyler Busby, Apr 14 2024

A093170 Primes of the form 60*R_k + 7, where R_k is the repunit (A002275) of length k.

Original entry on oeis.org

7, 67, 666667, 66666667, 666666667, 66666666667, 66666666666666666667, 66666666666666666666667, 66666666666666666666666666666666666666667, 666666666666666666666666666666666666666666666666666666666666667
Offset: 1

Views

Author

Rick L. Shepherd, Mar 26 2004

Keywords

Comments

Primes of the form (2*10^k + 1)/3. - Vincenzo Librandi, Nov 16 2010
Occur in the factorization of some of the numbers of the form 13...3 not in A093671, cf. second Kamada link. - M. F. Hasler, Sep 14 2014

Links

Crossrefs

Cf. A002275, A056657 (corresponding k), A093671, A096507.

Programs

  • Maple
    A093170:=n->`if`(isprime((2*10^n+1)/3),(2*10^n+1)/3,NULL): seq(A093170(n), n=1..70); # Wesley Ivan Hurt, Sep 14 2014
  • Mathematica
    Select[Table[FromDigits[PadLeft[{7},n,6]],{n,70}],PrimeQ] (* Harvey P. Dale, Jan 26 2013 *)

Formula

a(n) = (20*10^A056657(n)+1)/3 = (2*10^A096507(n)+1)/3.

Extensions

Edited by Ray Chandler, Feb 23 2012

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

A105323 Numbers of the form 41*(2*10^n+1) where (2*10^n+1)/3 is prime (n is in the sequence A096507).

Original entry on oeis.org

861, 8241, 82000041, 8200000041, 82000000041, 8200000000041, 8200000000000000000041, 8200000000000000000000041, 8200000000000000000000000000000000000000041
Offset: 1

Views

Author

Farideh Firoozbakht, Apr 16 2005

Keywords

Comments

A105323=41*A093170=41*(2*10^A096507+1)=41*(2*10^(A056657+1)+1). If m is in the sequence then d(m)*reversal(m)=sigma(m) (see A104907). So this sequence is a subsequence of A104907.

Examples

			861 is in the sequence because 861=41*(2*10^1+1); (2*10^1+1)/3=7 and 7 is prime.

Crossrefs

Cf. A104907, A056657, A096507, A093170.

Programs

  • Mathematica
    Do[If[PrimeQ[(2*10^n + 1)/3], Print[41*(2*10^n + 1)]], {n, 63}]

A064028 Sum of the unitary divisors of n!.

Original entry on oeis.org

1, 3, 12, 36, 216, 1020, 8160, 61920, 507744, 4383392, 52600704, 624249600, 8739494400, 109190390400, 1583122968000, 25318378008000, 455730804144000, 8193040840252800, 163860816805056000, 3256371347261760000, 67204676251838361600, 1366492477414792734720
Offset: 1

Views

Author

Labos Elemer, Sep 11 2001

Keywords

Examples

			n=6, 6! = 720, sum of the 8 unitary ones of its 30 divisors is 1020, a(6) = 720+1+16+45+9+80+5+144 = 1020.

Links

Crossrefs

Cf. A034448, A046656, A056657, A056171, A056172, A000203, A000142, A062569.

Programs

  • Mathematica
    usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma/@ (Range[17]!) (* Amiram Eldar, Jun 23 2019 *)
  • PARI
    valp(n,p)=my(s); while(n\=p, s+=n); s
a(n)=my(s=1); forprime(p=2,n, s*=p^valp(n,p)+1); s \\ Charles R Greathouse IV, Jan 26 2023

Formula

a(n) = usigma(n!) = A034448(A000142(n)).
a(n)/n! <= 2 (while usigma(n)/n and sigma(n!)/n! are unbounded; Wall, 1984). - Amiram Eldar, Feb 08 2022

A064138 Sum of non-unitary divisors of n!.

Original entry on oeis.org

0, 0, 0, 24, 144, 1398, 11184, 97200, 973296, 10950696, 131408352, 1593191808, 22304685312, 333297226080, 5103130001760, 81686161277280, 1470350902991040, 26490792085668288, 529815841713365760, 10635027891469974720
Offset: 1

Views

Author

Labos Elemer, Sep 11 2001

Keywords

Examples

			For n = 6, 6! = 720, the sum of its 30 divisors is 2418, the sum of the 8 unitary divisors is 1020, so the remaining 22 divisors give a(6) = 1398.

Links

Crossrefs

Cf. A034448, A048105, A046656, A056657, A056171, A056172, A000203, A000142, A062569, A063955, A063960.

Programs

  • Mathematica
    f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := p^e + 1; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n!]) - Times @@ f2 @@@ fct; a[1] = 0; Array[a, 20] (* Amiram Eldar, Apr 01 2024 *)
  • PARI
    usigma(n)= { local(f,s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) }
{ n=0; f=1; for (n=1, 100, f*=n; write("b064138.txt", n, " ", sigma(f) - usigma(f)); ) } \\ Harry J. Smith, Sep 08 2009

Formula

a(n) = sigma(n!) - usigma(n!) = A000203(n!) - A034448(A000142(n)) = A062569(n) - A034448(n!) = A048105(n!).

Extensions

Term corrected and more terms added by Harry J. Smith, Sep 08 2009
Showing 1-6 of 6 results.

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