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.

Previous Showing 11-17 of 17 results.

A366636 Number of distinct prime divisors of 7^n + 1.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 3, 3, 3, 4, 3, 5, 3, 3, 5, 3, 2, 5, 3, 4, 6, 5, 2, 4, 4, 4, 4, 6, 2, 8, 4, 4, 6, 5, 9, 8, 3, 3, 7, 6, 5, 6, 8, 5, 10, 6, 2, 6, 10, 8, 6, 5, 5, 8, 10, 8, 7, 6, 5, 9, 2, 5, 12, 4, 7, 11, 4, 5, 6, 8, 3, 9, 4, 3, 9, 7, 10, 8, 5, 6, 8, 5, 3, 12
Offset: 0

Views

Author

Sean A. Irvine, Oct 15 2023

Keywords

Links

Crossrefs

Cf. A034491, A001221, A057937, A227575, A366632, A366637, A366638, A366639.
Cf. A046799, A366580, A366605, A366615, A366627, A366655, A366664, A119704, A366686, A366712.

Programs

  • Mathematica
    PrimeNu[7^Range[0,84] + 1] (* Paul F. Marrero Romero, Nov 11 2023 *)
  • PARI
    for(n = 0, 100, print1(omega(7^n + 1), ", "))

Formula

a(n) = omega(7^n+1) = A001221(A034491(n)).

A366664 Number of distinct prime divisors of 9^n + 1.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 3, 4, 2, 4, 3, 4, 6, 4, 4, 5, 2, 4, 4, 4, 5, 7, 5, 4, 4, 6, 4, 5, 6, 4, 7, 5, 2, 6, 5, 8, 8, 5, 6, 7, 5, 5, 10, 7, 6, 8, 4, 4, 6, 9, 6, 8, 7, 6, 9, 7, 9, 9, 5, 3, 11, 6, 4, 11, 6, 7, 9, 9, 7, 6, 9, 5, 6, 6, 6, 11, 4, 8, 7, 5, 4, 7, 5, 5, 11
Offset: 0

Views

Author

Sean A. Irvine, Oct 15 2023

Keywords

Links

Crossrefs

Cf. A062396, A001221, A057935, A366660, A366665, A366666, A366667.
Cf. A046799, A366580, A366605, A366615, A366627, A366636, A366655, A119704, A366686, A366712.

Programs

  • Mathematica
    PrimeNu[9^Range[0,90]+1] (* Harvey P. Dale, Jul 04 2024 *)
  • PARI
    for(n = 0, 100, print1(omega(9^n + 1), ", "))

Formula

a(n) = omega(9^n+1) = A001221(A062396(n)).
a(n) = A366580(2*n). - Max Alekseyev, Jan 08 2024

A087021 Number of distinct prime factors of n-th cyclic number.

Original entry on oeis.org

4, 8, 9, 8, 10, 8, 10, 21, 23, 19, 19, 15, 16, 12, 11, 33, 31, 19, 24, 22, 24, 18, 14, 33, 39, 23, 36, 13, 13, 19, 36, 32, 29, 27, 25, 11, 20, 56, 37, 46, 25, 22, 21, 16, 47, 25, 33, 22, 55, 32, 25
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 30 2003

Keywords

Comments

A004042(n) factorized with Dario Alpern's ECM.
Extended using factors of 10^(A001913(n)-1)-1, see Kamada link.

Examples

			A004042(2) = 142857 = 37*13*11*3^3, therefore a(1) =
#{3,11,13,37} = 4.

Links

Crossrefs

Cf. A001913, A087021-A087026.

Formula

a(n) = A001221(A004042(n+1)).
For n>1, let p = A001913(n). If p is a base-10 Wieferich prime, then a(n) = A102347(p-1) + 2; otherwise a(n) = A102347(p-1) + 1. Also, we have A102347(p-1) = A102347((p-1)/2) + A119704((p-1)/2). - Max Alekseyev, Apr 26 2022

Extensions

a(3) corrected, a(12)-a(42) added by Ray Chandler, Nov 16 2011
a(43)-a(51) from Max Alekseyev, May 13 2022

A366668 Sum of the divisors of 10^n+1.

Original entry on oeis.org

3, 12, 102, 1344, 10212, 109104, 1010004, 10909104, 105882372, 1413350400, 10102223208, 114737461440, 1021097900424, 10921790676000, 104844305394000, 1355394166984704, 10073631600468000, 110177492439680640, 1010002989998020008, 10909090909090909104
Offset: 0

Views

Author

Sean A. Irvine, Oct 15 2023

Keywords

Examples

			a(3)=1344 because 10^3+1 has divisors {1, 7, 11, 13, 77, 91, 143, 1001}.

Links

Crossrefs

Cf. A000203, A003021, A057934, A062397, A102146, A119704, A344897, A366666.

Programs

  • Maple
    a:=n->numtheory[sigma](10^n+1):
seq(a(n), n=0..100);
  • Mathematica
    DivisorSigma[1, 10^Range[0,19] + 1] (* Paul F. Marrero Romero, Nov 12 2023 *)

Formula

a(n) = sigma(10^n+1) = A000203(A062397(n)).

A366669 a(n) = phi(10^n+1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 10, 100, 720, 9792, 90900, 990000, 9090900, 94117632, 681410880, 9897840000, 86925373920, 979102080000, 9080325951840, 95255567232000, 712493107200000, 9926748531589120, 90004044661864320, 989999010000000000, 9090909090909090900, 97910150554895155200
Offset: 0

Views

Author

Sean A. Irvine, Oct 15 2023

Keywords

Links

Crossrefs

Cf. A000010, A003021, A057934, A062397, A119704, A295503, A344897, A366668.
Cf. A053285, A366579, A366608, A366618, A366630, A366639, A366658, A366667, A366690, A366716.

Programs

  • Mathematica
    EulerPhi[10^Range[0,20] + 1] (* Paul F. Marrero Romero, Nov 10 2023 *)
  • PARI
    {a(n) = eulerphi(10^n+1)}

Formula

a(n) = A000010(A062397(n)). - Paul F. Marrero Romero, Nov 10 2023

A368418 Numbers X such that X^2 + Y^2 = 10^(2*k) + 1, with X > Y > 0 and k is the decimal digit length of X-1.

Original entry on oeis.org

10, 76, 100, 980, 1000, 8824, 10000, 76249, 87551, 98020, 100000, 753424, 766424, 999800, 1000000, 7209049, 7241380, 8220640, 8463640, 9801980, 9879740, 9990280, 10000000, 77053825, 78173720, 80404255, 83754376, 84711551, 86600176, 90880001, 93094625, 93728480
Offset: 1

Views

Author

A.H.M. Smeets, Dec 24 2023

Keywords

Comments

The values X and Y are used in finding A368416.
The number of terms for a given k is 2^(f-1), where f = A119704(2*k) is the number of prime factors of 10^(2*k) + 1.

Examples

			10 is a term since X = 10, Y = 1, k = 1 and 10^2 + 1^2 = 101.
76 is a term since X = 76, Y = 65, k = 2 and 76^2 + 65^2 = 10001.
980 is a term since X = 980, Y = 199, k = 3 and 980^2 + 199^2 = 1000001.

References

  • Frits Beukers, "Getallen - Een inleiding" (In Dutch), Epsilon Uitgaven, Amsterdam (2015).

Links

Crossrefs

Cf. A098608, A119704, A368416.

A086564 Smallest k such that 10^k + 1 has n distinct prime divisors.

Original entry on oeis.org

0, 4, 3, 11, 9, 36, 15, 33, 39, 69, 63, 45, 171, 117, 243, 105, 150, 135, 165, 255, 231, 210
Offset: 1

Views

Author

Amarnath Murthy, Aug 31 2003

Keywords

Comments

a(27) = 330; a(28) = 315; a(23) >= 347. - Max Alekseyev, Jun 18 2023
a(23) <= 385. - Jon E. Schoenfield, Jun 18 2023

Examples

			a(5) = 9 = log_10(1000000001 - 1).

Crossrefs

Smallest inverse of A119704.
Cf. A086563.

Programs

  • Mathematica
    Module[{nn=70,pn},pn=PrimeNu/@(10^Range[0,nn]+1);Flatten[Table[Position[pn,n,1,1],{n,12}]]]-1 (* Harvey P. Dale, Jun 10 2023 *)

Formula

a(n) = log_10(A086563(n) - 1).

Extensions

More terms from Sascha Kurz, Sep 22 2003
a(10)-a(12) from David Wasserman, Mar 28 2005
a(13)-a(22) from Max Alekseyev, Apr 27 2010
Previous Showing 11-17 of 17 results.

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