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.

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A059910 a(n) = |{m : multiplicative order of n mod m = 5}|.

Original entry on oeis.org

0, 1, 4, 6, 9, 4, 4, 6, 20, 9, 8, 2, 6, 6, 12, 44, 5, 6, 18, 14, 12, 4, 4, 2, 56, 13, 20, 4, 6, 2, 40, 6, 18, 12, 12, 44, 63, 6, 28, 4, 16, 14, 8, 2, 18, 12, 28, 14, 70, 3, 42, 12, 42, 6, 24, 8, 56, 44, 60, 6, 60, 2, 4, 90, 21, 20, 24, 2, 18, 60, 88, 6, 12, 2, 28, 26, 6, 28, 8, 14, 170
Offset: 1

Views

Author

Vladeta Jovovic, Feb 08 2001

Keywords

Comments

The multiplicative order of a mod m, gcd(a,m) = 1, is the smallest natural number d for which a^d = 1 (mod m).

Crossrefs

Cf. A059907-A059909, A059911, A059499, A059885-A059892, A002326, A053446-A053452, A002329, A055205, A048691, A048785.

Programs

  • Mathematica
    a[n_] := Subtract @@ DivisorSigma[0, {n^5-1, n-1}]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jan 25 2025 *)
  • PARI
    a(n) = if(n == 1, 0, numdiv(n^5-1) - numdiv(n-1)); \\ Amiram Eldar, Jan 25 2025

Formula

a(n) = tau(n^5-1)-tau(n-1), where tau(n) = number of divisors of n A000005. Generally, if b(n, r) = |{m : multiplicative order of n mod m = r}| then b(n, r) = Sum_{d|r} mu(d)*tau(n^(r/d)-1), where mu(n) = Moebius function A008683.

A069531 Smallest positive k such that 10^k + 1 is divisible by n, or 0 if no such number exists.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 3, 0, 0, 0, 8, 0, 9, 0, 0, 0, 11, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 23, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 29, 0, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22, 0, 3, 0, 0, 0, 0, 0, 48, 0, 0, 0, 2
Offset: 1

Views

Author

Amarnath Murthy, Apr 01 2002

Keywords

Comments

When a(n) is not zero, it is a divisor of phi(n). If n is a prime with primitive root 10 (cf. A001913) then a(n) = (n-1)/2.

Examples

			a(7) = a(13) = 3 as 1001 is divisible by 7 and 13. a(17) = 8 as 17 divides 100000001 = 10^8 + 1.

Links

Crossrefs

Cf. A069521 to A069530.
Cf. A000010, A001913, A002329.

Programs

  • PARI
    A069531(n) = { fordiv(eulerphi(n),k,if(!((1+(10^k))%n),return(k))); (0); }; \\ Antti Karttunen, Aug 23 2019

Extensions

More terms from Vladeta Jovovic, Apr 03 2002

A050979 Haupt-exponents of 7 modulo integers relatively prime to 7.

Original entry on oeis.org

1, 1, 2, 4, 1, 2, 3, 4, 10, 2, 12, 4, 2, 16, 3, 3, 4, 10, 22, 2, 4, 12, 9, 7, 4, 15, 4, 10, 16, 6, 9, 3, 12, 4, 40, 6, 10, 12, 22, 23, 2, 4, 16, 12, 26, 9, 20, 3, 7, 29, 4, 60, 15, 8, 12, 10, 66, 16, 22, 70, 6, 24, 9, 4, 6, 12, 78, 4, 27, 40, 41, 16, 6, 7, 10, 88, 12, 22, 15, 23, 12
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Links

Crossrefs

Cf. A002326 (base 2), A002329, A050977 (base 5), A053450.

A175545 Numbers n (relatively prime to 10) such that the decimal form of the period of 1/n is prime.

Original entry on oeis.org

3, 27, 33, 333, 369, 909, 2151, 2439, 2997, 3333, 27027, 33333, 37683, 41841, 76923, 90909, 142857, 194841, 243603, 333333
Offset: 1

Views

Author

Michel Lagneau, Jun 24 2010

Keywords

Comments

This sequence is infinite because the numbers 3, 33, 333, ... generate the decimal form 3. The correspondant primes of this sequence such that :
{3, 37, 3, 3, 271, 11, 4649, 41, 333667, 3} are included in the sequence A178505.
The Maple program below is very slow for the numbers > 3333.

Examples

			27 is in the sequence because 1/27 = 0.037 037 ... and 37 is prime.
2997 is in the sequence because 1/2997 = 0.000333667 000333667 ... and 333667 is prime.

References

  • H. Rademacher and O. Toeplitz, Von Zahlen und Figuren (Springer 1930, reprinted 1968), ch. 19, 'Die periodischen Dezimalbrueche'.

Links

Crossrefs

Cf. A178505, A045572, A002329.

Programs

  • Maple
    with(numtheory): Digits:=4000:nn:=4000:for n from 3 by 2 to nn do:z:=evalf(1/n): indic:=0:for p from 1 to nn do:if irem(10^p, n) = 1 and gcd(n, 5) = 1 and indic=0 then pp:=p:indic:=1:z1:=floor(z*10^pp): else fi:od:if indic=1 and type(z1,prime)=true then print(n):else fi:od:

Extensions

Extended and name corrected by T. D. Noe, Nov 18 2010
a(17)-a(20) from Ray Chandler, Apr 17 2017

A175550 Period of the decimal expansion of 1/F as F runs through the Fibonacci numbers greater than 1 and not divisible by 2 or 5.

Original entry on oeis.org

1, 6, 6, 44, 232, 84, 138, 133, 336, 396, 28656, 3016, 84, 514228, 335824, 152214, 67830, 4440, 261744, 504628, 108373609, 47124, 3295440, 2971215072, 49349664, 45240, 4438362040, 203028, 3599596, 10841042784, 104340657248, 252736776688
Offset: 1

Views

Author

Michel Lagneau, Jun 26 2010

Keywords

Comments

The Fibonacci numbers contributing to this sequence are {3, 13, 21, 89, 233, ...}, i.e., Fibonacci(k) for k = 4, 7, 8, 11, 13, ... (A229829, starting with A229829(3)).

Examples

			For n = 1, the 1st Fibonacci number > 1 and coprime to 2 and 5 is Fibonacci(4) = 3, and period(1/3) = 1, so a(1) = 1.
For n = 2, the 2nd Fibonacci number > 1 and coprime to 2 and 5 is Fibonacci(7) = 13, and period (1/13) = 6, so a(2) = 6.

Crossrefs

Cf. A000045, A175561, A178505, A045572, A002329, A229829.

Programs

  • Maple
    with(combinat, fibonacci):nn:= 50:for q from 1 to nn do:n:=fibonacci(q):indic:=0:for p from 1 to n do:if irem(10^p, n) = 1 and gcd(n, 5) = 1 and indic=0 then printf(`%d, `, p):indic:=1:else fi:od:od:
  • Mathematica
    Table[MultiplicativeOrder[10, n/Times @@ ({2, 5}^IntegerExponent[n, {2, 5}])], {n, Select[Fibonacci[Range[3, 70]], CoprimeQ[#, 10] &]}] (* Amiram Eldar, May 27 2024 *)

Extensions

a(15) onwards from Robert G. Wilson v, Jun 29 2010

A216415 a(n) = smallest positive m such that 2n-1 | 10^m-1, or 0 if no such m exists.

Original entry on oeis.org

1, 1, 0, 6, 1, 2, 6, 0, 16, 18, 6, 22, 0, 3, 28, 15, 2, 0, 3, 6, 5, 21, 0, 46, 42, 16, 13, 0, 18, 58, 60, 6, 0, 33, 22, 35, 8, 0, 6, 13, 9, 41, 0, 28, 44, 6, 15, 0, 96, 2, 4, 34, 0, 53, 108, 3, 112, 0, 6, 48, 22, 5, 0, 42, 21, 130, 18, 0, 8, 46, 46, 6, 0, 42, 148, 75, 16, 0, 78, 13, 66, 81, 0, 166, 78, 18, 43, 0, 58, 178, 180, 60, 0, 16, 6, 95, 192, 0, 98, 99
Offset: 1

Views

Author

V. Raman, Sep 07 2012

Keywords

Comments

This is yet another version of the sequences defined in A002329, A007732, A070682, A084680. - N. J. A. Sloane, Sep 08 2012
a(n) gives the multiplicative order of 10 mod (2n-1), if it is finite, or 0 if not defined.

Links

Crossrefs

Cf. A002326, A002329, A007732, A070682, A084680. - N. J. A. Sloane, Sep 08 2012

Programs

  • PARI
    for(i=0,200,i++;if(i%5==0,print1(0","),print1(znorder(Mod(10,i))","))) \\ V. Raman, Nov 22 2012
  • PARI
    for(i=0,200,i++;m=0;for(x=1,i,if(((10^x-1))%i==0,m=x;break));print1(m",")) \\ V. Raman, Nov 22 2012

A050976 Haupt-exponents of 4 modulo integers relatively prime to 4.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 2, 4, 9, 3, 11, 10, 9, 14, 5, 5, 6, 18, 6, 10, 7, 6, 23, 21, 4, 26, 10, 9, 29, 30, 3, 6, 33, 11, 35, 9, 10, 15, 39, 27, 41, 4, 14, 11, 6, 5, 18, 24, 15, 50, 51, 6, 53, 18, 18, 14, 22, 6, 12, 55, 10, 50, 7, 7, 65, 9, 18, 34, 69, 23, 30, 14, 21, 74, 15, 12, 10, 26
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Links

Crossrefs

Cf. A002326, A002329. Presumably this is a duplicate of A053447.

A050978 Haupt-exponents of 6 modulo integers relatively prime to 6.

Original entry on oeis.org

1, 2, 10, 12, 16, 9, 11, 5, 14, 6, 2, 4, 40, 3, 23, 14, 26, 10, 58, 60, 12, 33, 35, 36, 10, 78, 82, 16, 88, 12, 9, 12, 10, 102, 106, 108, 112, 11, 16, 110, 25, 126, 130, 18, 136, 23, 60, 14, 37, 150, 6, 156, 22, 27, 83, 156, 43, 10, 178, 60, 4, 80, 19, 96, 14, 198, 14
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Links

Crossrefs

Cf. A002326, A002329, A053449.

A050980 Haupt-exponents of 8 modulo integers relatively prime to 8.

Original entry on oeis.org

2, 4, 1, 2, 10, 4, 4, 8, 6, 2, 11, 20, 6, 28, 5, 10, 4, 12, 4, 20, 14, 4, 23, 7, 8, 52, 20, 6, 58, 20, 2, 4, 22, 22, 35, 3, 20, 10, 13, 18, 82, 8, 28, 11, 4, 10, 12, 16, 10, 100, 17, 4, 106, 12, 12, 28, 44, 4, 8, 110, 20, 100, 7, 14, 130, 6, 12, 68, 46, 46, 20, 28, 14, 148, 5
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Links

Crossrefs

Cf. A002326, A002329. Presumably this is a duplicate of A053451.

A050981 Haupt-exponents of 9 modulo integers relatively prime to 9.

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 5, 3, 3, 2, 8, 9, 2, 5, 11, 10, 3, 3, 14, 15, 4, 8, 6, 9, 9, 2, 4, 21, 5, 11, 23, 21, 10, 3, 26, 10, 3, 14, 29, 5, 15, 8, 6, 11, 8, 6, 35, 6, 9, 9, 15, 39, 2, 4, 41, 8, 21, 5, 44, 3, 11, 23, 18, 24, 21, 10, 50, 17, 3, 26, 53, 27, 10, 6, 56, 22, 14, 29, 24, 5, 5, 15
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Links

Crossrefs

Cf. A002326, A002329, A053452.
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