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.

A002174 Values taken by reduced totient function psi(n).

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 28, 30, 32, 36, 40, 42, 44, 46, 48, 52, 54, 56, 58, 60, 64, 66, 70, 72, 78, 80, 82, 84, 88, 90, 92, 96, 100, 102, 104, 106, 108, 110, 112, 116, 120, 126, 128, 130, 132, 136, 138, 140, 144, 148, 150, 156, 160, 162, 164, 166, 168
Offset: 1

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Author

Keywords

Comments

If p is a Sophie Germain prime (A005384), then 2p is here. - T. D. Noe, Aug 13 2008
Terms of A002322, sorted and multiple values taken just once. - Vladimir Joseph Stephan Orlovsky, Jul 21 2009
a(2445343) = 10^7, suggesting that Luca & Pomerance's lower bound may be closer to the truth than the upper bound. The fit exponent log a(n)/log n - 1 = 0.0957... in this case. - Charles R Greathouse IV, Jul 02 2017

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Crossrefs

Programs

  • Mathematica
    lst={}; Do[AppendTo[lst, CarmichaelLambda[n]], {n, 6*7!}]; lst; Take[Union[lst], 123] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2009 *)
    (* warning: there seems to be no guarantee that no terms near the end are omitted! - Joerg Arndt, Dec 23 2014 *)
    TakeWhile[Union@ Table[CarmichaelLambda@ n, {n, 10^6}], # <= 168 &] (* Michael De Vlieger, Mar 19 2016 *)
  • PARI
    list(lim)=my(v=List([1]),u,t); forprime(p=3,lim\3+1, u=List(); listput(u,p-1); while((t=u[#u]*p)<=lim, listput(u,t)); for(j=1,#v, for(i=1,#u, t=lcm(u[i],v[j]); if(t<=lim && t!=v[j], listput(v,t)))); v=List(Set(v))); forprime(p=lim\3+2,lim+1, listput(v,p-1)); v=List(Set(v)); for(i=1,#v, t=2*v[i]; if(t>lim, break); listput(v,t); while((t*=2)<=lim, listput(v,t))); Set(v) \\ Charles R Greathouse IV, Jun 23 2017
    
  • PARI
    is(n)=if(n%2, return(n==1)); my(f=factor(n),pe); for(i=1,#f~, if(n%(f[i,1]-1)==0, next); pe=f[i,1]^f[i,2]; forstep(q=2*pe+1,n+1,2*pe, if(n%(q-1)==0 && isprime(q), next(2))); return(0)); 1 \\ Charles R Greathouse IV, Jun 25 2017

Formula

n (log n)^0.086 << a(n) << n (log n)^0.36 where << is the Vinogradov symbol, see Luca & Pomerance. - Charles R Greathouse IV, Dec 28 2013

Extensions

More terms from T. D. Noe, Aug 13 2008

A143407 Largest number k such that the reduced totient function psi(k) = A002174(n).

Original entry on oeis.org

2, 24, 240, 504, 480, 264, 65520, 16320, 28728, 13200, 552, 131040, 6960, 171864, 32640, 138181680, 1082400, 151704, 5520, 1128, 4455360, 12720, 86184, 13920, 1416, 6814407600, 65280, 776664, 18744, 20174525280, 39816, 36801600, 1992
Offset: 1

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Author

T. D. Noe, Aug 13 2008

Keywords

Comments

For each of the values in A002174, there are only a finite number of numbers k such that psi(k)=A002174(n). This sequence gives the largest such k. Sequence A002396 gives the least k. The number of such k is given in A143408. When A002174(n) is twice a Sophie Germain prime, then a(n) is particularly small.

Crossrefs

Cf. A002322 (reduced totient function), A002174, A002396, A143408.

Formula

a(n) = A006863(A002174(n)/2) for n>1.

A304480 a(n) is the least m such that lambda(k) >= n for all k >= m where lambda is A002322, the Carmichael lambda function.

Original entry on oeis.org

1, 3, 25, 25, 241, 241, 505, 505, 505, 505, 505, 505, 65521, 65521, 65521, 65521, 65521, 65521, 65521, 65521, 65521, 65521, 65521, 65521, 131041, 131041, 131041, 131041, 131041, 131041, 171865, 171865, 171865, 171865, 171865, 171865, 138181681, 138181681, 138181681, 138181681, 138181681, 138181681
Offset: 1

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Author

Michel Marcus, May 13 2018

Keywords

Crossrefs

Programs

  • PARI
    minvl(n, v) = {vgt = select(x->(x>=n), v, 1); first = vgt[1]; vgtd = vector(#vgt-1, k, vgt[k+1] - vgt[k]); vgtdr = Vecrev(vgtd); vokdiff = select(x->x!=1, vgtdr, 1); if (#vokdiff, #v - vokdiff[1]+1, first);}
    lista(nn) = {v = read("v002322.txt"); for (n=1, nn, print1(minvl(n, v), ", "););}

Extensions

a(32) and beyond from Seiichi Manyama, May 24 2020

A141162 Smallest k such that lambda(k) = n, or 0 if there is no such k.

Original entry on oeis.org

1, 3, 0, 5, 0, 7, 0, 32, 0, 11, 0, 13, 0, 0, 0, 17, 0, 19, 0, 25, 0, 23, 0, 224, 0, 0, 0, 29, 0, 31, 0, 128, 0, 0, 0, 37, 0, 0, 0, 41, 0, 43, 0, 115, 0, 47, 0, 119, 0, 0, 0, 53, 0, 81, 0, 928, 0, 59, 0, 61, 0, 0, 0, 256, 0, 67, 0, 0, 0, 71, 0, 73, 0, 0, 0, 0, 0, 79, 0, 187, 0, 83, 0, 203, 0, 0, 0, 89, 0, 209, 0, 235, 0, 0, 0, 97, 0
Offset: 1

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Author

Michel Lagneau, Mar 17 2011

Keywords

Comments

Sequence A002174 gives the n such that a(n) > 0. Removing the zeros from this sequence produces A002396. Note that some n appear only for large k. For example, 728 does not appear until k=49184. See A143407 for the largest k that produces a particular value of the lambda function. See A143408 for the number of times each value occurs. - T. D. Noe, Mar 17 2011

Examples

			a(8) = 32 because lambda(32) = 8.
		

Crossrefs

Cf. A002174, A002322 (Carmichael lambda function), A002396, A143407, A143408.

Programs

  • Maple
    with(numtheory):for k from 1 to 100 do:id:=0:for n from 1 to 1000 while(id=0)
      do: if lambda(n) = k then id:=1:printf(`%d, `,n):else fi:od:if id=0 then printf(`%d, `,0):else fi:od:
  • Mathematica
    nn = 100; t = Table[0, {nn}]; Do[c = CarmichaelLambda[k]; If[c <= nn && t[[c]] == 0, t[[c]] = k], {k, 1000}]; t

Formula

a(A002174(n)) = A002396(n).

A302099 Decompose the multiplicative group of integers modulo N as a product of cyclic groups C_{k_1} x C_{k_2} x ... x C_{k_m}, where k_i divides k_j for i < j, then a(n) is the smallest N such that the product contains a copy of C_{2n}.

Original entry on oeis.org

3, 5, 7, 32, 11, 13, 1247, 17, 19, 25, 23, 224, 4187, 29, 31, 128, 14111, 37, 43739, 41, 43, 115, 47, 119, 15251, 53, 81, 928, 59, 61, 116003, 256, 67, 70555, 71, 73, 33227, 174269, 79, 187, 83, 203, 74563, 89, 209, 235, 186497, 97, 67571, 101, 103
Offset: 1

Views

Author

Jianing Song, Apr 01 2018

Keywords

Comments

a(n) exists for all n: by Dirichlet's theorem on arithmetic progressions, there must exist two primes with the form 2a*n + 1 and 2b*n + 1 where at least one of a,b is coprime to 2n, then the multiplicative group of integers modulo (2a*n + 1)(2b*n + 1) is isomorphic to C_{2*n} x C_{2ab*n}.
Factorizations of a(n) where 2n is not a term in A002174: a(7) = 29*43, a(13) = 53*79, a(17) = 103*137, a(19) = 191*229, a(25) = 101*151, a(31) = 311*373, a(34) = 5*103*137, a(37) = 149*223, a(38) = 229*761, a(43) = 173*431, a(47) = 283*659, a(49) = 7^3*197. - Jianing Song, Apr 29 2018 [Corrected on Sep 15 2018]
It may appear that for odd n, A046072(a(n)) = 1 or 2, but this is not generally true. The smallest counterexample is a(85) = 1542013, as the multiplicative group of integers modulo 1542013 is isomorphic to C_2 x C_170 x C_4080. - Jianing Song, Sep 15 2018

Examples

			For n = 7 the multiplicative group of integers modulo 1247 is isomorphic to C_14 x C_84, and 1247 is the smallest number that contains a copy of C_14 in the product of cyclic groups, so a(7) = 1247.
For n = 34 the multiplicative group of integers modulo 70555 is isomorphic to C_2 x C_68 x C_408, and 70555 is the smallest number that contains a copy of C_68 in the product of cyclic groups, so a(34) = 70555. - _Jianing Song_, Sep 15 2018
		

Crossrefs

Programs

  • PARI
    a(n)=my(i=3, Z=[2]); while(prod(j=1, #Z, 1-(Z[j]==2*n)), i++&&Z=znstar(i)[2]); i \\ Jianing Song, Sep 15 2018

Extensions

Some terms corrected by Jianing Song, Apr 29 2018
Some terms corrected by Jianing Song, Sep 15 2018
Showing 1-5 of 5 results.