A322315 -id:A322315 - cp's OEIS frontend

A079612 Largest number m such that a^n == 1 (mod m) whenever a is coprime to m.

2, 24, 2, 240, 2, 504, 2, 480, 2, 264, 2, 65520, 2, 24, 2, 16320, 2, 28728, 2, 13200, 2, 552, 2, 131040, 2, 24, 2, 6960, 2, 171864, 2, 32640, 2, 24, 2, 138181680, 2, 24, 2, 1082400, 2, 151704, 2, 5520, 2, 1128, 2, 4455360, 2, 264, 2, 12720, 2, 86184, 2, 13920

Offset: 1

N. J. A. Sloane, Jan 29 2003

nonn

a(m) divides the Jordan function J_m(n) for all n except when n is a prime dividing a(m) or m=2, n=4; it is the largest number dividing all but finitely many values of J_m(n). For m > 0, a(m) also divides Sum_{k=1}^n J_m(k) for n >= the largest exceptional value. - Franklin T. Adams-Watters, Dec 10 2005

The numbers m with this property are the divisors of a(n) that are not divisors of a(r) for r

R. C. Vaughan and T. D. Wooley, Waring's problem: a survey, pp. 285-324 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003. (The function K(n), see p. 303.)

Antti Karttunen, Table of n, a(n) for n = 1..16384
Shigeki Akiyama and Hajime Kaneko, Curious congruences on cyclotomic polynomials, arXiv:2204.11267 [math.NT], 2022. See Proposition 1 p. 5-6.
P. J. Cameron and D. A. Preece, Notes on primitive lambda-roots, 2009. See lambda*() in theorem 5.2 (b) p. 8.
Joris van der Hoeven and Grégoire Lecerf, Sparse polynomial interpolation. Exploring fast heuristic algorithms over finite fields, Simon Fraser University (BC Canada) / Institut Polytechnique de Paris (France, 2019) hal-02382117.
R. C. Vaughan and T. D. Wooley, Waring's problem: a survey, The function K(n), see p. 19.

Cf. A006863 (bisection except for initial term); A059379 (Jordan function).
Cf. A075180, A115000, A115001, A115002, A115003.
Cf. A143407, A143408, A185633, A322315.

a(n) = {if (n%2, 2, res = 1; forprime(p=2, n+1, if (!(n % (p-1)), t = valuation(n, p); if (p==2, if (t, res *= p^(t+2)), res *= p^(t+1)););); res;);} \\ Michel Marcus, May 12 2018

a(n) = 2 for n odd; for n even, a(n) = product of 2^(t+2) (where 2^t exactly divides n) and p^(t+1) (where p runs through all odd primes such that p-1 divides n and p^t exactly divides n).
From Antti Karttunen, Dec 19 2018: (Start)
a(n) = A185633(n)*(2-A000035(n)).
It also seems that for n > 1, a(n) = 2*A075180(n-1). (End)
We have 2*A075180(2n-1) = A006863(n) by definition, and A006863(n) = a(2n) by the comments in A006863. Hence a(n) = 2*A075180(n-1) for all even n. For all odd n > 1, we have a(n) = 2, which is also equal to 2*A075180(n-1). So the formula above is true. - Jianing Song, Apr 05 2021

Edited by Franklin T. Adams-Watters, Dec 10 2005
Definition corrected by T. D. Noe, Aug 13 2008
Rather arbitrary term a(0) removed by Max Alekseyev, May 27 2010

A185633 For odd n, a(n) = 2; for even n, a(n) = denominator of Bernoulli(n)/n; The number 2 alternating with the elements of A006953.

Original entry on oeis.org

2, 12, 2, 120, 2, 252, 2, 240, 2, 132, 2, 32760, 2, 12, 2, 8160, 2, 14364, 2, 6600, 2, 276, 2, 65520, 2, 12, 2, 3480, 2, 85932, 2, 16320, 2, 12, 2, 69090840, 2, 12, 2, 541200, 2, 75852, 2, 2760, 2, 564, 2, 2227680, 2, 132, 2, 6360

Offset: 1

Paul Curtz, Dec 18 2012

nonn

There is an integer sequence b(n) = A053657(n)/2^(n-1) = 1, 1, 6, 6, 360, 360, 45360, 45360, 5443200, 5443200,... which consists of the duplicated entries of A202367.

The ratios of this sequence are b(n+1)/b(n) = 1, 6, 1, 60, 1, 126 .... = a(n)/2, which is a variant of A036283.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A006953, A007395 (bisections).

Cf. A006863, A027760, A067513, A322312, A322315 (rgs-transform).

A185633 := proc(n)
    A053657(n+1)/A053657(n) ;
end proc: # R. J. Mathar, Dec 19 2012

max = 52; s = Expand[Normal[Series[(-Log[1-x]/x)^z, {x, 0, max}]]]; a[n_, k_] := Denominator[Coefficient[s, x^n*z^k]]; A053657 = Prepend[LCM @@@ Table[a[n, k], {n, max}, {k, n}], 1]; a[n_] := A053657[[n+1]]/A053657[[n]]; Table[a[n], {n, 1, max}] (* Jean-François Alcover, Dec 20 2012 *)

A185633(n) = if(n%2,2,denominator(bernfrac(n)/(n))); \\ Antti Karttunen, Dec 03 2018

A185633(n) = { my(m=1); fordiv(n, d, if(isprime(1+d), m *= (1+d)^(1+valuation(n,1+d)))); (m); }; \\ Antti Karttunen, Dec 03 2018

a(n) = A053657(n+1)/A053657(n).
a(2*n) = 2*A036283(n).
From Antti Karttunen, Dec 03 2018: (Start)
a(n) = Product_{d|n} [(1+d)^(1+A286561(n,1+d))]^A010051(1+d) - after Peter J. Cameron's Mar 25 2002 comment in A006863.
A007947(a(n)) = A027760(n)
A001221(a(n)) = A067513(n).
A181819(a(n)) = A322312(n).
(End)

Name edited by Antti Karttunen, Dec 03 2018

A322313 Lexicographically earliest such sequence a that a(i) = a(j) => A322312(i) = A322312(j) for all i, j.

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 2, 1, 8, 1, 9, 1, 9, 1, 6, 1, 10, 1, 2, 1, 11, 1, 12, 1, 13, 1, 2, 1, 14, 1, 2, 1, 10, 1, 15, 1, 11, 1, 6, 1, 16, 1, 6, 1, 11, 1, 17, 1, 18, 1, 6, 1, 19, 1, 2, 1, 20, 1, 12, 1, 3, 1, 21, 1, 22, 1, 2, 1, 3, 1, 23, 1, 16, 1, 6, 1, 24, 1, 2, 1, 25, 1, 26, 1, 11, 1, 2, 1, 27, 1, 2, 1, 28, 1, 23, 1, 18, 1

Offset: 1

Antti Karttunen, Dec 03 2018

nonn

Restricted growth sequence transform of A322312.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A322312, A322315.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A322312(n) = { my(m=1,p); fordiv(n,d,p=1+d; if(isprime(p), for(i=0,oo,if(n%(p^i),m *= prime(i);break)))); (m); };
v322313 = rgs_transform(vector(up_to, n, A322312(n)));
A322313(n) = v322313[n];

A323156 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = A323155(n) for all n, except f(1) = 0.

Original entry on oeis.org

1, 2, 3, 4, 2, 5, 2, 6, 3, 2, 2, 7, 2, 8, 3, 6, 2, 9, 2, 10, 3, 2, 2, 11, 2, 2, 3, 12, 2, 13, 2, 14, 3, 2, 2, 15, 2, 16, 3, 17, 2, 18, 2, 19, 3, 2, 2, 20, 2, 2, 3, 4, 2, 21, 2, 22, 3, 2, 2, 23, 2, 24, 3, 14, 2, 5, 2, 25, 3, 8, 2, 26, 2, 27, 3, 28, 2, 5, 2, 29, 3, 2, 2, 30, 2, 2, 3, 31, 2, 32, 2, 4, 3, 2, 2, 33, 2, 34, 3, 10, 2, 35, 2, 36, 3

Offset: 1

Antti Karttunen, Jan 09 2019

nonn

For all i, j:
  a(i) = a(j) => A072627(i) = A072627(j),
  a(i) = a(j) => A323157(i) = A323157(j) => A322977(i) = A322977(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A323157, also A322315.

Cf. A072627, A322977.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A323155(n) = { my(m=1); fordiv(n, d, if(isprime(d-1), m *= (d-1)^(1+valuation(n,d-1)))); (m); };
v323156 = rgs_transform(vector(up_to, n, if(1==n,0,A323155(n))));
A323156(n) = v323156[n];

Showing 1-4 of 4 results.

cp's OEIS Frontend

A079612 Largest number m such that a^n == 1 (mod m) whenever a is coprime to m.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

PARI

Formula

Extensions

A185633 For odd n, a(n) = 2; for even n, a(n) = denominator of Bernoulli(n)/n; The number 2 alternating with the elements of A006953.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A322313 Lexicographically earliest such sequence a that a(i) = a(j) => A322312(i) = A322312(j) for all i, j.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A323156 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = A323155(n) for all n, except f(1) = 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI