A023506 -id:A023506 - cp's OEIS frontend

A361180 Primes p such that the odd part of p - 1 is upper-bounded by the dyadic valuation of p - 1.

3, 5, 17, 97, 193, 257, 641, 769, 12289, 18433, 40961, 65537, 114689, 147457, 163841, 786433, 1179649, 5767169, 7340033, 13631489, 23068673, 167772161, 469762049, 2013265921, 2281701377, 3221225473, 3489660929, 12348030977, 77309411329, 206158430209, 2061584302081, 2748779069441

Offset: 1

Lorenzo Sauras Altuzarra, Mar 03 2023

nonn

Primes of the form k*2^m + 1 where k <= m and k is odd. - David A. Corneth, Mar 03 2023

Primes prime(k) such that A057023(k) <= A023506(k). - Michel Marcus, Mar 09 2023

			3 is a term because the odd part of 2 is 1, the dyadic valuation of 2 is 1 and 1 <= 1.
641 = 5*2^7 + 1 is a term because the odd part of 640 is 5, the dyadic valuation of 640 is 7 and 5 <= 7.

Cf. A000040 (primes), A000265 (odd part), A007814 (dyadic valuation).

Cf. A057023, A023506.

# Maple program due to David A. Corneth, Mar 03 2023
aList := proc(upto)
   local i, j, p, R:
   R := {}:
   for i from 1 to ilog2(upto) do
      for j from 1 to min(i, floor(upto/2^i)) do
         p := j*2^i+1:
         if isprime(p) then R := `union`(R, {p}): fi: od: od:
   R: end:
aList(10^12);

isok(p) = if (isprime(p), my(m=valuation(p-1,2)); (p-1)/2^m <= m); \\ Michel Marcus, Mar 03 2023

upto(n) = {my(res = List()); for(i = 1, logint(n, 2), forstep(j = 1, min(i, n>>i), 2, if(isprime((j<David A. Corneth, Mar 03 2023

a(17)..a(27) from Michel Marcus, Mar 03 2023

More terms from David A. Corneth, Mar 03 2023

A376349 Number of isomorphism classes k of groups G of order p*2^n when G contains a unique Sylow p subgroup and the maximal 2^m dividing p-1 is such that 2^m >= 2^n.

Original entry on oeis.org

1, 2, 5, 15, 54, 247, 1684, 21820, 1118964

Offset: 0

Miles Englezou, Sep 19 2024

A Sylow p subgroup is a subgroup of order p^r that necessarily exists when r is a maximal power of p. It is not necessarily unique, but when it is unique it is normal in G.
The condition that G of order p*2^n contains a unique Sylow p subgroup places an upper bound on the number of isomorphism classes of G; it is equivalent to stating that the minimal 2^r such that 2^r == 1 (mod p) be such that 2^r > 2^n. The condition that the maximal 2^m dividing p-1, i.e. for p == 1 (mod 2^m), is such that 2^m >= 2^n ensures a lower bound which is equal to the upper bound. See the Miles Englezou link for a proof.
If we relax the two conditions and just consider an arbitrary odd prime p and the number of isomorphism classes for |G| = p*2^n, it is likely that the set of such numbers is unique to p. Since every odd prime has a minimal 2^r such that 2^r == 1 (mod p) (a consequence of Fermat's little theorem), when 2^r = 2^n for |G| = p*2^n, the number of isomorphism classes will differ from a(n) due to the existence of groups where the Sylow p subgroup is not unique.

			a(2) = 5 since D_(p*2^2), C_(p*2^2), C_(p*2^1) x C_2, and two semidirect products C_p : C_4 are all the groups of order p*2^2 for p satisfying the two conditions.
Table showing minimal 2^r and maximal 2^m (as defined in the Comments) for some primes:
---------------------------------------------------------------------------
p |      Minimal 2^r == 1 (mod p)       |   Maximal 2^m, p == 1 (mod 2^m)  |
---------------------------------------------------------------------------
2 |             2^0  = 1                |              2^0 = 1             |
3 |             2^2  = 4                |              2^1 = 2             |
5 |             2^4  = 16               |              2^2 = 4             |
7 |             2^3  = 8                |              2^1 = 2             |
11|             2^10 = 1024             |              2^1 = 2             |
13|             2^12 = 4096             |              2^2 = 4             |
17|             2^8  = 256              |              2^4 = 16            |
19|             2^18 = 262144           |              2^1 = 2             |
23|             2^11 = 2048             |              2^1 = 2             |
29|             2^28 = 268435456        |              2^2 = 4             |
31|             2^5  = 32               |              2^1 = 2             |
37|             2^36 = 68719476736      |              2^2 = 4             |
---------------------------------------------------------------------------
Table of primes satisfying 2^r > 2^n, and 2^m >= 2^n:
-------------------------------------------------------------------------------
   2^n   |                          primes                           |   a(n)  |
-------------------------------------------------------------------------------
2^0 = 1  |  all primes                                    = A000040  | 1       |
2^1 = 2  |  all primes > 2                                = A065091  | 2       |
2^2 = 4  |  5, 13, 17, 29, 37, 41, 53, ...                = A002144  | 5       |
2^3 = 8  |  17, 41, 73, 89, 97, 113, 137, ...             = A007519  | 15      |
2^4 = 16 |  17, 97, 113, 193, 241, 257, 337 ...           = A094407  | 54      |
2^5 = 32 |  97, 193, 257, 353, 449, 577, 641, ...         = A133870  | 247     |
2^6 = 64 |  193, 257, 449, 577, 641, 769, 1153, ...       = A142925  | 1684    |
2^7 = 128|  257, 641, 769, 1153, 1409, 2689, 3329, ...    = A208177  | 21820   |
2^8 = 256|  257, 769, 3329, 7937, 9473, 14081, 14593 ...  = A105131  | 1118964 |
-------------------------------------------------------------------------------

Miles Englezou, Proof

Cf. A000001, A000040, A065091, A002144, A007519, A094407, A133870, A142925, A208177, A105131, A139669, A014664, A023506, A058384.

S:=[];
for i in [0..8] do
    n:=7681*2^i; # 7681 is an appropriate prime for reproducing up to a(8)
    S:=Concatenation(S,[NrSmallGroups(n)]);
od;
Print(S);

a(n) = A000001(p*2^(n)) for every p satisfying the two conditions mentioned in Comments.

A279229 Odd orders n for which a complete dihedral Hamiltonian cycle system of the cocktail graph exists.

Original entry on oeis.org

21, 33, 45, 57, 65, 69, 77, 85, 93, 105, 117, 123, 129, 133, 141, 145, 153, 161, 165, 177, 185, 189, 201, 209, 213, 217, 219, 221, 225, 237, 245, 249, 253, 261, 265, 267, 273, 285, 287, 291, 297, 301, 305, 309, 321, 325, 329, 333, 341, 345, 357

Offset: 1

R. J. Mathar, Jan 04 2017

M. Buratti and F. Merola, Dihedral Hamiltonian cycle systems of the Cocktail Party Graph, J. Combin. Des. 21 (1) (2013) 1-23, Section 3.

isA000961 := proc(n)
    local pf;
    if n = 1 then
        return true;
    end if;
    pf := ifactors(n)[2] ;
    if nops(pf) > 1 then
        false;
    else
        true;
    end if ;
end proc:
A023506 := proc(p)
    padic[ordp](p-1,2) ;
end proc:
isA279229 := proc(n)
    local ct2,p,l ;
    if type(n,'even') then
        false;
    elif isA000961(n) then
        false;
    else
        ct2 := 0 ;
        for pf in ifactors(n)[2] do
            l := A023506(op(1,pf)) ;
            ct2 := ct2+l*op(2,pf) ;
        end do:
        type(ct2,'even') ;
    end if;
end proc:
for n from 2 to 2000 do
    if isA279229(n) then
        printf("%d,",n);
    end if;
end do:

A023506[p_] := IntegerExponent[p - 1, 2];
isA279229[n_] := Module[{ct2, l}, Which[EvenQ[n], False, PrimePowerQ[n], False, True, ct2 = 0; Do[l = A023506[pf[[1]]]; ct2 = ct2 + l*pf[[2]], {pf, FactorInteger[n]}]; EvenQ[ct2]]];
Select[Range[2, 400], isA279229] (* Jean-François Alcover, Oct 28 2023, after R. J. Mathar's program *)

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A361180 Primes p such that the odd part of p - 1 is upper-bounded by the dyadic valuation of p - 1.

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A376349 Number of isomorphism classes k of groups G of order p*2^n when G contains a unique Sylow p subgroup and the maximal 2^m dividing p-1 is such that 2^m >= 2^n.

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A279229 Odd orders n for which a complete dihedral Hamiltonian cycle system of the cocktail graph exists.

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