A080148 -id:A080148 - cp's OEIS frontend

A139669 Number of isomorphism classes of finite groups of order 11*2^n.

1, 2, 4, 12, 42, 195, 1387, 19324, 1083472

Offset: 0

Anthony D. Elmendorf (aelmendo(AT)calumet.purdue.edu), Jun 12 2008

This appears to be the smallest possible number of groups of order q*2^n for an odd number q.
Apparently, a(n) is also the number of isomorphism classes of finite groups of order 19*2^n and, more generally, of order p*2^n for primes p such that p is congruent to 3 modulo 4 and p+1 is not a power of 2.
Comment from Miles Englezou, Sep 26 2024: (Start)
The comment above which starts "Apparently, ... power of 2." is not true. (For a proof see the Miles Englezou link). However, it is true that a(0) to a(8) are the smallest possible number of groups of order q*2^n for an odd number q, and this can be generalized in the way stated below. (For further details see the Miles Englezou link).
A correct generalization of the 9 terms:
 The number of groups of order q*2^n is the least possible for prime q such that q == 3 (mod 4) and where the least 2^m such that 2^m == 1 (mod q) is greater than 2^n. Or put another way, if A014664(A080148(n)) > n, then for q = A000040(A080148(n)) the number of groups of order q*2^n is the least possible. (End)

			a(2) is the number of groups of order 11*2^2=44, which is 4 and also the number of groups of order 19*2^2=76, 23*2^2=92, etc.

J. H. Conway et al., The Symmetries of Things, Peters, 2008, p. 206.

John H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica.
Miles Englezou, Proofs

Cf. A000001, A002145, A014664, A080148.

A139669 := n -> GroupTheory[NumGroups](11*2^n);

a(n) = A000001(11*2^n). - Max Alekseyev, Apr 26 2010

a(8) from Max Alekseyev, Dec 24 2014

A081728 Length of periods of Euler numbers modulo prime(n).

Original entry on oeis.org

1, 2, 2, 6, 10, 6, 8, 18, 22, 14, 30, 18, 20, 42, 46, 26, 58, 30, 66, 70, 36, 78, 82, 44, 48, 50, 102, 106, 54, 56, 126, 130, 68, 138, 74, 150, 78, 162, 166, 86, 178, 90, 190, 96, 98, 198, 210, 222, 226, 114, 116, 238, 120, 250, 128, 262, 134, 270, 138, 140, 282, 146

Offset: 1

Benoit Cloitre, Apr 06 2003

nonn

As proved by Kummer, if the actual signed Euler numbers (A122045) are used, then the period is prime(n)-1 for n>1. - T. D. Noe, Mar 16 2007

			A000364 modulo 5=prime(3) gives : 1,1,0,1,0,1,0,1,0,1,0,... with period (1,0) of length 2, hence a(3)=2.

Cf. A000364, A045326, A080148.

f[n_] := Block[{p = Prime[n], t, d = Divisors[p - 1], dk, k = 1},t = Mod[Table[Abs@EulerE[2i], {i, 2, p}], p];While[dk = d[[k]];Nand @@ Equal @@@ Partition[Partition[t, dk], 2, 1], k++ ];dk];Array[f, 63] (* Ray Chandler, Mar 15 2007 *)

a(n)=prime(n)-1 if prime(n) == 2 or 3 (mod 4)

More terms from John W. Layman, Jul 29 2005

Extended by Ray Chandler, Mar 15 2007

A129842 Primes p such that (p^2 - 3p - 2)/2 is prime.

Original entry on oeis.org

7, 11, 19, 23, 31, 43, 47, 67, 79, 83, 107, 127, 151, 167, 199, 211, 227, 239, 251, 271, 283, 307, 359, 419, 431, 439, 443, 467, 479, 523, 547, 563, 587, 599, 643, 647, 719, 743, 827, 859, 883, 887, 947, 991, 1031, 1039, 1103, 1171, 1259, 1303, 1399, 1423

Offset: 1

J. M. Bergot, May 22 2007

nonn

These primes are all of the form 4k+3 (cf. A002145, A080148). - R. J. Mathar, Jun 08 2007

			For p = 31, one half of 31^2 - 3*31 - 2 equals 433, which is prime.

Max Alekseyev, Table of n, a(n) for n=1..230

a:=proc(n) local b: b:=ithprime(n): if isprime((b^2-3*b-2)/2)=true then b else fi end: seq(a(n),n=1..300); # Emeric Deutsch, May 25 2007
isA129842 := proc(p) if isprime(p) then isprime((p^2-3*p-2)/2) ; else false ; fi ; end: for n from 1 to 880 do p := ithprime(n) : if isA129842(p) then printf("%d, ",p) ; fi ; od : # R. J. Mathar, Jun 08 2007

Select[Prime[Range[2, 224]], PrimeQ[(#^2 - 3*# - 2)/2] &] (* Stefan Steinerberger, Jun 23 2007 *)

More terms from Emeric Deutsch and R. J. Mathar, May 25 2007

A228432 Sum_{i=1..floor(prime(n)/4)} floor(sqrt(i*prime(n))).

Original entry on oeis.org

0, 0, 2, 2, 7, 14, 24, 25, 37, 70, 71, 114, 140, 143, 170, 234, 274, 310, 357, 399, 444, 498, 552, 660, 784, 850, 856, 926, 990, 1064, 1310, 1395, 1564, 1574, 1850, 1859, 2054, 2173, 2277, 2494, 2623, 2730, 2986, 3104, 3234, 3246, 3656, 4085, 4235, 4370

Offset: 1

Michel Marcus, Nov 11 2013

nonn

If p = prime(n) in A002145 and n>3, or said differently, if n in A080148 and n>1, then a(n) = A081115(n).

			For n=7, p=17 and a(7) = floor(sqrt(17)) + floor(sqrt(34)) + floor(sqrt(51)) + floor(sqrt(68)) = 4+5+7+8 = 24.

S. A. Shirali, A family portrait of primes -- a case study in discrimination, Math. Mag., Vol. 70, No. 4 (Oct. 1997), pp. 263-272.

Cf. A002145, A080148, A081115.

Table[p = Prime[n]; Sum[Floor[Sqrt[i*p]], {i, Floor[p/4]}], {n, 100}] (* T. D. Noe, Nov 13 2013 *)

a(n) = p = prime(n); sum(i=1, p\4, sqrtint(i*p));

A249132 Smallest noncomposite k such that prime(n) is the largest prime factor of k^2+1, or 0 if no such k exists.

Original entry on oeis.org

1, 0, 2, 0, 0, 5, 13, 0, 0, 17, 0, 31, 73, 0, 0, 23, 0, 11, 0, 0, 173, 0, 0, 233, 463, 293, 0, 0, 251, 919, 0, 0, 37, 0, 193, 0, 443, 0, 0, 599, 0, 19, 0, 467, 211, 0, 0, 0, 0, 107, 89, 0, 659, 0, 241, 0, 2503, 0, 337, 53, 0, 3671, 0, 0

Offset: 1

Juri-Stepan Gerasimov, Oct 22 2014

nonn

a(A080148(m)) = 0. - Joerg Arndt, Oct 22 2014

			a(1)=1 is in this sequence because 1 is in A008578 and the largest prime factor of 1^2+1 = 2 is prime(1).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A080148, A185389, A223702, A223705.

A249132:= proc(n) local p,i,k,a,b;
   p:= ithprime(n);
   if p mod 4 = 3 then return 0 fi;
   a:= numtheory:-msqrt(-1,p);
   if a < p/2 then b:= p-a
   else b:= a; a:= p-a
   fi;
   for i from 0 do
    for k in [a+i*p,b+i*p] do
      if isprime(k) and p = max(numtheory:-factorset(k^2+1)) then
        return(k)
      fi
    od
   od
end proc:
1,seq(A249132(n),n=2..100); # Robert Israel, Nov 10 2014

a249132[n_Integer] := Module[{t = Table[0, {n}], k, s, p}, Do[If[Mod[Prime[k], 4] == 3, t[[k]] = -1], {k, n}]; k = 0; While[Times @@ t == 0, k++; s = FactorInteger[k^2 + 1][[-1, 1]]; p = PrimePi[s]; If[p <= n && t[[p]] == 0 && ! CompositeQ[k], t[[p]] = k]]; t /. -1 -> 0]; a249132[120] (* Michael De Vlieger, Nov 11 2014, adapted from A223702 *)

A120311 Numbers k such that pi(k) == 0 (mod 4), where pi() = A000720, and such that prime(k) == 3 (mod 4).

Original entry on oeis.org

8, 9, 19, 20, 22, 38, 39, 54, 56, 58, 72, 90, 91, 92, 93, 94, 95, 96, 107, 131, 132, 133, 153, 154, 155, 156, 173, 175, 177, 193, 195, 224, 225, 239, 240, 266, 281, 282, 311, 340, 344, 346, 360, 363, 364, 365, 383, 385, 386, 387

Offset: 1

Roger L. Bagula, Jun 20 2006

nonn

Cf. A000720. Intersection of A080148 and A120309.

a = Flatten[Table[If[Mod[PrimePi[n], 4] == 0&& Mod[Prime[n], 4] -3 == 0, n, {}], {n, 1, 200}]]
Select[Range[400],Mod[PrimePi[#],4]==0&&Mod[Prime[#],4]==3&] (* Harvey P. Dale, Apr 17 2019 *)

Edited and more terms added by Michel Marcus, Jun 01 2013

Offset corrected by Michel Marcus, Feb 10 2021

cp's OEIS Frontend

A139669 Number of isomorphism classes of finite groups of order 11*2^n.

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A081728 Length of periods of Euler numbers modulo prime(n).

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A129842 Primes p such that (p^2 - 3p - 2)/2 is prime.

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A228432 Sum_{i=1..floor(prime(n)/4)} floor(sqrt(i*prime(n))).

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A249132 Smallest noncomposite k such that prime(n) is the largest prime factor of k^2+1, or 0 if no such k exists.

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A120311 Numbers k such that pi(k) == 0 (mod 4), where pi() = A000720, and such that prime(k) == 3 (mod 4).

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