Des MacHale - cp's OEIS frontend

A355591 a(n) = (product of the first n odd primes) - (sum of the first n odd primes).

1, 0, 7, 90, 1129, 14976, 255199, 4849770, 111546337, 3234846488, 100280244907, 3710369067210, 152125131763369, 6541380665834736, 307444891294245379, 16294579238595021986, 961380175077106319097, 58644190679703485491136, 3929160775540133527938979

Offset: 0

Des MacHale and Bernard Schott, Jul 12 2022

nonn

The parity of a(n) is the opposite of the parity of n.

			a(4) = (3*5*7*11) - (3+5+7+11) = 1129.

Robert Israel, Table of n, a(n) for n = 0..348

Cf. A000040, A059841, A070826, A071148, A355590.

a:= n-> (l-> mul(i,i=l)-add(i,i=l))([ithprime(i)$i=2..n+1]):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 12 2022

FoldList[Times, 1, p = Prime[Range[2, 20]]] - Prepend[Accumulate[p], 0] (* Amiram Eldar, Jul 14 2022 *)

a(n) = my(vp=primes(n+1)); vecprod(vp)/2 - vecsum(vp) + 2; \\ Michel Marcus, Jul 12 2022

from itertools import count, islice
from sympy import nextprime
def agen():
    p, s, primen = 1, 0, 2
    while True:
        yield p - s; primen = nextprime(primen); p *= primen; s += primen
print(list(islice(agen(), 19))) # Michael S. Branicky, Jul 12 2022

a(n) = A070826(n+1) - A071148(n).

More terms from Michael S. Branicky, Jul 12 2022

A355590 a(n) = (product of the first n primes) - (sum of the first n primes).

Original entry on oeis.org

1, 0, 1, 20, 193, 2282, 29989, 510452, 9699613, 223092770, 6469693101, 200560489970, 7420738134613, 304250263526972, 13082761331669749, 614889782588491082, 32589158477190044349, 1922760350154212638630, 117288381359406970982769, 7858321551080267055878522

Offset: 0

Des MacHale and Bernard Schott, Jul 08 2022

nonn

The parity of a(n) is the opposite of the parity of n.

			a(4) = (2*3*5*7) - (2+3+5+7) = 193.

Cf. A000040, A002110, A007504, A059841.

FoldList[Times, 1, p = Prime[Range[20]]] - Prepend[Accumulate[p], 0] (* Amiram Eldar, Jul 08 2022 *)

a(n) = my(vp=primes(n)); vecprod(vp) - vecsum(vp); \\ Michel Marcus, Jul 08 2022

from itertools import count, islice
from sympy import nextprime
def agen():
    p, s, primen = 1, 0, 0
    while True:
        yield p - s; primen = nextprime(primen); p *= primen; s += primen
print(list(islice(agen(), 20))) # Michael S. Branicky, Jul 08 2022

a(n) = A002110(n) - A007504(n).

A341825 Number of finite groups G with |Aut(G)| = n.

Original entry on oeis.org

2, 3, 0, 4, 0, 6, 0, 7, 0, 2, 0, 9, 0, 0, 0, 11, 0, 4, 0, 7, 0, 2, 0, 22, 0, 0, 0, 2, 0, 2, 0, 19, 0, 0, 0, 12, 0, 0, 0, 14, 0, 7, 0, 3, 0, 2, 0

Offset: 1

Des MacHale, Mar 02 2021

The smallest odd index n > 1 for which a(n) > 0 is 2187 = 3^7 (see A340521).

There exist even indices n that are not values taken by totient function phi (A002202) for which a(n) > 0. For example, John Bray has produced a group G that is the semidirect product 19:9 of order 3^2*19 = 171 such that |Aut(G)| = 1026 = 2*3^3*19.

			a(6) = 6, because there are six groups G with |Aut(G)| = 6. Four  cyclic groups: Aut(C_7) = Aut(C_9) = Aut(C_14) = Aut(C_18) ~~ C_6, and also Aut(C_2 x C_2) = Aut(S_3) ~~ S_3, where ~~ stands for “isomorphic to”. - _Bernard Schott_, Mar 02 2021
a(8) = 7, because there are seven groups G with |Aut(G)| = 8.

D. MacHale and R. Sheehy, Finite groups with few automorphisms, Mathematical Proceedings of the Royal Irish Academy, Vol. 104A, No. 2 (December 2004), 231-238.

Cf. A002202, A137315, A340521, A341823, A341824.

a(2) = 3, a(p) = 0 if p odd prime.

a(A002202(n)) > 0, since |Aut(C_n)| = phi(n).

A341824 Number of groups of order 2^n which occur as Aut(G) for some finite group G.

Original entry on oeis.org

1, 1, 2, 3, 4, 9, 14, 33

Offset: 0

Des MacHale, Feb 26 2021

The number of groups of order 2^n is A000679(n); the percentage of the 2-groups which occur as automorphism groups appears to decrease as n increases: 100, 100, 100, 60, 28.5, 17.6, 5.2, 1.4, ...

Jianing Song remarks that it is also interesting to consider infinite groups, and asks if there is an infinite group G with Aut(G) isomorphic to C_8. Des MacHale, Mar 03 2021, replies that at present this is not known. [Comment edited by N. J. A. Sloane, Mar 07 2021]

			a(5) = 9 because there are nine groups of order 32 which occur as automorphism groups of finite groups.
From _Bernard Schott_, Feb 26 2021: (Start)
Aut(C_15) = Aut(C_16) = Aut(C_20) = Aut(C_30) ~~ C_4 x C_2 where ~~ stands for "isomorphic to".
Aut(C_4 x C_2) = Aut(D_4) ~~ D_4 where D_4 is the dihedral group of the square.
Aut(C_24) ~~ C_2 x C_2 x C_2 = (C_2)^3.
There exist five groups of order 8 (A054397), the three groups C_4 x C_2, D_4, C_2 x C_2 x C_2 occur as automorphim groups of order 8, but the cyclic group C_8 and the quaternions group Q_8 never occur as Aut(G) for some finite G, so a(3) = 3. (End)

J. Flynn, D. MacHale, E. A. O'Brien and R. Sheehy, Finite Groups whose Automorphism Groups are 2-groups, Proc. Royal Irish Academy, 94A, (2) 1994, 137-145.

Cf. A000679, A054397, A340521, A341823, A341825.

a(n) <= A000679(n). - Des MacHale, Mar 02 2021

Offset modified by Jianing Song, Mar 02 2021

A341823 Number of finite groups G with |Aut(G)| = 2^n.

Original entry on oeis.org

2, 3, 4, 7, 11, 19, 34, 70

Offset: 0

Des MacHale, Feb 20 2021

This sequence is infinite, but the amount of computation needed to consider the large number of groups of order 2^8 suggests it may be hard to find the next term.

			a(3) = 7, because there are seven finite groups G with |Aut(G)| = 8. Four cyclic groups: Aut(C_15) = Aut(C_16) = Aut(C_20) = Aut(C_30) ~~ C_4 x C_2, also Aut(C_4 x C_2) = Aut(D_4) ~~ D_4, with D_4 is the dihedral group of the square, finally Aut(C_24) ~~ C_2 x C_2 x C_2 = (C_2)^3 where ~~ stands for “isomorphic to". - _Bernard Schott_, Mar 04 2021

J. Flynn, D. MacHale, E. A. O'Brien and R. Sheehy, Finite Groups whose Automorphism Groups are 2-groups, Proc. Royal Irish Academy, 94A, (2) 1994, 137-145.

Cf. A341824, A341825.

Subsequence of A340521.

Offset modified by Bernard Schott, Mar 04 2021

A341298 Orders of complete groups.

Original entry on oeis.org

1, 6, 20, 24, 42, 54, 110, 120, 144, 156, 168, 216, 252, 272, 320, 324, 336, 342, 384, 432, 480, 486, 500, 506, 660, 720, 800, 812, 840, 864, 930, 936, 960, 972, 1008, 1012, 1080

Offset: 1

Bob Heffernan and Des MacHale, Feb 10 2021

A finite group G is called complete if Aut G = Inn G and Z(G) = {1} i.e. G has no outer automorphisms and the center of G is trivial.
The symmetric group S(n) of order n! is complete for n not equal to 2 or 6.
If p is an odd prime, there is a complete group of order p(p-1) and a complete group of order p^m*(p^m - p^(m-1)) for each m.
Dark in 1975 discovered a nontrivial complete group G of odd order. It has order 788953370457 = 3*19*7^12. [Corrected by Jianing Song, Aug 25 2023]
Recently, Dark showed that the smallest possible nontrivial complete group G of odd order has order 352947 = 3*7^6. [In fact, for every prime p == 1 (mod 3), there exists a complete group of order 3*p^6, and it occurs as the automorphism group of a group of order 3*p^5. This means that there are infinitely many odd terms in this sequence. See the M. John Curran and Rex S. Dark link. - Jianing Song, Aug 25 2023]
From Jianing Song, Aug 25 2023: (Start)
The holomorph (see the Wikipedia link) of an abelian group of odd order is a complete group. See Theorem 3.2, Page 618 of the W. Peremans link.
No prime power (A246655) is a term. See the first Groupprops link.
The automorphism group of a complete group is isomorphic to itself. The converse is not true, as shown by the counterexamples D_8 and D_12. In contrast with the fact that the holomorph of a complete group is isomorphic to the external direct product of two copies of it (see the second Groupprops link), the holomorph of D_8 (SmallGroup(64,134)) is not isomorphic to D_8 X D_8 = SmallGroup(64,226), and the holomorph of D_12 (SmallGroup(144,154)) is not isomorphic to D_12 X D_12 = SmallGroup(144,192). (End)

			a(3) = 20 because 20 is the third number for which there is a complete group of that order.

M. John Curran and Rex S. Dark, Complete Groups of Order 3p^6, Advances in Group Theory and Applications, 2 (2016), pp. 1-12.
R. S. Dark, A complete group of odd order, Mathematical Proc. Cambridge Philosophical Society, Vol. 77, No. 1, January 1975, pp. 21-28.
Groupprops, Group isomorphic to its automorphism group
Groupprops, Holomorph of a group
W. Peremans, Completeness of Holomorphs, Nederl. Akad. Wetensch. Proc. Ser. A, 60. (1957) 608-619.
Jianing Song, List of complete groups with order <= 1080
Wikipedia, Holomorph

a(36) and a(37) added by Jianing Song, Aug 25 2023

A341297 Let T(G) be the sum of the degrees of the irreducible complex matrix representations of a finite group G, and let M(G) = |G|*(|G|-T(G)). Sequence gives allowable values of M(G).

Original entry on oeis.org

12, 16, 40, 48, 64, 72, 84, 136, 144

Offset: 1

Des MacHale, Feb 05 2021

B. Dolan, D. MacHale and P. MacHale, Counting Commutativities in Finite Algebraic Systems, Irish Math. Soc. Bulletin, Number 77, Summer 2016, 61-70.

Cf. A341296.

A341296 Let T(G) be the sum of the degrees of the irreducible complex matrix representations of a finite group G, and let N(G) = |G|*T(G). Sequence gives allowable values of N(G).

Original entry on oeis.org

24, 48, 60, 72, 96, 112, 120, 160, 189, 192, 240, 264, 288

Offset: 1

Des MacHale, Feb 05 2021

B. Dolan, D. MacHale and P. MacHale, Counting Commutativities in Finite Algebraic Systems, Irish Math. Soc. Bulletin, Number 77, Summer 2016, 61-70.

Cf. A341297.

A341295 Allowable values for the total number of times we can have ab != ba in the Cayley table of a finite non-abelian group.

Original entry on oeis.org

18, 24, 60, 72, 96, 126, 144, 162, 216, 240, 288, 300, 330, 336, 360, 384, 408, 432, 450, 456, 468, 480, 504, 540, 576, 600, 630, 648, 672

Offset: 1

Des MacHale, Feb 05 2021

a(n) is always even.

B. Dolan, D. MacHale and P. MacHale, Counting Commutativities in Finite Algebraic Systems, Irish Math. Soc. Bulletin, Number 77, Summer 2016, 61-70.

Cf. A341294.

A341294 Allowable values for the total number of times we can have ab=ba in the Cayley table of a finite non-abelian group.

Original entry on oeis.org

18, 40, 48, 70, 72, 100, 105, 108, 112, 120, 154, 160, 162, 168, 192, 208, 216, 270, 273, 280, 288, 294, 297, 300, 324, 340, 352, 360, 364, 384, 385, 400, 418, 432

Offset: 1

Des MacHale, Feb 05 2021

B. Dolan, D. MacHale and P. MacHale, Counting Commutativities in Finite Algebraic Systems, Irish Math. Soc. Bulletin, Number 77, Summer 2016, 61-70.

Cf. A341295.

cp's OEIS Frontend

User: Des MacHale

A355591 a(n) = (product of the first n odd primes) - (sum of the first n odd primes).

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A355590 a(n) = (product of the first n primes) - (sum of the first n primes).

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A341825 Number of finite groups G with |Aut(G)| = n.

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A341824 Number of groups of order 2^n which occur as Aut(G) for some finite group G.

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A341823 Number of finite groups G with |Aut(G)| = 2^n.

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A341298 Orders of complete groups.

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A341297 Let T(G) be the sum of the degrees of the irreducible complex matrix representations of a finite group G, and let M(G) = |G|*(|G|-T(G)). Sequence gives allowable values of M(G).

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A341296 Let T(G) be the sum of the degrees of the irreducible complex matrix representations of a finite group G, and let N(G) = |G|*T(G). Sequence gives allowable values of N(G).

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A341295 Allowable values for the total number of times we can have ab != ba in the Cayley table of a finite non-abelian group.

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