A064899 -id:A064899 - cp's OEIS frontend

A068198 Duplicate of A064899.

6, 10, 14, 21, 22, 26, 34, 38, 39, 46, 55, 57, 58, 62, 74, 82, 86, 93, 94, 106, 111, 118

Offset: 1

dead

A064911 If n is semiprime (or 2-almost prime) then 1 else 0.

0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Patrick De Geest, Oct 13 2001

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65545 (first 10000 terms from Reinhard Zumkeller)
Eric Weisstein's World of Mathematics, Semiprime
Eric Weisstein's World of Mathematics, Prime zeta function primezeta(s).
Index entries for characteristic functions
Index entries for sequences computed from exponents in factorization of n

Cf. A010051, A064899-A064910, A053409, A046413, A101040, A105700, A280710, A302047, A302048, A302049, A353475, A353476, A353477, A353478, A353479, A353480, A353481.

a064911 = a010051 . a032742 -- Reinhard Zumkeller, Mar 13 2011

with(numtheory):
a:= n-> `if`(bigomega(n)=2, 1, 0):
seq(a(n), n=1..120);  # Alois P. Heinz, Mar 16 2011

Table[If[PrimeOmega[n] == 2, 1, 0], {n, 105}] (* Jayanta Basu, May 25 2013 *)

a(n)=bigomega(n)==2 \\ Charles R Greathouse IV, Mar 13 2011

a(n) = 1 iff n is in A001358 (semiprimes), a(n) = 0 iff n is in A100959 (non-semiprimes). - Reinhard Zumkeller, Nov 24 2004
Dirichlet g.f.: (primezeta(2s) + primezeta(s)^2)/2. - Franklin T. Adams-Watters, Jun 09 2006
a(n) = A057427(A174956(n)); a(n)*A072000(n) = A174956(n). - Reinhard Zumkeller, Apr 03 2010
a(n) = A010051(A032742(n)) (i.e., largest proper divisor is prime). - Reinhard Zumkeller, Mar 13 2011
From Antti Karttunen, Apr 24 2018 & Apr 22 2022: (Start)
a(n) = A280710(n) + A302048(n) = A101040(n) - A010051(n).
a(n) = A353478(n) + A353480(n) = A353477(n) + A353478(n) + A353479(n).
a(n) = A353475(n) + A353476(n).
(End)
a(n) = [Omega(n) = 2], where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jul 22 2025

Edited by M. F. Hasler, Oct 18 2017

A051532 The abelian orders (or abelian numbers): numbers m such that every group of order m is abelian.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 29, 31, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 99, 101, 103, 107, 109, 113, 115, 119, 121, 123, 127, 131, 133, 137, 139, 141, 143, 145, 149, 151, 153, 157, 159, 161

Offset: 1

Des MacHale, Dec 11 1999

Except for a(2)=2 and a(4)=4, all of the terms in the sequence are odd. This is because of the existence of a non-abelian dihedral group of order 2m for each m > 2.
Cubefree terms of A056867; A212793(a(n)) = 1. - Reinhard Zumkeller, Jun 28 2013
See similar comment with "squarefree terms" in A003277 (Donald J. McCarthy link). - Bernard Schott, Feb 20 2023

			a(4) = 4 because every group of order 4 is abelian.
These two abelian groups of order 4 are the cyclic group C_4 and the Klein four-group = C_2 X C_2, the smallest non-cyclic abelian group. - _Bernard Schott_, Feb 21 2023

W. R. Scott, Group Theory, Dover, 1987, page 217.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Donald J. McCarthy, A survey of partial converses to Lagrange's theorem on finite groups, Transactions of the New York Academy of Sciences, 1971, page 592.
J. Pakianathan and K. Shankar, Nilpotent Numbers, Amer. Math. Monthly, 107, August-September 2000, pp. 631-634.
Index entries for sequences related to groups

Subsequence of A056867 and supersequence of A003277.
Complement of A060652.
Intersection of A004709 and A056867.
Cf. A064899, A212793.

a051532 n = a051532_list !! (n-1)
a051532_list = filter ((== 1) . a212793) a056867_list
-- Reinhard Zumkeller, Jun 28 2013

okQ[n_] := Module[{f, lf, p, e, v}, f = FactorInteger[n]; lf = Length[f]; p = f[[All, 1]]; e = f[[All, 2]]; If[AnyTrue[e, # > 2&], Return[False]]; v = p^e; For[i = 1, i <= lf, i++, For[j = i+1, j <= lf, j++, If[Mod[v[[i]], p[[j]]]==1 || Mod[v[[j]], p[[i]]]==1, Return[False]]]]; Return[True]];
Select[Range[200], okQ] (* Jean-François Alcover, May 03 2012, after PARI, updated Jan 10 2020 *)

is(n)=my(f=factor(n),v=vector(#f[,1])); for(i=1,#v, if(f[i,2]>2, return(0), v[i]=f[i,1]^f[i,2])); for(i=1,#v, for(j=i+1,#v, if(v[i]%f[j,1]==1 || v[j]%f[i,1]==1, return(0)))); 1 \\ Charles R Greathouse IV, Feb 13 2011

from sympy import factorint
def ok(n):
    if n == 1: return True
    f = factorint(n)
    p, e = f.keys(), f.values()
    if max(e) >= 3: return False
    return all((pi**k)%pj!=1 for pi in p for pj in p if pj!=pi for k in range(1, f[pi]+1))
print([k for k in range(1, 162) if ok(k)]) # Michael S. Branicky, Feb 20 2023

m must be cubefree and its prime divisors must satisfy certain congruences.

Let the prime factorization of m be p1^e1 * ... * pr^er. Then m is in this sequence if ei < 3 for all i and pi^k is not congruent to 1 (mod pj) for all i and j and 1 <= k <= ei. - T. D. Noe, Mar 25 2007

A064910 Smallest semiprime p*q such that q >= p and q mod p = n.

Original entry on oeis.org

4, 6, 15, 65, 77, 133, 91, 319, 209, 341, 299, 481, 493, 799, 527, 1007, 1139, 2449, 703, 3611, 989, 1541, 1643, 3589, 1537, 2407, 2747, 2759, 1829, 3811, 1891, 4633, 2993, 3959, 2627, 4033, 2701, 6157, 3239, 9073, 3569, 5461, 4183, 6439, 5141, 6533

Offset: 0

Patrick De Geest, Oct 13 2001

John Cerkan, Table of n, a(n) for n = 0..10000
John Cerkan, Python 2.7 program

Cf. A001358 (p2 mod p1 = 0), A064899-A064909, A064911, A053409, A046413.

nsp[n_Integer] := nsp[n] = Block[{sp = n + 1}, While[PrimeOmega[sp] != 2, sp++]; sp]; a[n_Integer] := Block[{sp = 4}, While[ fi = FactorInteger@ sp; Mod[fi[[-1, 1]], fi[[1, 1]]] != n, sp = nsp[sp]]; sp]; Array[a, 46, 0] (* Robert G. Wilson v, Aug 20 2025 *)

Name amended by John Cerkan, Apr 12 2018

A064900 Semiprimes p1*p2 such that p2 > p1 and p2 mod p1 = 2.

Original entry on oeis.org

15, 33, 35, 51, 69, 85, 87, 123, 141, 143, 159, 161, 177, 185, 213, 235, 249, 259, 267, 303, 321, 323, 335, 339, 393, 411, 447, 485, 501, 519, 533, 535, 537, 553, 573, 591, 635, 681, 685, 699, 717, 749, 753, 771, 785, 789, 807, 835, 843, 869, 871, 879, 899

Offset: 1

Patrick De Geest, Oct 13 2001

nonn

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A001358 (p2 mod p1 = 0), A064899-A064911.

okQ[n_] := Module[{p, e}, {p, e} = Transpose[FactorInteger[n]]; e == {1, 1} && Mod[p[[2]], p[[1]]] == 2]; Select[Range[1000], okQ]
Select[Range[1000],PrimeOmega[#]==2&&Mod[FactorInteger[#][[;;,1]][[2]],FactorInteger[#][[;;,1]][[1]]]==2&]//Quiet (* Harvey P. Dale, Jan 09 2025 *)

isok(n) = (bigomega(n)==2) && (f=factor(n)) && (#f~ == 2) && ((f[2,1] % f[1,1]) == 2); \\ Michel Marcus, May 20 2017

Name clarified by Sean A. Irvine, Jul 31 2023

A069209 Orders of non-Abelian Z-groups.

Original entry on oeis.org

6, 10, 12, 14, 18, 20, 21, 22, 24, 26, 28, 30, 34, 36, 38, 39, 40, 42, 44, 46, 48, 50, 52, 54, 55, 56, 57, 58, 60, 62, 63, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 93, 94, 96, 98, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 116, 117, 118

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), Apr 14 2002

nonn

Z-groups are groups in which all Sylow subgroups are cyclic. n belongs to this sequence iff n is divisible by two distinct primes p and q, such that p divides q-1. This sequence contains sequence A064899 and it is a subsequence of sequence A056868.

Numbers n such that there is more than one Z-group of order n. - Eric M. Schmidt, Sep 15 2014

Eric M. Schmidt, Table of n, a(n) for n = 1..10000
Wikipedia, Z-group.

Cf. A064899, A056868, A247371.

filter:= proc(n) local F,p,q; F:= numtheory:-factorset(n);
  for p in F do if member(1,map(`modp`,F,p)) then return true fi od:
  false
end proc:
select(filter, [$1..1000]); # Robert Israel, Sep 15 2014

filterQ[n_] := With[{pp = FactorInteger[n][[All, 1]]}, AnyTrue[pp, MemberQ[pp, q_ /; Divisible[q - 1, #]]&]];
Select[Range[2, 200], filterQ] (* Jean-François Alcover, Jul 18 2020 *)

def is_A069209(n) : return any((q-1)%p==0 for p,q in Combinations(prime_divisors(n),2)) # Eric M. Schmidt, Sep 15 2014

Edited and extended by, and missing term 78 added by, Eric M. Schmidt, Sep 15 2014

A064909 Semiprimes p1*p2 such that p2 > p1 and p2 mod p1 = 11.

Original entry on oeis.org

481, 1157, 1343, 1921, 2171, 2263, 2369, 2509, 3077, 3097, 3427, 3523, 3683, 4171, 4537, 4541, 4811, 5213, 5263, 5389, 5543, 6107, 6227, 6707, 7123, 7241, 8279, 8593, 8621, 8717, 8857, 8873, 9353, 9607, 10411, 10537, 11359, 11461, 11567, 11747, 11761, 11819

Offset: 1

Patrick De Geest, Oct 13 2001

nonn

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A001358 (p2 mod p1 = 0), A064899-A064911.

pmp11Q[n_]:=Module[{fi=FactorInteger[n][[All,1]]},Mod[fi[[2]],fi[[1]]] == 11]; Select[ Range[12000],PrimeNu[#]==PrimeOmega[#]==2&&pmp11Q[#]&] (* Harvey P. Dale, Jun 25 2018 *)

isok(n) = my(f = factor(n)); (#f~ == 2) && (vecmax(f[,2]) < 2) && ((f[2,1] % f[1,1]) == 11);

from sympy import factorint
def is_A064909(n):
    f = factorint(n)
    return (sum([f[i] for i in f]) == 2) and (max(f) % min(f) == 11)
def first_A064909(n):
    x = 1
    an = []
    while len(an) < n:
        if is_A064909(x): an.append(x)
        x += 2
    return an # John Cerkan, Apr 14 2018

Offset and name fixed by John Cerkan, Apr 12 2018

A350586 Numbers m with exactly 2 groups of order m, where one is abelian and the other is nonabelian.

Original entry on oeis.org

6, 10, 14, 21, 22, 26, 34, 38, 39, 46, 55, 57, 58, 62, 74, 82, 86, 93, 94, 105, 106, 111, 118, 122, 129, 134, 142, 146, 155, 158, 165, 166, 178, 183, 194, 195, 201, 202, 203, 205, 206, 214, 218, 219, 226, 231, 237, 253, 254, 262, 274, 278, 285, 291, 298, 301, 302

Offset: 1

Bernard Schott, Jan 07 2022

nonn

Differs from A064899 that is a subsequence: a(20) = 105 while A064899(20) = 106.
When m = 2*p, p odd prime, abelian group is C_{2*p} and nonabelian group is D_{2*p} ~ C_p : C_2.
When m = p*q, pIn both cases, C, D mean cyclic, dihedral groups of the stated order; the symbols ~ and : mean isomorphic to and semidirect product respectively.
A number m is a term iff m is squarefree and m has exactly one pair of prime factors (p, q) such that q == 1 (mod p). - David Radcliffe, Jul 30 2025

			There is only one group of order 1, 2, 3, 5 and the two groups of order 4 are abelian; hence 6 is the smallest term because the two groups of order 6 are the abelian and cyclic group C_6, while the nonabelian group is the symmetric group S_3 isomorphic to dihedral group D_6.
The smallest odd term is 21, the two corresponding groups are C_21 and semi-direct product C_7 : C_3.
The smallest term of the form p*q*r, p < q < r primes, is 105, the two corresponding groups are C_105 and semi-direct product C_35 : C_3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
J.H. Conway, H. Dietrich, and E.A. O’Brien, Counting Groups: Gnus, Moas, and other Exotica, The Mathematical Intelligencer 30, 6-15 (2008), doi:10.1007/BF02985731.

Equals A054395 \ A350322.
Subsequence of A060650 and of A005117.
Cf. A000001, A064899.

is(n,f=factor(n))=my(p=f[,1],s); if(#p && vecmax(f[,2])>1, return(0)); for(i=2,#p, for(j=1,i-1, if(p[i]%p[j]==1 && s++>1, return(0)))); s==1 \\ Charles R Greathouse IV, Jan 08 2022

list(lim)=my(v=List()); forsquarefree(n=6,lim\1, my(p=n[2][,1],s); for(i=2,#p, for(j=1,i-1, if(p[i]%p[j]==1 && s++>1, next(3)))); if(s==1, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Jan 08 2022

from sympy import factorint
def is_ok(m):
    f = factorint(m)
    if any(e > 1 for e in f.values()): return False # m must be squarefree
    return sum(q % p == 1 for p in f for q in f) == 1 # David Radcliffe, Jul 30 2025

More terms from Jinyuan Wang, Jan 08 2022

A068919 Squarefree orders of non-Abelian groups.

Original entry on oeis.org

6, 10, 14, 21, 22, 26, 30, 34, 38, 39, 42, 46, 55, 57, 58, 62, 66, 70, 74, 78, 82, 86, 93, 94, 102, 105, 106, 110, 111, 114, 118, 122, 129, 130, 134, 138, 142, 146, 154, 155, 158, 165, 166, 170, 174, 178, 182, 183, 186, 190, 194, 195, 201, 202, 203, 205, 206

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), Mar 29 2002

nonn

This set is the difference A005117 \ A003277, that is, the squarefree numbers that are not cyclic (A003277). - Jeppe Stig Nielsen, Sep 25 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Intersection of A060652 and A005117.
Cf. A064899.
Also intersection of A056868 and A005117. - Reinhard Zumkeller, Jun 28 2013
Cf. A008966.

a068919 n = a068919_list !! (n-1)
a068919_list = filter ((== 1) . a008966) a056868_list
-- Reinhard Zumkeller, Jun 28 2013

Complement[Select[Range[206], SquareFreeQ[#] &],
Select[Range[206], FiniteGroupCount[#] == 1 &]] (* Geoffrey Critzer, Nov 28 2015 *)

is(n)=issquarefree(n)&&gcd(n,eulerphi(n))!=1 \\ Jeppe Stig Nielsen, Sep 25 2017

A064901 Semiprimes p1*p2 such that p2 > p1 and p2 mod p1 = 3.

Original entry on oeis.org

65, 115, 119, 215, 217, 265, 365, 377, 413, 415, 511, 515, 517, 565, 629, 707, 779, 815, 865, 965, 1099, 1115, 1165, 1207, 1243, 1315, 1391, 1393, 1415, 1465, 1501, 1565, 1589, 1687, 1727, 1765, 1769, 1865, 1883, 1915, 1969, 1981, 2165, 2177, 2215

Offset: 1

Patrick De Geest, Oct 13 2001

nonn

The semiprimes must be squarefree, since p1 does not divide p2. - Michael De Vlieger, Apr 12 2018

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A001358 (p2 mod p1 = 0), A006881, A064899-A064911.

Select[Range@ 2215, And[#[[All, -1]] == {1, 1}, Mod[#2, #1] == 3 & @@ #[[All, 1]]] &@ FactorInteger[#] &] (* Michael De Vlieger, Apr 12 2018 *)

isok(n) = my(f = factor(n)); (#f~ == 2) && (vecmax(f[,2]) < 2) && ((f[2,1] % f[1,1]) == 3); \\ Michel Marcus, Apr 16 2018

from sympy import factorint
def is_A064901(n):
    f = factorint(n)
    return (sum([f[i] for i in f]) == 2) and (max(f) % min(f) == 3)
def first_A064901(n):
    x = 1
    an = []
    while len(an) < n:
        if is_A064901(x): an.append(x)
        x += 2
    return an # John Cerkan, Apr 14 2018

Name clarified by John Cerkan, Apr 13 2018

Showing 1-10 of 17 results. Next

cp's OEIS Frontend

A068198 Duplicate of A064899.

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A064911 If n is semiprime (or 2-almost prime) then 1 else 0.

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A051532 The abelian orders (or abelian numbers): numbers m such that every group of order m is abelian.

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A064910 Smallest semiprime p*q such that q >= p and q mod p = n.

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A064900 Semiprimes p1*p2 such that p2 > p1 and p2 mod p1 = 2.

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A069209 Orders of non-Abelian Z-groups.

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A064909 Semiprimes p1*p2 such that p2 > p1 and p2 mod p1 = 11.

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A350586 Numbers m with exactly 2 groups of order m, where one is abelian and the other is nonabelian.

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