A054753
Numbers which are the product of a prime and the square of a different prime (p^2 * q).
Original entry on oeis.org
12, 18, 20, 28, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99, 116, 117, 124, 147, 148, 153, 164, 171, 172, 175, 188, 207, 212, 236, 242, 244, 245, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 338, 356, 363, 369, 387, 388, 404, 412, 423, 425, 428, 436, 452
Offset: 1
a(1) = 12 because 12 = 2^2*3 is the smallest number of the form p^2*q.
- Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
- Guilhem Castagnos, Antoine Joux, Fabien Laguillaumie, and Phong Q. Nguyen, Factoring pq^2 with quadratic forms: nice cryptanalyses, Advances in Cryptology - ASIACRYPT 2009. Lecture Notes in Computer Science Volume 5912 (2009), pp. 469-486.
- Diophante, A 350, Les Nombres d'Einstein (in French).
- Mathematics Stack Exchange, Sequence of numbers with prime factorization pq^2
- René Peralta and Eiji Okamoto, Faster factoring of integers of a special form (1996).
- Index to sequences related to prime signature
Numbers with 6 divisors (
A030515) which are not 5th powers of primes (
A050997).
Table giving for each subsequence the corresponding number of groups of order p^2*q, from
Bernard Schott, Jan 23 2022
-------------------------------------------------------------------------------
-------------------------------------------------------------------------------
|
A000001(p^2*q)| (q+9)/2 | 5 | 5 | 4 | 3 | 2 |
-------------------------------------------------------------------------------
-
Select[Range[12,452], {1,2}==Sort[Last/@FactorInteger[ # ]]&] (* Zak Seidov, Jul 19 2009 *)
With[{nn=60},Take[Union[Flatten[{#[[1]]#[[2]]^2,#[[1]]^2 #[[2]]}&/@ Subsets[ Prime[Range[nn]],{2}]]],nn]] (* Harvey P. Dale, Dec 15 2014 *)
-
is(n)=vecsort(factor(n)[,2])==[1,2]~ \\ Charles R Greathouse IV, Dec 30 2014
-
for(n=1, 1e3, if(numdiv(n) - bigomega(n) == 3, print1(n, ", "))) \\ Altug Alkan, Nov 24 2015
-
from sympy import factorint
def ok(n): return sorted(factorint(n).values()) == [1, 2]
print([k for k in range(453) if ok(k)]) # Michael S. Branicky, Dec 18 2021
-
from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A054753(n):
def bisection(f,kmin=0,kmax=1):
while f(kmax) > kmax: kmax <<= 1
kmin = kmax >> 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
def f(x): return n+x-sum(primepi(x//p**2) for p in primerange(isqrt(x)+1))+primepi(integer_nthroot(x,3)[0])
return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025
Link added and incorrect Mathematica code removed by
David Bevan, Sep 17 2011
A183558
Number of partitions of n containing a clique of size 1.
Original entry on oeis.org
0, 1, 1, 2, 3, 6, 7, 13, 16, 25, 33, 49, 61, 90, 113, 156, 198, 269, 334, 448, 556, 726, 902, 1163, 1428, 1827, 2237, 2817, 3443, 4302, 5219, 6478, 7833, 9632, 11616, 14197, 17031, 20712, 24769, 29925, 35688, 42920, 50980, 61059, 72318, 86206, 101837, 120941
Offset: 0
a(5) = 6, because 6 partitions of 5 contain (at least) one clique of size 1: [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5].
From _Gus Wiseman_, Apr 19 2019: (Start)
The a(1) = 1 through a(8) = 16 partitions are the following. The Heinz numbers of these partitions are given by A052485 (weak numbers).
(1) (2) (3) (4) (5) (6) (7) (8)
(21) (31) (32) (42) (43) (53)
(211) (41) (51) (52) (62)
(221) (321) (61) (71)
(311) (411) (322) (332)
(2111) (3111) (331) (422)
(21111) (421) (431)
(511) (521)
(2221) (611)
(3211) (3221)
(4111) (4211)
(31111) (5111)
(211111) (32111)
(41111)
(311111)
(2111111)
(End)
Cf.
A000041,
A007690,
A183559,
A183560,
A183561,
A183562,
A183563,
A183564,
A183565,
A183566,
A183567.
-
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
add((l->`if`(j=1, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..50);
-
max = 50; f = (1 - Product[1 - x^j + x^(2*j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; CoefficientList[s, x] (* Jean-François Alcover, Oct 01 2014. Edited by Gus Wiseman, Apr 19 2019 *)
A090858
Number of partitions of n such that there is exactly one part which occurs twice, while all other parts occur only once.
Original entry on oeis.org
0, 0, 1, 0, 2, 2, 2, 4, 6, 7, 8, 13, 15, 21, 25, 30, 39, 50, 58, 74, 89, 105, 129, 156, 185, 221, 264, 309, 366, 433, 505, 593, 696, 805, 941, 1090, 1258, 1458, 1684, 1933, 2225, 2555, 2922, 3346, 3823, 4349, 4961, 5644, 6402, 7267, 8234, 9309, 10525, 11886, 13393
Offset: 0
a(7) = 4 because we have 4 such partitions of 7: [1,1,2,3], [1,1,5], [2,2,3], [1,3,3].
From _Gus Wiseman_, Apr 19 2019: (Start)
The a(2) = 1 through a(11) = 13 partitions described in the name are the following (empty columns not shown). The Heinz numbers of these partitions are given by A060687.
(11) (22) (221) (33) (322) (44) (441) (55) (443)
(211) (311) (411) (331) (332) (522) (433) (533)
(511) (422) (711) (442) (551)
(3211) (611) (3321) (622) (722)
(3221) (4221) (811) (911)
(4211) (4311) (5221) (4322)
(5211) (5311) (4331)
(6211) (4421)
(5411)
(6221)
(6311)
(7211)
(43211)
The a(2) = 1 through a(10) = 8 partitions described in Emeric Deutsch's comment are the following (empty columns not shown). The Heinz numbers of these partitions are given by A325284.
(2) (22) (32) (222) (322) (332) (432) (3322)
(31) (311) (3111) (331) (431) (3222) (3331)
(421) (2222) (4221) (22222)
(31111) (3311) (4311) (42211)
(4211) (33111) (43111)
(311111) (42111) (331111)
(3111111) (421111)
(31111111)
(End)
-
g:=sum(x^(k*(k+1)/2)*((1-x^k)/x^(k-1)/(1-x)-k)/product(1-x^i,i=1..k),k=1..15): gser:=series(g,x=0,64): seq(coeff(gser,x,n),n=1..54); # Emeric Deutsch, Apr 18 2006
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n>i*(i+3-2*t)/2, 0,
`if`(n=0, t, b(n, i-1, t)+`if`(i>n, 0, b(n-i, i-1, t)+
`if`(t=1 or 2*i>n, 0, b(n-2*i, i-1, 1)))))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..100); # Alois P. Heinz, Dec 28 2015
-
b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 3 - 2*t)/2, 0, If[n == 0, t, b[n, i - 1, t] + If[i > n, 0, b[n - i, i - 1, t] + If[t == 1 || 2*i > n, 0, b[n - 2*i, i - 1, 1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 100} ] (* Jean-François Alcover, Jan 20 2016, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Length[#]-Length[Union[#]]==1&]],{n,0,30}] (* Gus Wiseman, Apr 19 2019 *)
-
alist(n)=concat([0,0],Vec(sum(k=1,n\2,(x^(2*k)+x*O(x^n))/(1+x^k)*prod(j=1,n-2*k,1+x^j+x*O(x^n))))) \\ Franklin T. Adams-Watters, Nov 02 2015
More terms from Pab Ter (pabrlos(AT)yahoo.com), May 26 2004
A096156
Numbers with ordered prime signature (2,1).
Original entry on oeis.org
12, 20, 28, 44, 45, 52, 63, 68, 76, 92, 99, 116, 117, 124, 148, 153, 164, 171, 172, 175, 188, 207, 212, 236, 244, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 356, 369, 387, 388, 404, 412, 423, 425, 428, 436, 452, 475, 477, 508, 524, 531, 539, 548, 549
Offset: 1
a(2) = 20 because 20 = 2*2*5 and 2 < 5.
Note that 18 = 2*3^2 is not in the sequence, even though it has prime signature (2,1), because its ordered prime signature is (1,2) (A095990). Prime signatures correspond to partitions of Omega(n), while ordered prime signatures correspond to compositions of Omega(n).
-
Take[ Sort[ Flatten[ Table[ Prime[p]^2 Prime[q], {q, 2, 33}, {p, q - 1}]]], 54] (* Robert G. Wilson v, Jul 28 2004 *)
Select[Range[10^5],FactorInteger[#][[All,2]]=={2,1}&] (* Enrique Pérez Herrero, Jun 27 2012 *)
-
list(lim)=my(v=List()); forprime(q=3, lim\4, forprime(p=2, min(sqrtint(lim\q), q-1), listput(v, p^2*q))); Set(v) \\ Charles R Greathouse IV, Feb 26 2014
-
from sympy import factorint
def ok(n): return list(factorint(n).values()) == [2, 1]
print([k for k in range(550) if ok(k)]) # Michael S. Branicky, Dec 20 2021
A325240
Numbers whose minimum prime exponent is 2.
Original entry on oeis.org
4, 9, 25, 36, 49, 72, 100, 108, 121, 144, 169, 196, 200, 225, 288, 289, 324, 361, 392, 400, 441, 484, 500, 529, 576, 675, 676, 784, 800, 841, 900, 961, 968, 972, 1089, 1125, 1152, 1156, 1225, 1323, 1352, 1369, 1372, 1444, 1521, 1568, 1600, 1681, 1764, 1800
Offset: 1
The sequence of terms together with their prime indices begins:
4: {1,1}
9: {2,2}
25: {3,3}
36: {1,1,2,2}
49: {4,4}
72: {1,1,1,2,2}
100: {1,1,3,3}
108: {1,1,2,2,2}
121: {5,5}
144: {1,1,1,1,2,2}
169: {6,6}
196: {1,1,4,4}
200: {1,1,1,3,3}
225: {2,2,3,3}
288: {1,1,1,1,1,2,2}
289: {7,7}
324: {1,1,2,2,2,2}
361: {8,8}
392: {1,1,1,4,4}
400: {1,1,1,1,3,3}
Maximum instead of minimum gives
A067259.
Cf.
A001221,
A001222,
A001358,
A001694,
A007774,
A036966,
A051903,
A052485,
A118914,
A244515,
A325241.
-
Select[Range[1000],Min@@FactorInteger[#][[All,2]]==2&]
-
is(n)={my(e=factor(n)[,2]); n>1 && vecmin(e) == 2; } \\ Amiram Eldar, Jan 30 2023
-
from math import isqrt, gcd
from sympy import integer_nthroot, factorint, mobius
def A325240(n):
def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
def bisection(f,kmin=0,kmax=1):
while f(kmax) > kmax: kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
def f(x):
c, l = n+x, 0
j = isqrt(x)
while j>1:
k2 = integer_nthroot(x//j**2,3)[0]+1
w = squarefreepi(k2-1)
c -= j*(w-l)
l, j = w, isqrt(x//k2**3)
c -= squarefreepi(integer_nthroot(x,3)[0])-l
for w in range(1,integer_nthroot(x,5)[0]+1):
if all(d<=1 for d in factorint(w).values()):
for y in range(1,integer_nthroot(z:=x//w**5,4)[0]+1):
if gcd(w,y)==1 and all(d<=1 for d in factorint(y).values()):
c += integer_nthroot(z//y**4,3)[0]
return c
return bisection(f,n,n**2) # Chai Wah Wu, Oct 02 2024
A325259
Numbers with one fewer distinct prime exponents than distinct prime factors.
Original entry on oeis.org
6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 120, 122, 123, 126, 129, 132, 133, 134, 140, 141, 142, 143, 145, 146, 150, 155, 156, 158, 159
Offset: 1
The sequence of terms together with their prime indices begins:
6: {1,2}
10: {1,3}
14: {1,4}
15: {2,3}
21: {2,4}
22: {1,5}
26: {1,6}
33: {2,5}
34: {1,7}
35: {3,4}
36: {1,1,2,2}
38: {1,8}
39: {2,6}
46: {1,9}
51: {2,7}
55: {3,5}
57: {2,8}
58: {1,10}
60: {1,1,2,3}
62: {1,11}
Cf.
A056239,
A060687,
A090858,
A112798,
A116608,
A118914,
A130091,
A323023,
A325241,
A325242,
A325244,
A325270,
A325281.
A325284
Numbers whose prime indices form an initial interval with a single hole: (1, 2, ..., x, x + 2, ..., m - 1, m), where x can be 0 but must be less than m - 1.
Original entry on oeis.org
3, 9, 10, 15, 20, 27, 40, 42, 45, 50, 70, 75, 80, 81, 84, 100, 105, 126, 135, 140, 160, 168, 200, 225, 243, 250, 252, 280, 294, 315, 320, 330, 336, 350, 375, 378, 400, 405, 462, 490, 500, 504, 525, 560, 588, 640, 660, 672, 675, 700, 729, 735, 756, 770, 800
Offset: 1
The sequence of terms together with their prime indices begins:
3: {2}
9: {2,2}
10: {1,3}
15: {2,3}
20: {1,1,3}
27: {2,2,2}
40: {1,1,1,3}
42: {1,2,4}
45: {2,2,3}
50: {1,3,3}
70: {1,3,4}
75: {2,3,3}
80: {1,1,1,1,3}
81: {2,2,2,2}
84: {1,1,2,4}
100: {1,1,3,3}
105: {2,3,4}
126: {1,2,2,4}
135: {2,2,2,3}
140: {1,1,3,4}
Cf.
A055932,
A056239,
A061395,
A090858,
A112798,
A124010,
A127002,
A130091,
A325241,
A325251,
A325259,
A325270.
-
Select[Range[100],Length[Complement[Range[PrimePi[FactorInteger[#][[-1,1]]]],PrimePi/@First/@FactorInteger[#]]]==1&]
A376249
Numbers that are not prime powers and have a unique largest prime exponent that is larger than the second-largest prime exponent by 1.
Original entry on oeis.org
12, 18, 20, 28, 44, 45, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 108, 116, 117, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 171, 172, 175, 188, 198, 200, 204, 207, 212, 220, 228, 234, 236, 242, 244, 245, 260, 261, 268, 275, 276, 279, 284, 292, 294
Offset: 1
-
q[k_] := Module[{e = Sort[FactorInteger[k][[;; , 2]]]}, Length[e] > 1 && e[[-1]] == e[[-2]] + 1]; Select[Range[300], q]
-
is(k) = {my(e = vecsort(factor(k)[, 2])); #e > 1 && e[#e] == e[#e-1] + 1;}
A325279
Number of integer partitions of n whose maximum multiplicity is one greater than their minimum multiplicity.
Original entry on oeis.org
0, 0, 0, 0, 1, 2, 1, 5, 6, 9, 10, 18, 18, 31, 34, 48, 57, 80, 86, 122, 138, 183, 211, 275, 311, 402, 461, 576, 663, 825, 942, 1163, 1334, 1621, 1865, 2248, 2566, 3084, 3532, 4193, 4794, 5674, 6472, 7617, 8685, 10153, 11576, 13483, 15320, 17790, 20200, 23342
Offset: 0
The a(4) = 1 through a(11) = 18 partitions:
(211) (221) (411) (322) (332) (441) (433) (443)
(311) (331) (422) (522) (442) (533)
(511) (611) (711) (622) (551)
(3211) (3221) (3321) (811) (722)
(22111) (4211) (4221) (5221) (911)
(22211) (4311) (5311) (4322)
(5211) (6211) (4331)
(32211) (33211) (4421)
(33111) (42211) (5411)
(2221111) (6221)
(6311)
(7211)
(33221)
(33311)
(43211)
(44111)
(52211)
(2222111)
-
Table[Length[Select[IntegerPartitions[n],Max@@Length/@Split[#]-Min@@Length/@Split[#]==1&]],{n,0,30}]
A376251
Numbers that have a second-largest exponent in their prime factorization and it is smaller by 1 than the largest exponent.
Original entry on oeis.org
12, 18, 20, 28, 44, 45, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 108, 116, 117, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 171, 172, 175, 180, 188, 198, 200, 204, 207, 212, 220, 228, 234, 236, 242, 244, 245, 252, 260, 261, 268, 275, 276, 279
Offset: 1
-
q[k_] := Module[{e = Union[FactorInteger[k][[;; , 2]]]}, Length[e] > 1 && e[[-2]] + 1 == e[[-1]]]; Select[Range[300], q]
-
is(k) = {my(e = Set(factor(k)[, 2])); #e > 1 && e[#e-1] + 1 == e[#e]; }
Showing 1-10 of 10 results.
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