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Numbers which are the product of a prime and the square of a different prime (p^2 * q).
+10
83
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
COMMENTS
Numbers with prime signature (2,1) = union of numbers with ordered prime signature (1,2) and numbers with ordered prime signature (2,1) (restating second part of above comment). - Daniel Forgues, Feb 05 2011 For k>1, Sum_{n>=1} 1/a(n)^k = P(k) * P(2*k) - P(3*k), where P is the prime zeta function. - Enrique Pérez Herrero, Jun 27 2012 These numbers are called "Nombres d'Einstein" on the French site "Diophante" (see link) because a(n) = m * c^2 where m and c are two different primes.
The numbers 44 = 2^2 * 11 and 45 = 3^2 * 5 are the two smallest consecutive "Einstein numbers"; 603, 604, 605 are the three smallest consecutive integers in this sequence. It's not possible to get more than five such consecutive numbers (proof in the link); the first set of five such consecutive numbers begins at the 17-digit number 10093613546512321. Where does the first sequence of four consecutive "Einstein numbers" begin? (End) [corrected by Jon E. Schoenfield, Sep 20 2017] The first set of four consecutive integers in this sequence begins at the 11-digit number 17042641441. (Each such set must include two even numbers, one of which is of the form 2^2*q, the other of the form p^2*2; a quick search, taking the factorizations of consecutive integers before and after numbers of the latter form, shows that the number of sets of four consecutive k-digit integers in this sequence is 1, 7, 12, 18 for k = 11, 12, 13, 14, respectively.) - Jon E. Schoenfield, Sep 16 2017 The first 13 sets of 5 consecutive integers in this sequence have as their first terms 10093613546512321, 14414905793929921, 266667848769941521, 562672865058083521, 1579571757660876721, 1841337567664174321, 2737837351207392721, 4456162869973433521, 4683238426747860721, 4993613853242910721, 5037980611623036721, 5174116847290255921, 5344962129269790721. Each of these numbers except for the last is 7^2 times a prime; the last is 23^2 times a prime. - Jon E. Schoenfield, Sep 17 2017
EXAMPLE
a(1) = 12 because 12 = 2^2*3 is the smallest number of the form p^2*q.
MATHEMATICA
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 *)
PROG
(PARI) for(n=1, 1e3, if(numdiv(n) - bigomega(n) == 3, print1(n, ", "))) \\ Altug Alkan, Nov 24 2015 (Python)
from sympy import factorint
def ok(n): return sorted(factorint(n).values()) == [1, 2]
CROSSREFS
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 | -------------------------------------------------------------------------------
EXTENSIONS
Link added and incorrect Mathematica code removed by David Bevan, Sep 17 2011
Number of partitions of n containing a clique of size 1.
+10
29
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
COMMENTS
All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.
FORMULA
G.f.: (1-Product_{j>0} (1-x^(j)+x^(2*j))) / (Product_{j>0} (1-x^j)).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n). (End)
EXAMPLE
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].
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)
MAPLE
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);
MATHEMATICA
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 *)
CROSSREFS
Cf. A000041, A007690, A183559, A183560, A183561, A183562, A183563, A183564, A183565, A183566, A183567.
Number of partitions of n such that there is exactly one part which occurs twice, while all other parts occur only once.
+10
24
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
COMMENTS
Number of solutions (p(1),p(2),...,p(n)), p(i)>=0,i=1..n, to p(1)+2*p(2)+...+n*p(n)=n such that |{i: p(i)<>0}| = p(1)+p(2)+...+p(n)-1.
Also number of partitions of n such that if k is the largest part, then, with exactly one exception, all the integers 1,2,...,k occur as parts. Example: a(7)=4 because we have [4,2,1], [3,3,1], [3,2,2] and [3,1,1,1,1]. - Emeric Deutsch, Apr 18 2006
FORMULA
G.f.: sum(x^(k(k+1)/2)[(1-x^k)/x^(k-1)/(1-x)-k]/product(1-x^i,i=1..k), k=1..infinity). - Emeric Deutsch, Apr 18 2006 a(n) ~ c * exp(Pi*sqrt(n/3)) / n^(1/4), where c = 3^(1/4) * (1 - log(2)) / (2*Pi) = 0.064273294789... - Vaclav Kotesovec, May 24 2018
EXAMPLE
a(7) = 4 because we have 4 such partitions of 7: [1,1,2,3], [1,1,5], [2,2,3], [1,3,3].
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)
MAPLE
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):
MATHEMATICA
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 *)
PROG
(PARI) 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
EXTENSIONS
More terms from Pab Ter (pabrlos(AT)yahoo.com), May 26 2004
Numbers with ordered prime signature (2,1).
+10
16
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
COMMENTS
Numbers of the form p^2 * q where p and q are primes with p < q.
There are pairs that differ by 1, which is not the case in A095990, beginning with 44 and 45, 116 and 117, 171 and 172, 332 and 333, etc.
EXAMPLE
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).
MATHEMATICA
Take[ Sort[ Flatten[ Table[ Prime[p]^2 Prime[q], {q, 2, 33}, {p, q - 1}]]], 54] (* Robert G. Wilson v, Jul 28 2004 *)
PROG
(PARI) 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 (Python)
from sympy import factorint
def ok(n): return list(factorint(n).values()) == [2, 1]
Numbers whose minimum prime exponent is 2.
+10
6
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
COMMENTS
Or barely powerful numbers, a subset of powerful numbers A001694. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose minimum multiplicity is 2 (counted by A244515).
FORMULA
Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Product_{p prime} (1 + 1/(p^2*(p-1))) = A082695 - A065483 = 0.6038122832... . - Amiram Eldar, Jan 30 2023
EXAMPLE
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}
MATHEMATICA
Select[Range[1000], Min@@FactorInteger[#][[All, 2]]==2&]
PROG
(PARI) is(n)={my(e=factor(n)[, 2]); n>1 && vecmin(e) == 2; } \\ Amiram Eldar, Jan 30 2023 (Python)
from math import isqrt, gcd
from sympy import integer_nthroot, factorint, mobius
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
CROSSREFS
Maximum instead of minimum gives A067259. Cf. A001221, A001222, A001358, A001694, A007774, A036966, A051903, A052485, A118914, A244515, A325241.
Numbers with one fewer distinct prime exponents than distinct prime factors.
+10
6
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
COMMENTS
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions with one fewer distinct multiplicities than distinct parts. The enumeration of these partitions by sum is given by A325244.
EXAMPLE
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}
MATHEMATICA
Select[Range[100], PrimeNu[#]==Length[Union[Last/@FactorInteger[#]]]+1&]
CROSSREFS
Cf. A056239, A060687, A090858, A112798, A116608, A118914, A130091, A323023, A325241, A325242, A325244, A325270, A325281.
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.
+10
4
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
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose distinct parts form an initial interval with a single hole. The enumeration of these partitions by sum is given by A090858.
EXAMPLE
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}
MATHEMATICA
Select[Range[100], Length[Complement[Range[PrimePi[FactorInteger[#][[-1, 1]]]], PrimePi/@First/@FactorInteger[#]]]==1&]
CROSSREFS
Cf. A055932, A056239, A061395, A090858, A112798, A124010, A127002, A130091, A325241, A325251, A325259, A325270.
Numbers that are not prime powers and have a unique largest prime exponent that is larger than the second-largest prime exponent by 1.
+10
2
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
COMMENTS
First differs from A325241 at n = 36: A325241(36) = 2^2 * 3^2 * 5 is not a term of this sequence. Also, a(71) = 360 = 2^3 * 3^2 * 5 is the least term that is not a term of A325241. Numbers whose unordered prime signature (i.e., sorted, see A118914) ends with two consecutive integers: {..., k, k+1} for some k >= 1. The asymptotic density of this sequence is Sum_{k >= 1, p prime} (d(k+1, p) - d(k, p))/p^(k+1) = 0.21831645263800520483..., where d(k, p) = 0 for k = 1, and (1-1/p)/((1-1/p^k)*zeta(k)) for k > 1, is the density of terms that have in their prime factorization a prime p with the largest exponent that is > k.
MATHEMATICA
q[k_] := Module[{e = Sort[FactorInteger[k][[;; , 2]]]}, Length[e] > 1 && e[[-1]] == e[[-2]] + 1]; Select[Range[300], q]
PROG
(PARI) is(k) = {my(e = vecsort(factor(k)[, 2])); #e > 1 && e[#e] == e[#e-1] + 1; }
Number of integer partitions of n whose maximum multiplicity is one greater than their minimum multiplicity.
+10
1
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
COMMENTS
The Heinz numbers of these partitions are given by A325241. For example, the partition (44111) has two multiplicities (2 and 3) which differ by 1, so is counted under a(11).
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Max@@Length/@Split[#]-Min@@Length/@Split[#]==1&]], {n, 0, 30}]
Numbers that have a second-largest exponent in their prime factorization and it is smaller by 1 than the largest exponent.
+10
1
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
COMMENTS
First differs from its subsequence A325241 at n = 74: a(74) = 360 = 2^3 * 3^2 * 5 is not a term of A325241. The asymptotic density of this sequence is Sum_{k>=2} d(k) = 0.24179287499021146826..., where d(2) = 1/zeta(3) - 1/zeta(2), and d(k) = 1/zeta(k+1) - 1/zeta(k) + 1/zeta(k-1) - Product_{p prime} (1 - 1/p^(k-1) + 1/p^k - 1/p^(k+1)) for k >= 3.
MATHEMATICA
q[k_] := Module[{e = Union[FactorInteger[k][[;; , 2]]]}, Length[e] > 1 && e[[-2]] + 1 == e[[-1]]]; Select[Range[300], q]
PROG
(PARI) is(k) = {my(e = Set(factor(k)[, 2])); #e > 1 && e[#e-1] + 1 == e[#e]; }
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