Displaying 91-100 of 108 results found.
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Numbers whose prime indices are all powers of the same squarefree number.
+10
2
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 36, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58, 59, 62, 63, 64, 67, 68, 72, 73, 76, 79, 80, 81, 82, 83, 84, 86, 88, 92
COMMENTS
The complement is {15, 30, 33, 35, 37, 39, 45, ...}. First differs from A318991 at a(33) = 38, A318991(33) = 37. A multiset multisystem is a finite multiset of finite multisets. 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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. The dual of a multiset multisystem has, for each vertex, one block consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The sequence lists all MM-numbers of multiset multisystems whose dual is constant, i.e. of the form {x,x,x,...,x} for some multiset x.
EXAMPLE
The prime indices of 756 are {1,1,2,2,2,4}, which are all powers of 2, so 756 belongs to the sequence.
The prime indices of 841 are {10,10}, which are all powers of 10, so 841 belongs to the sequence.
The prime indices of 2645 are {3,9,9}, which are all powers of 3, so 2645 belongs to the sequence.
The prime indices of 3178 are {1,4,49}, which are all powers of squarefree numbers but not of the same squarefree number, so 3178 does not belong to the sequence.
The prime indices of 30599 are {12,144}, which are all powers of the same number 12, but this number is not squarefree, so 30599 does not belong to the sequence.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (3,2), (3,2,1), (5,2), (4,3), (6,2), (3,2,2), (7,2), (5,3), (3,2,1,1), (6,3), (5,2,1), (9,2), (4,3,1), (3,3,2), (5,4), (6,2,1), (7,3), (10,2), (3,2,2,1), (6,4), (11,2), (8,3), (5,2,2).
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
powsqfQ[n_]:=SameQ@@Last/@FactorInteger[n];
sqfker[n_]:=Times@@First/@FactorInteger[n];
Select[Range[100], And[And@@powsqfQ/@primeMS[#], SameQ@@sqfker/@DeleteCases[primeMS[#], 1]]&]
CROSSREFS
Cf. A000688, A000961, A001597, A005117, A023893, A052410, A056239, A072720, A072774, A302242, A302593, A318400, A322847, A322901, A322912.
Number of strict integer partitions of n using 1 and numbers that are not perfect powers.
+10
2
1, 1, 1, 2, 1, 2, 3, 3, 4, 4, 5, 6, 7, 9, 10, 12, 14, 16, 20, 22, 26, 31, 34, 40, 46, 51, 59, 66, 75, 86, 96, 110, 123, 139, 157, 176, 199, 221, 248, 278, 309, 346, 385, 427, 476, 528, 586, 650, 719, 795, 880, 973, 1074, 1186, 1307, 1439, 1584, 1744, 1915, 2104
FORMULA
O.g.f.: (1 + x) * Product_{n in A007916} (1 + x^n).
EXAMPLE
A list of all strict integer partitions using 1 and numbers that are not perfect powers begins:
1: (1) 8: (5,2,1) 12: (12) 14: (14)
2: (2) 9: (7,2) 12: (11,1) 14: (13,1)
3: (3) 9: (6,3) 12: (10,2) 14: (12,2)
3: (2,1) 9: (6,2,1) 12: (7,5) 14: (11,3)
4: (3,1) 9: (5,3,1) 12: (7,3,2) 14: (11,2,1)
5: (5) 10: (10) 12: (6,5,1) 14: (10,3,1)
5: (3,2) 10: (7,3) 12: (6,3,2,1) 14: (7,6,1)
6: (6) 10: (7,2,1) 13: (13) 14: (7,5,2)
6: (5,1) 10: (6,3,1) 13: (12,1) 14: (6,5,3)
6: (3,2,1) 10: (5,3,2) 13: (11,2) 14: (6,5,2,1)
7: (7) 11: (11) 13: (10,3) 15: (15)
7: (6,1) 11: (10,1) 13: (10,2,1) 15: (14,1)
7: (5,2) 11: (7,3,1) 13: (7,6) 15: (13,2)
8: (7,1) 11: (6,5) 13: (7,5,1) 15: (12,3)
8: (6,2) 11: (6,3,2) 13: (7,3,2,1) 15: (12,2,1)
8: (5,3) 11: (5,3,2,1) 13: (6,5,2) 15: (11,3,1)
MATHEMATICA
perpowQ[n_]:=GCD@@FactorInteger[n][[All, 2]]>1;
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&And@@Not/@perpowQ/@#&]], {n, 65}]
Triangle read by rows: T(n,k) = 0 if all positive integers can be colored with two colors without any positive integer x being the same color as n*x or k*x; otherwise, T(n,k) = 1 (for 2 <= k <= n).
+10
2
0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
COMMENTS
T(n,k) = 1 if and only if there exists at least one pair of positive integers (x, y) such that n^x = k^y and x+y is odd. Otherwise, T(n,k) = 0.
If n is an element of A007916, then T(n,k) = 1 if and only if k is a perfect power of n^2.
FORMULA
T(d^(2x), d^(2y-1)) = 1 for all positive integers d > 1, x, y.
T( A000302(n), A004171(k)) = T( A001019(n), A013708(k)) = T( A001025(n), A013709(k)) = T( A009969(n), A013710(k)) = T( A009980(n), A013711(k)) = T( A087752(n), A013712(k)) = T( A089357(n), A013713(k)) = T( A089683(n), A013714(k)) = T( A098608(n), A013715(k)) = 1 for all n >= 1, k >= 0.
EXAMPLE
Triangle T(n,k) begins:
n\k 2 3 4 5 6 7 8 9 10 11 ...
2 0
3 0 0
4 1 0 0
5 0 0 0 0
6 0 0 0 0 0
7 0 0 0 0 0 0
8 0 0 1 0 0 0 0
9 0 1 0 0 0 0 0 0
10 0 0 0 0 0 0 0 0 0
11 0 0 0 0 0 0 0 0 0 0
...
If we color all positive integers whose 2-adic order and 3-adic order add up to an even number in color A and the rest in color B, every positive integer will be a different color from its double and triple. Therefore, T(3, 2) = 0.
PROG
(Python)
def T(n, k):
parity_check = [False]
i = 0
while True:
while not n % k:
n /= k
parity_check[i] = not parity_check[i]
if k % n:
return 0
elif n == 1:
x, y = True, not parity_check[0]
for j in range(1, i + 1):
x, y = y, x ^ (y and parity_check[j])
return y + 0
else:
n, k = k, n
parity_check.append(False)
i += 1
print([T(n, k) for n in range(2, 14) for k in range(2, n + 1)])
(Python)
def T(n, k):
nk = n*k
is_odd = 0
while True:
while not n % k:
n /= k
if k % n:
return 0
elif n == 1:
while not nk % k:
nk /= k
is_odd = 0 if is_odd else 1
return is_odd
else:
n, k = k, n
print([T(n, k) for n in range(2, 14) for k in range(2, n + 1)])
CROSSREFS
Cf. A000302, A001019, A001025, A004171, A007814, A007916, A009969, A009980, A013708- A013715, A052409, A052410, A087752, A089357, A089683, A098608, A346460, A346461.
Reduced root factorial of n: product of the smallest integer root of numbers from 1 to n.
+10
1
1, 1, 2, 6, 12, 60, 360, 2520, 5040, 15120, 151200, 1663200, 19958400, 259459200, 3632428800, 54486432000, 108972864000, 1852538688000, 33345696384000, 633568231296000, 12671364625920000, 266098657144320000, 5854170457175040000, 134645920515025920000
COMMENTS
A "binomial" style a(m+n)/(a(m)*a(n)) is not always an integer, as for instance at m = n = 18 (unlike ordinary factorials or A048803). - Hal M. Switkay, Jul 22 2024
EXAMPLE
a(8) = 1*2*3*2*5*6*7*2 = 5040.
MAPLE
b:= proc(n) option remember; (l-> (t-> mul(i[1]^(i[2]/t),
i=l))(igcd(seq(i[2], i=l))))(ifactors(n)[2])
end:
a:= proc(n) option remember; `if`(n<1, 1, a(n-1)*b(n)) end:
a(1) = 1; for n > 1 a(n) = smallest number of the form n^r (with r rational != 1) not included earlier.
+10
1
1, 4, 9, 2, 25, 36, 49, 16, 3, 100, 121, 144, 169, 196, 225, 8, 289, 324, 361, 400, 441, 484, 529, 576, 5, 676, 81, 784, 841, 900, 961, 64, 1089, 1156, 1225, 6, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 7, 2500, 2601, 2704
FORMULA
a(n) = n^((b - (-1)^b) / b), b = gcd(b_1, ..., b_r) with prime factorization n = p_1^b_1*...*p_r^b_r. - Sascha Kurz, Aug 14 2002
EXAMPLE
a(15) = 15^2 = 225, but a(16) = 8 = 16^(3/4).
MAPLE
for n from 2 to 150 do a := ifactors(n); b := a[2][1][2]:for j from 2 to nops(a[2]) do b := gcd(b, a[2][j][2]); od; bb := floor(evalf(n^(1/b))); if(b mod 2=1) then c[n] := bb^(b+1) else c[n] := bb^(b-1); fi; od:c[1]=1:seq(c[j], j=1..150);
If b(n) is the smallest positive integer and c(n) is the largest positive integer such that n = b(n)^c(n), then a(n) = b(n)^c(n+1).
+10
1
1, 2, 9, 2, 5, 6, 343, 4, 3, 10, 11, 12, 13, 14, 50625, 2, 17, 18, 19, 20, 21, 22, 23, 576, 5, 17576, 3, 28, 29, 30, 28629151, 2, 33, 34, 1225, 6, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 2304, 7, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 62523502209, 2, 65, 66
EXAMPLE
b(7) = 7, since 7 = 7^1. And c(8) = 3, since 8 = 2^3.
So a(7) = 7^3 = 343.
Largest integer power m (with 0 to stand for infinity) for which a representation of the form n = k^m exists (for some k >= 1) multiplied by -1 when k is prime.
+10
1
0, -1, -1, -2, -1, 1, -1, -3, -2, 1, -1, 1, -1, 1, 1, -4, -1, 1, -1, 1, 1, 1, -1, 1, -2, 1, -3, 1, -1, 1, -1, -5, 1, 1, 1, 2, -1, 1, 1, 1, -1, 1, -1, 1, 1, 1, -1, 1, -2, 1, 1, 1, -1, 1, 1, 1, 1, 1, -1, 1, -1, 1, 1, -6, 1, 1, -1, 1, 1, 1, -1, 1, -1, 1, 1, 1, 1, 1, -1, 1, -4, 1, -1, 1, 1, 1, 1, 1
COMMENTS
This sequence reveals, among the positive integers, which are the unit, the primes, the perfect powers (with |a(n)| as largest exponent) telling whether these are perfect powers of either primes or composites and finally which are non-perfect powers composites per the following:
a(n) < -1: perfect powers of primes (largest exponent = |a(n)|)
a(n) = -1: primes (not perfect powers)
a(n) = 0: (standing for infinity): unit, perfect power of unit
a(n) = +1: composites (not perfect powers)
a(n) > +1: perfect powers of composites (largest exponent = |a(n)|).
FORMULA
a(n) = m * (-1)^{pi(k) - pi(k-1)} where m is the largest exponent of k^m = n for some k >= 1 and pi(k) is the prime counting function evaluated at k.
a(n) = A052409(n) * (-1)^{Pi(k(n)) - Pi(k(n)-1)}, with k(n) = A052410(n).
CROSSREFS
Cf. A052409 a(n) = largest integer power m for which a representation of the form n = k^m exists (for some k). Cf. A052410 Value of a in a^p=n, where p is the largest power given by A052409. Cf. A000961 Prime powers p^k (p prime, k >= 0). Cf. A001597 Perfect powers: m^k where m is an integer and k >= 2.
Triangle read by rows: T(m,n) = smallest odd prime p such that (m^p-n^p)/(m-n) is prime (0<n<m), or 0 if no such p exists.
+10
1
3, 3, 3, 0, 0, 3, 3, 5, 13, 3, 3, 0, 0, 0, 5, 5, 3, 3, 5, 3, 3, 3, 0, 3, 0, 19, 0, 7, 0, 3, 0, 0, 3, 0, 3, 7, 19, 0, 3, 0, 0, 0, 31, 0, 3, 17, 5, 3, 3, 5, 3, 5, 7, 5, 3, 3, 0, 0, 0, 3, 0, 3, 0, 0, 0, 3, 5, 3, 7, 5, 5, 3, 7, 3, 3, 251, 3, 17, 3, 0, 5, 0, 151, 0, 0, 0, 59, 0, 5, 0, 3, 3, 5, 0, 1097, 0, 0, 3, 3, 0, 0, 7, 0, 17, 3
COMMENTS
T(m,n) is 0 if and only if m and n are not coprime or A052409(m) and A052409(n) are not coprime. (The latter has some exceptions, like T(8,1) = 3. In fact, if p is a prime and does not equal to A052410(gcd( A052409(m), A052409(n))), then (m^p-n^p)/(m-n) is composite, so if it is not 0, then it is A052410(gcd( A052409(m), A052409(n))).) - Eric Chen, Nov 26 2014 a(i) = T(m,n) corresponds only to probable primes for (m,n) = {(15,4), (18,1), (19,18), (31,6), (37,22), (37,25), ...} (i={95, 137, 171, 441, 652, 655, ...}). With the exception of these six (m,n), all corresponding primes up to a(663) are definite primes. - Eric Chen, Nov 26 2014 a(n) is currently known up to n = 663, a(664) = T(37, 34) > 10000. - Eric Chen, Jun 01 2015 For n up to 1000, a(n) is currently unknown only for n = 664, 760, and 868. - Eric Chen, Jun 01 2015
EXAMPLE
Read by rows:
m\n 1 2 3 4 5 6 7 8 9 10 11
2 3
3 3 3
4 0 0 3
5 3 5 13 3
6 3 0 0 0 5
7 5 3 3 5 3 3
8 3 0 3 0 19 0 7
9 0 3 0 0 3 0 3 7
10 19 0 3 0 0 0 31 0 3
11 17 5 3 3 5 3 5 7 5 3
12 3 0 0 0 3 0 3 0 0 0 3
etc.
MATHEMATICA
t1[n_] := Floor[3/2 + Sqrt[2*n]]
m[n_] := Floor[(-1 + Sqrt[8*n-7])/2]
t2[n_] := n-m[n]*(m[n]+1)/2
b[n_] := GCD @@ Last /@ FactorInteger[n]
is[m_, n_] := GCD[m, n] == 1 && GCD[b[m], b[n]] == 1
Do[k=2, If[is[t1[n], t2[n]], While[ !PrimeQ[t1[n]^Prime[k] - t2[n]^Prime[k]], k++]; Print[Prime[k]], Print[0]], {n, 1, 663}] (* Eric Chen, Jun 01 2015 *)
PROG
(PARI) a052409(n) = my(k=ispower(n)); if(k, k, n>1);
a(m, n) = {if (gcd(m, n) != 1, return (0)); if (gcd(a052409(m), a052409(n)) != 1, return (0)); forprime(p=3, , if (isprime((m^p-n^p)/(m-n)), return (p)); ); }
tabl(nn) = {for (m=2, nn, for(n=1, m-1, print1(a(m, n), ", "); ); print(); ); } \\ Michel Marcus, Nov 19 2014 (PARI) t1(n)=floor(3/2+sqrt(2*n))
t2(n)=n-binomial(floor(1/2+sqrt(2*n)), 2)
b(n)=my(k=ispower(n)); if(k, k, n>1)
a(n)=if(gcd(t1(n), t2(n)) !=1 || gcd(b(t1(n)), b(t2(n))) !=1, 0, forprime(p=3, 2^24, if(ispseudoprime((t1(n)^p-t2(n)^p)/(t1(n)-t2(n))), return(p)))) \\ Eric Chen, Jun 01 2015
CROSSREFS
Cf. A000043 (2,1), A028491 (3,1), A057468 (3,2), A059801 (4,3), A004061 (5,1), A082182 (5,2), A121877 (5,3), A059802 (5,4), A004062 (6,1), A062572 (6,5), A004063 (7,1), A215487 (7,2), A128024 (7,3), A213073 (7,4), A128344 (7,5), A062573 (7,6), A128025 (8,3), A128345 (8,5), A062574 (8,7), A173718 (9,2), A128346 (9,5), A059803 (9,8), A004023 (10,1), A128026 (10,3), A062576 (10,9), A005808 (11,1), A210506 (11,2), A128027 (11,3), A216181 (11,4), A128347 (11,5), A062577 (11,10), A004064 (12,1), A128348 (12,5), A062578 (12,11).
Powers of squarefree numbers whose prime indices are relatively prime. Heinz numbers of uniform partitions with relatively prime parts.
+10
1
2, 4, 6, 8, 10, 14, 15, 16, 22, 26, 30, 32, 33, 34, 35, 36, 38, 42, 46, 51, 55, 58, 62, 64, 66, 69, 70, 74, 77, 78, 82, 85, 86, 93, 94, 95, 100, 102, 105, 106, 110, 114, 118, 119, 122, 123, 128, 130, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158, 161, 165
COMMENTS
A prime index of n is a number m such that prime(m) divides n. An integer partition is uniform if all parts appear with the same multiplicity. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The number of uniform partitions of n with relatively prime parts is A078374(n).
EXAMPLE
Sequence of all uniform relatively prime integer partitions begins (1), (11), (21), (111), (31), (41), (32), (1111), (51), (61), (321), (11111), (52), (71), (43), (2211).
MATHEMATICA
Select[Range[200], And[GCD@@PrimePi/@FactorInteger[#][[All, 1]]===1, SameQ@@FactorInteger[#][[All, 2]]]&]
CROSSREFS
Cf. A000009, A000837, A007916, A047966, A052409, A052410, A072774, A078374, A289023, A289509, A300486, A302491, A302796.
Triangle read by rows where T(n,k) is the number of integer partitions of n > 0 into powers of k > 0.
+10
1
1, 1, 2, 1, 2, 2, 1, 4, 2, 2, 1, 4, 2, 2, 2, 1, 6, 3, 2, 2, 2, 1, 6, 3, 2, 2, 2, 2, 1, 10, 3, 3, 2, 2, 2, 2, 1, 10, 5, 3, 2, 2, 2, 2, 2, 1, 14, 5, 3, 3, 2, 2, 2, 2, 2, 1, 14, 5, 3, 3, 2, 2, 2, 2, 2, 2, 1, 20, 7, 4, 3, 3, 2, 2, 2, 2, 2, 2, 1, 20, 7, 4, 3, 3, 2
EXAMPLE
Triangle begins:
1
1 2
1 2 2
1 4 2 2
1 4 2 2 2
1 6 3 2 2 2
1 6 3 2 2 2 2
1 10 3 3 2 2 2 2
1 10 5 3 2 2 2 2 2
1 14 5 3 3 2 2 2 2 2
1 14 5 3 3 2 2 2 2 2 2
1 20 7 4 3 3 2 2 2 2 2 2
1 20 7 4 3 3 2 2 2 2 2 2 2
Row n = 6 counts the following partitions:
(111111) (42) (33) (411) (51) (6)
(222) (3111) (111111) (111111) (111111)
(411) (111111)
(2211)
(21111)
(111111)
MATHEMATICA
Table[If[k==1, 1, Length[Select[IntegerPartitions[n], And@@(IntegerQ[Log[k, #]]&/@#)&]]], {n, 10}, {k, n}]
CROSSREFS
Same as A102430 except for the k = 1 column.
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