A052410 -id:A052410 - cp's OEIS frontend

A317994 Number of inequivalent leaf-colorings of the free pure symmetric multifunction with e-number n.

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 4, 2, 2, 2, 1, 4, 2, 2, 2, 2, 1, 2, 4, 2, 2, 2, 2, 2, 1, 2, 5, 4, 2, 2, 2, 2, 2, 1, 2, 5, 4, 2, 2, 2, 2, 2, 2, 1, 2, 5, 4, 2, 2, 2, 2, 2, 2, 1, 5, 2, 5, 4, 2, 2, 2, 2, 2, 2, 1, 5, 2, 5, 4, 2, 2, 4, 2, 2, 2, 2, 1, 5

Offset: 1

Gus Wiseman, Aug 18 2018

nonn

If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique free pure symmetric multifunction (with empty expressions allowed) e(n) with one atom by expressing n as a power of a number that is not a perfect power to a product of prime numbers: n = rad(x)^(prime(y_1) * ... * prime(y_k)) where rad = A007916. Then e(n) = e(x)[e(y_1), ..., e(y_k)]. For example, e(21025) = o[o[o]][o] because 21025 = rad(rad(1)^prime(rad(1)^prime(1)))^prime(1).

			Inequivalent representatives of the a(441) = 11 colorings of the expression e(441) = o[o,o][o] are the following.
  1[1,1][1]
  1[1,1][2]
  1[1,2][1]
  1[1,2][2]
  1[1,2][3]
  1[2,2][1]
  1[2,2][2]
  1[2,2][3]
  1[2,3][1]
  1[2,3][2]
  1[2,3][4]

Cf. A007916, A052409, A052410, A277576, A277996, A300626, A316112, A317056, A317658, A317765.

A377467 Number of perfect-powers x in the range 2^n < x < 2^(n+1).

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 4, 6, 7, 10, 15, 23, 31, 41, 60, 81, 117, 165, 230, 321, 452, 634, 891, 1252, 1766, 2486, 3504, 4935, 6958, 9815, 13849, 19537, 27577, 38932, 54971, 77640, 109667, 154921, 218878, 309276, 437046, 617657, 872967, 1233895, 1744152, 2465546, 3485477

Offset: 0

Gus Wiseman, Nov 04 2024

nonn

Perfect-powers (A001597) are numbers with a proper integer root, complement A007916.

Also the number of perfect-powers, except for powers of 2, with n bits.

			The perfect-powers in each prescribed range (rows):
    .
    .
    .
    9
   25   27
   36   49
   81  100  121  125
  144  169  196  216  225  243
  289  324  343  361  400  441  484
  529  576  625  676  729  784  841  900  961 1000
The binary expansions for n >= 3 (columns):
    1001  11001  100100  1010001  10010000  100100001
          11011  110001  1100100  10101001  101000100
                         1111001  11000100  101010111
                         1111101  11011000  101101001
                                  11100001  110010000
                                  11110011  110111001
                                            111100100

The version for squarefree numbers is A077643.
The version for prime-powers is A244508.
For primes instead of powers of 2 we have A377432, zeros A377436.
Including powers of 2 in the range gives A377435.
The version for non-perfect-powers is A377701.
The union of all numbers counted is A377702.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect-powers, differences A053289.
A007916 lists the non-perfect-powers, differences A375706.
A081676 gives the greatest perfect-power <= n.
A131605 lists perfect-powers that are not prime-powers.
A377468 gives the least perfect-power > n.
Cf. A000015, A013597, A014210, A014234, A023055, A031218, A045542, A052410, A065514, A069623, A216765, A246655, A345531.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[Length[Select[Range[2^n+1,2^(n+1)-1],perpowQ]],{n,0,15}]

from sympy import mobius, integer_nthroot
def A377467(n):
    def f(x): return int(1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    return f((1<Chai Wah Wu, Nov 05 2024

For n != 1, a(n) = A377435(n) - 1.

a(26)-a(46) from Chai Wah Wu, Nov 05 2024

A378356 Prime index of the next prime after the n-th perfect power.

Original entry on oeis.org

1, 3, 5, 5, 7, 10, 10, 12, 12, 16, 19, 23, 26, 31, 31, 32, 35, 40, 45, 48, 49, 54, 55, 62, 67, 69, 73, 79, 86, 93, 98, 100, 106, 115, 123, 130, 138, 147, 155, 163, 169, 173, 182, 192, 201, 211, 218, 220, 229, 241, 252, 264, 270, 275, 284, 296, 307, 310, 320

Offset: 1

Gus Wiseman, Dec 05 2024

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

First differences are A080769.
Union is A378365.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect powers, differences A053289.
A007916 lists the non perfect powers, differences A375706.
A069623 counts perfect powers <= n.
A076411 counts perfect powers < n.
A377432 counts perfect powers between primes, see A377434, A377436, A377466.
A378249 gives the least perfect power > prime(n), restriction of A377468.
Cf. A045542, A052410, A065514, A068361, A076412, A081676, A216765, A345531, A377283, A378035, A378250, A378251, A378253.

Table[PrimePi[NextPrime[n]],{n,Select[Range[1000],perpowQ]}]

a(n) = A000720(A001597(n)) + 1.

A378357 Distance from n to the least non perfect power >= n.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 24 2024

nonn

Perfect powers (A001597) are 1 and numbers with a proper integer root, complement A007916.

All terms are <= 2 because the only adjacent perfect powers are 8 and 9.

The version for prime numbers is A007920, subtraction of A159477 or A007918.
The version for perfect powers is A074984, subtraction of A377468.
The version for squarefree numbers is A081221, subtraction of A067535.
Subtracting from n gives A378358, opposite A378363.
The opposite version is A378364.
The version for nonsquarefree numbers is A378369, subtraction of A120327.
The version for prime powers is A378370, subtraction of A000015.
The version for non prime powers is A378371, subtraction of A378372.
The version for composite numbers is A378456, subtraction of A113646.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect powers, differences A053289, seconds A376559.
A007916 lists the non perfect powers, differences A375706, seconds A376562.
A069623 counts perfect powers <= n.
A076411 counts perfect powers < n.
A377432 counts perfect powers between primes, zeros A377436.
Cf. A013597, A013603, A013632, A031218, A045542, A052410, A081676, A131605, A188951, A216765, A375702, A375736.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[NestWhile[#+1&,n,#>1&&perpowQ[#]&]-n,{n,100}]

from sympy import perfect_power
def A378357(n): return 0 if n>1 and perfect_power(n)==False else 1 if perfect_power(n+1)==False else 2 # Chai Wah Wu, Nov 27 2024

a(n) = n - A378358(n).

A072720 Number of partitions of n into parts which are each powers of a single number (which may vary between partitions).

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 11, 15, 17, 23, 24, 34, 35, 43, 47, 57, 58, 73, 74, 91, 96, 112, 113, 139, 141, 163, 168, 197, 198, 235, 236, 272, 279, 317, 321, 378, 379, 427, 436, 501, 502, 575, 576, 653, 666, 742, 743, 851, 853, 952, 963, 1080, 1081, 1211, 1216, 1361

Offset: 0

Henry Bottomley, Jul 05 2002

nonn

First differs from A322912 at a(12) = 34, A322912(12) = 33.

			a(6)=10 since 6 can be written as 6 (powers of 6), 5+1 (5), 4+1+1 (4 or 2), 3+3 (3), 3+1+1+1 (3), 4+2 (2), 2+2+2 (2), 2+2+1+1 (2), 2+1+1+1+1 (2) and 1+1+1+1+1+1 (powers of anything).
From _Gus Wiseman_, Jan 01 2019: (Start)
The a(1) = 1 through a(8) = 15 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (41)     (33)      (61)       (44)
             (111)  (31)    (221)    (42)      (331)      (71)
                    (211)   (311)    (51)      (421)      (422)
                    (1111)  (2111)   (222)     (511)      (611)
                            (11111)  (411)     (2221)     (2222)
                                     (2211)    (4111)     (3311)
                                     (3111)    (22111)    (4211)
                                     (21111)   (31111)    (5111)
                                     (111111)  (211111)   (22211)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
(End)

Cf. A000123, A005704, A005705, A005706, A072721.

Cf. A018819, A023893, A052410, A102430, A322900, A322901, A322902, A322903, A322912.

radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
Table[Length[Select[IntegerPartitions[n],SameQ@@radbase/@DeleteCases[#,1]&]],{n,30}] (* Gus Wiseman, Jan 01 2019 *)

a(n) = a(n-1) + A072721(n). a(p) = a(p-1)+1 for p prime.

A303709 Number of periodic factorizations of n using elements of A007916 (numbers that are not perfect powers).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Gus Wiseman, Apr 29 2018

nonn

A periodic factorization of n is a finite multiset of positive integers greater than 1 whose product is n and whose multiplicities have a common divisor greater than 1. Note that a factorization of a number that is not a perfect power (A007916) is always aperiodic (A303386), so the indices of nonzero entries of this sequence all lie at perfect powers (A001597).

			The a(900) = 5 periodic factorizations are (2*2*3*3*5*5), (2*2*15*15), (3*3*10*10), (5*5*6*6), (30*30).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A000740, A000837, A001055, A001597, A007716, A007916, A052409, A052410, A281116, A303547, A303552, A303553, A303386, A303707, A303708, A303710.

radQ[n_]:=Or[n===1,GCD@@FactorInteger[n][[All,2]]===1];
facsr[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsr[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],radQ]}]];
Table[Length[Select[facsr[n],GCD@@Length/@Split[#]!=1&]],{n,200}]

gcd_of_multiplicities(lista) = { my(u=length(lista)); if(u<2, u, my(g=0, pe = lista[1], j=1); for(i=2,u,if(lista[i]==pe, j++, g = gcd(j,g); j=1; pe = lista[i])); gcd(g,j)); }; \\ the supplied lista (newfacs) should be monotonic
A303709(n, m=n, facs=List([])) = if(1==n, (1!=gcd_of_multiplicities(facs)), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m)&&!ispower(d), newfacs = List(facs); listput(newfacs,d); s += A303709(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Dec 06 2018

a(n) <= A303553(n) <= A001055(n). - Antti Karttunen, Dec 06 2018

Changed a(1) to 1 by Gus Wiseman, Dec 06 2018

A317765 Number of distinct subexpressions of the free pure symmetric multifunction (with empty expressions allowed) with e-number n.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 5, 3, 3, 4, 6, 4, 4, 5, 7, 2, 5, 5, 6, 8, 3, 6, 6, 7, 4, 9, 3, 4, 7, 7, 8, 4, 5, 10, 4, 3, 5, 8, 8, 9, 5, 6, 11, 5, 4, 6, 9, 9, 5, 10, 6, 7, 12, 6, 5, 7, 10, 10, 6, 11, 7, 8, 13, 3, 7, 6, 8, 11, 11, 7, 12, 8, 9, 14, 4, 8, 7, 9, 12, 12, 3, 8

Offset: 1

Gus Wiseman, Aug 18 2018

nonn

If n = 1 let e(n) be the leaf symbol "o". Given a positive integer n > 1 we construct a unique free pure symmetric multifunction (with empty expressions allowed) e(n) with one atom by expressing n as a power of a number that is not a perfect power to a product of prime numbers: n = rad(x)^(prime(y_1) * ... * prime(y_k)) where rad = A007916. Then e(n) = e(x)[e(y_1), ..., e(y_k)]. For example, e(21025) = o[o[o]][o] because 21025 = rad(rad(1)^prime(rad(1)^prime(1)))^prime(1).

			The a(12) = 4 subexpressions of o[o[]][] are {o, o[], o[o[]], o[o[]][]}.

Cf. A007916, A052409, A052410, A277576, A277996, A300626, A316112, A317056, A317658, A317713, A317994.

nn=1000;
radQ[n_]:=If[n===1,False,GCD@@FactorInteger[n][[All,2]]===1];
rad[n_]:=rad[n]=If[n===0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
Clear[radPi];Set@@@Array[radPi[rad[#]]==#&,nn];
exp[n_]:=If[n===1,"o",With[{g=GCD@@FactorInteger[n][[All,2]]},Apply[exp[radPi[Power[n,1/g]]],exp/@Flatten[Cases[FactorInteger[g],{p_?PrimeQ,k_}:>ConstantArray[PrimePi[p],k]]]]]];
Table[Length[Union[Cases[exp[n],_,{0,Infinity},Heads->True]]],{n,100}]

A322900 Number of integer partitions of n whose parts are all proper powers of the same number.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 5, 2, 5, 3, 7, 2, 11, 2, 9, 5, 11, 2, 16, 2, 18, 6, 17, 2, 27, 3, 23, 6, 30, 2, 38, 2, 37, 8, 39, 5, 58, 2, 49, 10, 66, 2, 74, 2, 78, 14, 77, 2, 109, 3, 100, 12, 118, 2, 131, 6, 146, 15, 143, 2, 190, 2, 169, 20, 203, 6, 224, 2, 242, 18, 248

Offset: 0

Gus Wiseman, Dec 30 2018

nonn

Such a partition contains either no 1's or only 1's.
A proper power of n is a number n^k for some positive integer k.
Also integer partitions whose parts all have the same radical base (A052410).

			The a(1) = 1 through a(14) = 9 integer partitions (A = 10, B = 11, C = 12, D = 13, E = 14):
  (1) (2)  (3)   (4)    (5)     (6)      (7)       (8)        (9)
      (11) (111) (22)   (11111) (33)     (1111111) (44)       (333)
                 (1111)         (42)               (422)      (111111111)
                                (222)              (2222)
                                (111111)           (11111111)
.
  (A)          (B)           (C)            (D)             (E)
  (55)         (11111111111) (66)           (1111111111111) (77)
  (82)                       (84)                           (842)
  (442)                      (93)                           (4442)
  (4222)                     (444)                          (8222)
  (22222)                    (822)                          (44222)
  (1111111111)               (3333)                         (422222)
                             (4422)                         (2222222)
                             (42222)                        (11111111111111)
                             (222222)
                             (111111111111)

a(n) = A072721(n) + 1.

Cf. A000961, A001597, A018819, A023893, A023894, A052409, A052410, A072720, A102430, A302593, A322901, A322902, A322903.

radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
Table[Length[Select[IntegerPartitions[n],SameQ@@radbase/@#&]],{n,30}]

A323054 Number of strict integer partitions of n with no 1's such that no part is a power of any other part.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 3, 3, 4, 4, 6, 6, 8, 9, 12, 13, 16, 19, 21, 25, 30, 36, 40, 47, 53, 63, 71, 83, 94, 107, 121, 140, 159, 180, 204, 233, 260, 296, 334, 377, 421, 474, 532, 598, 668, 750, 835, 933, 1038, 1163, 1292, 1435, 1597, 1771, 1966, 2180, 2421, 2673

Offset: 0

Gus Wiseman, Jan 04 2019

nonn

			The a(2) = 1 through a(13) = 8 strict integer partitions (A = 10, B = 11, C = 12, D = 13):
  (2)  (3)  (4)  (5)   (6)  (7)   (8)   (9)   (A)    (B)    (C)    (D)
                 (32)       (43)  (53)  (54)  (64)   (65)   (75)   (76)
                            (52)  (62)  (63)  (73)   (74)   (84)   (85)
                                        (72)  (532)  (83)   (A2)   (94)
                                                     (92)   (543)  (A3)
                                                     (632)  (732)  (B2)
                                                                   (643)
                                                                   (652)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..350

Cf. A001597, A007916, A025147, A052410, A078374, A087897, A120641, A275972, A303362, A323053, A323087, A323088.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],And[FreeQ[#,1],UnsameQ@@#,stableQ[#,IntegerQ[Log[#1,#2]]&]]&]],{n,30}]

A367587 Least element in row n of A367858 (multiset multiplicity cokernel).

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 1, 6, 4, 3, 1, 7, 1, 8, 1, 4, 5, 9, 1, 3, 6, 2, 1, 10, 3, 11, 1, 5, 7, 4, 2, 12, 8, 6, 1, 13, 4, 14, 1, 2, 9, 15, 1, 4, 1, 7, 1, 16, 1, 5, 1, 8, 10, 17, 1, 18, 11, 2, 1, 6, 5, 19, 1, 9, 4, 20, 1, 21, 12, 2, 1, 5, 6, 22, 1, 2

Offset: 1

Gus Wiseman, Dec 03 2023

nonn

We define the multiset multiplicity cokernel MMC(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then max(S) has multiplicity |S| in MMC(m). For example, MMC({1,1,2,2,3,4,5}) = {2,2,5,5,5}, and MMC({1,2,3,4,5,5,5,5}) = {4,4,4,4,5}. As an operation on multisets MMC is represented by A367858, and as an operation on their ranks it is represented by A367859.

Indices of first appearances are A008578.
Depends only on rootless base A052410, see A007916.
For kernel instead of cokernel we have A055396.
For maximum instead of minimum element we have A061395.
The opposite version is A367583.
Row-minima of A367858.
A007947 gives squarefree kernel.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A124010 lists prime multiplicities (prime signature), sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A238747 gives prime metasignature, sorted A353742.
A304038 lists distinct prime indices, length A001221, sum A066328.
A367579 lists MMK, rank A367580, sum A367581, max A367583, min A055396.
Cf. A000720, A027746, A051904, A071625, A072774, A130091, A367582, A367584, A367585, A367586.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
mmc[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Max@@Select[q,Count[q,#]==i&],{i,mts}]]];
Table[If[n==1,0,Min@@mmc[prix[n]]],{n,100}]

a(n) = A055396(A367859(n)).
a(n^k) = a(n) for all positive integers n and k.
If n is a power of a squarefree number, a(n) = A061395(n).

cp's OEIS Frontend

A317994 Number of inequivalent leaf-colorings of the free pure symmetric multifunction with e-number n.

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A377467 Number of perfect-powers x in the range 2^n < x < 2^(n+1).

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A378356 Prime index of the next prime after the n-th perfect power.

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A378357 Distance from n to the least non perfect power >= n.

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A072720 Number of partitions of n into parts which are each powers of a single number (which may vary between partitions).

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A303709 Number of periodic factorizations of n using elements of A007916 (numbers that are not perfect powers).

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A317765 Number of distinct subexpressions of the free pure symmetric multifunction (with empty expressions allowed) with e-number n.

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A322900 Number of integer partitions of n whose parts are all proper powers of the same number.

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