A007916 -id:A007916 - cp's OEIS frontend

A303707 Number of factorizations of n using elements of A007916 (numbers that are not perfect powers).

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 5, 1, 1, 2, 2, 2, 5, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 1, 2, 5, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 2, 5, 1, 5, 1, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Apr 29 2018

nonn

First differs from A081707 at a(60) = 9, A081707(60) = 8.

			The a(60) = 9 factorizations are (2*2*3*5), (2*2*15), (2*3*10), (2*5*6), (2*30), (3*20), (5*12), (6*10), (60).

Cf. A000837, A001055, A001597, A007716, A007916, A045778, A052409, A052410, A162247, A281113, A281116, A303708, A303709, 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[facsr[n]],{n,100}]

Dirichlet g.f.: Product_{n in A007916} 1/(1 - n^s).

A376562 Second differences of consecutive non-perfect-powers (A007916). First differences of A375706.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 01 2024

sign

Non-perfect-powers (A007916) are numbers without a proper integer root.

			The non-perfect powers (A007916) are:
  2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, ...
with first differences (A375706):
  1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, ...
with first differences (A376562):
  1, -1, 0, 2, -2, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 1, -1, 0, ...

The version for A000002 is A376604, first differences of A054354.
For first differences we had A375706, ones A375740, complement A375714.
Positions of zeros are A376588, complement A376589.
Runs of non-perfect-powers:
- length: A375702 = A053289(n+1) - 1
- first: A375703 (same as A216765 with 2 exceptions)
- last: A375704 (same as A045542 with 8 removed)
- sum: A375705
A000961 lists prime-powers inclusive, exclusive A246655.
A007916 lists non-perfect-powers, complement A001597.
A112344 counts integer partitions into perfect-powers, factorizations A294068.
A333254 gives run-lengths of differences between consecutive primes.
For non-perfect-powers: A375706 (first differences), A376588 (inflections and undulations), A376589 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).
Cf. A025475, A052410, A053707, A064113, A069623, A093555, A174965, A182853, A336416, A336417, A361102.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Differences[Select[Range[100],radQ],2]

from itertools import count
from sympy import mobius, integer_nthroot, perfect_power
def A376562(n):
    def f(x): return int(n+1-sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    r = m+((k:=next(i for i in count(1) if not perfect_power(m+i)))<<1)
    return next(i for i in count(1-k) if not perfect_power(r+i)) # Chai Wah Wu, Oct 02 2024

A277562 Numbers of the form c(x_1)^c(x_2)^...^c(x_k) where each c(i) = A007916(i) is a non-perfect-power, k >= 2, and the exponents are nested from the right.

Original entry on oeis.org

16, 81, 256, 512, 625, 1296, 2401, 6561, 10000, 14641, 19683, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, 234256, 279841, 331776, 390625, 456976, 614656, 707281, 810000, 923521, 1185921, 1336336, 1500625, 1679616, 1874161, 1953125, 2085136, 2313441, 2560000, 2825761, 3111696, 3418801

Offset: 1

Gus Wiseman, Oct 19 2016

nonn

Non-perfect-powers, or NPPs (A007916), are numbers whose prime multiplicities are relatively prime. As discussed in A007916, the expansion of a positive integer into a tower of NPPs is unique and always possible. 65536=2^2^2^2 is the smallest number that requires a tower of height more than 3.

			       16 = 2^2^2        81 = 3^2^2       256 = 2^2^3       512 = 2^3^2
      625 = 5^2^2      1296 = 6^2^2      2401 = 7^2^2      6561 = 3^2^3
    10000 = 10^2^2    14641 = 11^2^2    19683 = 3^3^2     20736 = 12^2^2
    28561 = 13^2^2    38416 = 14^2^2    50625 = 15^2^2
    65536 = 2^2^2^2   83521 = 17^2^2   104976 = 18^2^2   130321 = 19^2^2
   160000 = 20^2^2   194481 = 21^2^2   234256 = 22^2^2   279841 = 23^2^2
   331776 = 24^2^2   390625 = 5^2^3    456976 = 26^2^2   614656 = 28^2^2
   707281 = 29^2^2   810000 = 30^2^2   923521 = 31^2^2  1185921 = 33^2^2
  1336336 = 34^2^2  1500625 = 35^2^2  1679616 = 6^2^3   1874161 = 37^2^2
  1953125 = 5^3^2   2085136 = 38^2^2  2313441 = 39^2^2  2560000 = 40^2^2
  2825761 = 41^2^2  3111696 = 42^2^2  3418801 = 43^2^2  3748096 = 44^2^2
  4100625 = 45^2^2  4477456 = 46^2^2  4879681 = 47^2^2  5308416 = 48^2^2
  5764801 = 7^2^3   6250000 = 50^2^2  6765201 = 51^2^2  7311616 = 52^2^2
  7890481 = 53^2^2  8503056 = 54^2^2  9150625 = 55^2^2  9834496 = 56^2^2

David A. Corneth, Table of n, a(n) for n = 1..10025 (terms <= 10^16)
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy)

Cf. A007916, A001597, A164336, A164337, A106490 (Quetian Superfactorization).

radicalQ[1]:=False;
radicalQ[n_]:=SameQ[GCD@@FactorInteger[n][[All,2]],1];
hyperfactor[1]:={};
hyperfactor[n_?radicalQ]:={n};
hyperfactor[n_]:=With[{g=GCD@@FactorInteger[n][[All,2]]},Prepend[hyperfactor[g],Product[Apply[Power[#1,#2/g]&,r],{r,FactorInteger[n]}]]];
Select[Range[10^6],Length[hyperfactor[#]]>2&]

Edited by N. J. A. Sloane, Nov 09 2016

Offset changed to 1 by David A. Corneth, Apr 30 2024

A278028 Let {c(i)} = A007916 denote the sequence of numbers > 1 which are not perfect powers. Every positive integer n has a unique representation as a tower n = c(x_1)^c(x_2)^c(x_3)^...^c(x_k), where the exponents are nested from the right. The sequence is an irregular triangle read by rows, where the n-th row lists x_1, ..., x_k.

Original entry on oeis.org

1, 2, 1, 1, 3, 4, 5, 1, 2, 2, 1, 6, 7, 8, 9, 10, 11, 1, 1, 1, 12, 13, 14, 15, 16, 17, 18, 19, 3, 1, 20, 2, 2, 21, 22, 23, 24, 1, 3, 25, 26, 27, 4, 1, 28, 29, 30, 31

Offset: 1

N. J. A. Sloane, Nov 09 2016

Row lengths are A288636(n). - Gus Wiseman, Jun 12 2017

			Rows 2 through 32 are:
1,
2,
1, 1,
3,
4,
5,
1, 2,
2, 1,
6,
7,
8,
9,
10,
11,
1, 1, 1,
12,
13,
14,
15,
16,
17,
18,
19,
3, 1,
20,
2, 2,
21,
22,
23,
24,
1, 3,
...

N. J. A. Sloane, Table of n, a(n) for n = 1..20181
N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564

See A277564 for another version.

Cf. A007916, A089723, A277562, A288636.

A375740 Numbers k such that A007916(k+1) - A007916(k) = 1. In other words, the k-th non-perfect-power is 1 less than the next.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Gus Wiseman, Sep 10 2024

nonn

Positions in A007916 of numbers k such that k+1 is also a member.
Positions of 1's in A375706 (first differences of A007916).
Non-perfect-powers (A007916) are numbers with no proper integer roots.

			The non-perfect-powers are 2, 3, 5, 6, 7, 10, 11, 12, 13, ... which increase by one after positions 1, 3, 4, 6, ...

The version for non-prime-powers is A375713, differences A373672.
The complement is A375714, differences A375702.
The version for prime-powers is A375734, differences A373671.
The complement for non-prime-powers is A375928, differences A110969.
A000040 lists the prime numbers, differences A001223.
A000961 lists prime-powers (inclusive), differences A057820.
A001597 lists perfect-powers, differences A053289.
A002808 lists the composite numbers, differences A073783.
A018252 lists the nonprime numbers, differences A065310.
Non-perfect-powers:
- terms: A007916
- differences: A375706
- runs: sum A375705, A375703, A375704, A375702
- anti-runs: A375737, A375738, A375739, A375736.
Non-prime-powers (exclusive):
- terms: A361102
- differences: A375708
- runs: A373678, A373676, A373677, A110969
- anti-runs: A373679, A373575, A255346, A373672
Cf. A006549, A046933, A093555, A174965, A246655, A251092.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Join@@Position[Differences[Select[Range[100],radQ]],1]

from itertools import count, islice
from sympy import perfect_power
def A375740_gen(): # generator of terms
    a, b = -1, 0
    for n in count(2):
        c = not perfect_power(n)
        if c:
            a += 1
        if b&c:
            yield a
    b = c
A375740_list = list(islice(A375740_gen(), 52)) # Chai Wah Wu, Sep 11 2024

A153158 a(n) = A007916(n)^2.

Original entry on oeis.org

4, 9, 25, 36, 49, 100, 121, 144, 169, 196, 225, 289, 324, 361, 400, 441, 484, 529, 576, 676, 784, 841, 900, 961, 1089, 1156, 1225, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2500, 2601, 2704, 2809, 2916, 3025, 3136, 3249, 3364, 3481

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 19 2008

nonn

A378168(n) is the number of terms <= 10^n. - Chai Wah Wu, Nov 21 2024

			2^2 = 4, 3^2 = 9, 4^2 = 16 = 2^4 is not in the sequence, 5^2 = 25, 6^2 = 36, ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A007916, A153147, A153157, A153159, A153160, A062503, A076467, A378168, A378287.

a153158 n = a153158_list !! (n-1)
a153158_list = filter ((== 2) . foldl1 gcd . a124010_row) [2..]
-- Reinhard Zumkeller, Apr 13 2012

q:= n-> is(igcd(seq(i[2], i=ifactors(n)[2]))=2):
select(q, [i^2$i=2..60])[];  # Alois P. Heinz, Nov 26 2024

Select[Range[2,100],GCD@@Last/@FactorInteger@#==1&]^2

from sympy import mobius, integer_nthroot
def A153158(n):
    def f(x): return int(n+1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m**2 # Chai Wah Wu, Aug 13 2024

GCD(exponents in prime factorization of a(n)) = 2, cf. A124010. - Reinhard Zumkeller, Apr 13 2012
Sum_{n>=1} 1/a(n) = zeta(2) - 1 - Sum_{k>=2} mu(k)*(1 - zeta(2*k)) = 0.5444587396... - Amiram Eldar, Jul 02 2022
Intersection of A000290 and A378287. Squares that are not of the form m^k for some k>=3. - Chai Wah Wu, Nov 21 2024

Edited by Ray Chandler, Dec 22 2008

A277564 Let {c(i)} = A007916 denote the sequence of numbers > 1 which are not perfect powers. Every positive integer n has a unique representation as a tower n = c(x_1)^c(x_2)^c(x_3)^...^c(x_k), where the exponents are nested from the right. The sequence is an irregular triangle read by rows, where the n-th row lists n followed by x_1, ..., x_k.

Original entry on oeis.org

1, 2, 1, 3, 2, 4, 1, 1, 5, 3, 6, 4, 7, 5, 8, 1, 2, 9, 2, 1, 10, 6, 11, 7, 12, 8, 13, 9, 14, 10, 15, 11, 16, 1, 1, 1, 17, 12, 18, 13, 19, 14, 20, 15, 21, 16, 22, 17, 23, 18, 24, 19, 25, 3, 1, 26, 20, 27, 2, 2, 28, 21, 29, 22, 30, 23, 31, 24, 32, 1, 3, 33, 25, 34, 26, 35, 27, 36, 4, 1, 37, 28, 38, 29, 39, 30, 40, 31

Offset: 1

Gus Wiseman, Oct 20 2016

The row lengths are A288636(n) + 1. - Gus Wiseman, Jun 12 2017

See A278028 for a version in which row n simply lists x_1, x_2, ..., x_k (omitting the initial n).

			1 is represented by the empty sequence (), by convention.
Successive rows of the triangle are as follows (c(k) denotes the k-th non-prime-power, A007916(k)):
2, 1,
3, 2,
4, 1, 1,
5, 3,
6, 4, because 6 = c(4)
7, 5,
8, 1, 2, because 8 = 2^3 = c(1)^c(2)
9, 2, 1,
10, 6,
11, 7,
...
16, 1, 1, 1, because 16 = 2^4 = c(1)^4 = c(1)^(c(1)^2) = c[1]^(c[1]^c[1])
17, 12,
...
This sequence represents a bijection N -> Q where Q is the set of all finite sequences of positive integers: 1->(), 2->(1), 3->(2), 4->(1 1), 5->(3), 6->(4), 7->(5), 8->(1 2), 9->(2 1), ...

Gus Wiseman, Table of n, a(n) for n = 1..20131
N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564

Cf. A007916, A277562, A164336, A000961, A164337, A278028, A089723.

Maple
```
See link.
```

nn=10000;radicalQ[1]:=False;radicalQ[n_]:=SameQ[GCD@@FactorInteger[n][[All,2]],1];
hyperfactor[1]:={};hyperfactor[n_?radicalQ]:={n};hyperfactor[n_]:=With[{g=GCD@@FactorInteger[n][[All,2]]},Prepend[hyperfactor[g],Product[Apply[Power[#1,#2/g]&,r],{r,FactorInteger[n]}]]];
rad[0]:=1;rad[n_?Positive]:=rad[n]=NestWhile[#+1&,rad[n-1]+1,Not[radicalQ[#]]&];Set@@@Array[radPi[rad[#]]==#&,nn];
Flatten[Join[{#},radPi/@hyperfactor[#]]&/@Range[nn]]

Edited by N. J. A. Sloane, Nov 09 2016

A277576 a(1)=1; thereafter a(n) = A007916(a(n-1)).

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 15, 20, 26, 34, 43, 53, 63, 74, 86, 98, 111, 126, 142, 159, 177, 195, 214, 235, 258, 281, 305, 330, 356, 383, 411, 439, 468, 498, 530, 562, 595, 629, 663, 698, 734, 770, 807, 845, 883, 922, 962, 1003, 1045, 1087, 1130, 1174, 1218, 1263, 1309, 1356, 1404, 1453, 1502, 1552, 1603, 1654, 1706, 1759

Offset: 1

Gus Wiseman, Oct 20 2016

nonn

Non-perfect-powers (A007916) are numbers such that the exponents in their prime factorizations have GCD equal to 1. For each n we can construct a plane tree by replacing all positive integers at any level with their corresponding planar factorization sequences (A277564), and repeating this replacement until no numbers are left. The result will be a unique "pure" sequence or plane tree. Under this correspondence a(n) is the path tree ((((((...)))))) = string of n consecutive open brackets followed by the same number of closed brackets.

			The first forty plane trees:
()         11(((((())))))      ((()()()))         (((((((()())))))))
2(())         ((()(())))        ((((()(())))))     (()((())))
3((()))       (((())()))        (((((())()))))     ((((()))()))
(()())       ((((()()))))      ((((((()())))))) 34(((((((((())))))))))
5(((())))   15((((((()))))))    (((()))())         (((())(())))
((()()))     (()()())        26((((((((())))))))) ((()())())
7((((()))))   (((()(()))))      ((())(()))         ((((()()()))))
(()(()))     ((((())())))      (((()()())))       ((((((()(())))))))
((())())     (((((()())))))    (((((()(()))))))   (((((((())()))))))
(((()()))) 20(((((((())))))))  ((((((())())))))   ((((((((()()))))))))

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..2480 from Gus Wiseman)

Cf. A007916, A277564, A276625, A004111 (rooted trees), A007097 (rooted paths).

radicalQ[1]:=False;radicalQ[n_]:=SameQ[GCD@@FactorInteger[n][[All,2]],1];
rad[0]:=1;rad[n_?Positive]:=rad[n]=NestWhile[#+1&,rad[n-1]+1,Not[radicalQ[#]]&];
nn=2000;Scan[rad,Range[nn]];NestWhileList[rad,1,#

from itertools import islice
from sympy import mobius, integer_nthroot
def A277576_gen(): # generator of terms
    def iterfun(f,n=0):
        m, k = n, f(n)
        while m != k: m, k = k, f(k)
        return m
    def f(x): return int(1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    a = 1
    while True:
        yield a
        a = iterfun(lambda x:f(x)+a,a)
A277576_list = list(islice(A277576_gen(),40)) # Chai Wah Wu, Nov 21 2024

Edited by N. J. A. Sloane, Nov 09 2016

A289023 Position in the sequence of numbers that are not perfect powers (A007916) of the smallest positive integer x such that for some positive integer y we have n = x^y (A052410).

Original entry on oeis.org

1, 2, 1, 3, 4, 5, 1, 2, 6, 7, 8, 9, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 19, 3, 20, 2, 21, 22, 23, 24, 1, 25, 26, 27, 4, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 5, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 1, 54, 55, 56, 57, 58, 59, 60

Offset: 2

Gus Wiseman, Jun 22 2017

nonn

Every pair p of positive integers is of the form p = (a(n), A052409(n)) for exactly one n.

			a(27)=2 because the smallest root of 27 is 3, and 3 is the 2nd entry of A007916.
a(25)=3 because the smallest root of 25 is 5, and 5 is the 3rd entry of A007916.

Cf. A007916, A052409, A052410, A278028, A288636.

nn=100;
q=Table[Power[n,1/GCD@@FactorInteger[n][[All,2]]],{n,2,nn}];
q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}]

a(n) = if (ispower(n,,&r), x = r, x = n); sum(k=2, x, ispower(k)==0); \\ Michel Marcus, Jul 19 2017

For n>1 we have a(n) = A278028(n,1).

A303708 Number of aperiodic factorizations of n using elements of A007916 (numbers that are not perfect powers).

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 3, 1, 2, 2, 0, 1, 3, 1, 3, 2, 2, 1, 4, 0, 2, 0, 3, 1, 5, 1, 0, 2, 2, 2, 3, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 0, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 0, 2, 5, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 2, 5, 1, 5, 0, 2, 1, 9, 2, 2, 2, 4, 1, 9, 2

Offset: 1

Gus Wiseman, Apr 29 2018

nonn

An aperiodic factorization of n is a finite multiset of positive integers greater than 1 whose product is n and whose multiplicities are relatively prime.

The positions of zeros in this sequence are the prime powers A000961.

			The a(144) = 8 aperiodic factorizations are (2*2*2*3*6), (2*2*2*18), (2*2*3*12), (2*3*24), (2*6*12), (2*72), (3*48) and (6*24). Missing from this list are (12*12), (2*2*6*6) and (2*2*2*2*3*3).

Cf. A000740, A000837, A001055, A001597, A007716, A007916, A052409, A052410, A100953, A275870, A281116, A301700, A303386, A303431, A303546, A303551, A303707, A303709, 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,100}]

a(n) = Sum_{d in A007916, d|A052409(n)} mu(d) * A303707(n^(1/d)).

cp's OEIS Frontend

A303707 Number of factorizations of n using elements of A007916 (numbers that are not perfect powers).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A376562 Second differences of consecutive non-perfect-powers (A007916). First differences of A375706.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

A277562 Numbers of the form c(x_1)^c(x_2)^...^c(x_k) where each c(i) = A007916(i) is a non-perfect-power, k >= 2, and the exponents are nested from the right.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A375740 Numbers k such that A007916(k+1) - A007916(k) = 1. In other words, the k-th non-perfect-power is 1 less than the next.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

A153158 a(n) = A007916(n)^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Python

Formula

Extensions

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A277576 a(1)=1; thereafter a(n) = A007916(a(n-1)).

Views

Author

Keywords

Comments