A112344 -id:A112344 - cp's OEIS frontend

A053289 First differences of consecutive perfect powers (A001597).

3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, 33, 35, 19, 18, 39, 41, 43, 28, 17, 47, 49, 51, 53, 55, 57, 59, 61, 39, 24, 65, 67, 69, 71, 35, 38, 75, 77, 79, 81, 47, 36, 85, 87, 89, 23, 68, 71, 10, 12, 95, 97, 99, 101, 103, 40, 65, 107, 109, 100

Offset: 1

Labos Elemer, Mar 03 2000

nonn

Michel Waldschmidt writes: Conjecture 1.3 (Pillai). Let k be a positive integer. The equation x^p - y^q = k where the unknowns x, y, p and q take integer values, all >= 2, has only finitely many solutions (x,y,p,q). This means that in the increasing sequence of perfect powers [A001597] the difference between two consecutive terms [the present sequence] tends to infinity. It is not even known whether for, say, k=2, Pillai's equation has only finitely many solutions. A related open question is whether the number 6 occurs as a difference between two perfect powers. See Sierpiński [1970], problem 238a, p. 116. - Jonathan Vos Post, Feb 18 2008

Are there are any adjacent equal terms? - Gus Wiseman, Oct 08 2024

			Consecutive perfect powers are A001597(14) = 121, A001597(13) = 100, so a(13) = 121 - 100 = 21.

Wacław Sierpiński, 250 problems in elementary number theory, Modern Analytic and Computational Methods in Science and Mathematics, No. 26, American Elsevier, Warsaw, 1970, pp. 21, 115-116.
S. S. Pillai, On the equation 2^x - 3^y = 2^X - 3^Y, Bull, Calcutta Math. Soc. 37 (1945) 15-20.

Daniel Forgues and T. D. Noe, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Gaps between consecutive perfect powers, International Mathematical Forum, Vol. 11, No. 9 (2016), pp. 429-437.
Holly Krieger and Brady Haran, Catalan's Conjecture, Numberphile video (2018).
Michel Waldschmidt, Open Diophantine problems, arXiv:math/0312440 [math.NT], 2003-2004.

For non-perfect-powers (A007916) we have A375706.
The union is A023055.
For prime-powers (A000961 or A246655) we have A057820.
Sorted positions of first appearances are A376268, complement A376519.
For second differences we have A376559.
Ascending and descending points are A376560 and A376561.
A001597 lists perfect-powers.
A112344 counts integer partitions into perfect-powers, factorizations A294068.
A333254 gives run-lengths of differences between consecutive primes.
Cf. A025475, A069623, A219551.
Cf. A007921, A036263, A045542, A052410, A053707, A174965, A336416, A375735, A375736, A375740, A376562.

Differences@ Select[Range@ 3200, # == 1 || GCD @@ FactorInteger[#][[All, 2]] > 1 &] (* Michael De Vlieger, Jun 30 2016, after Ant King at A001597 *)

from sympy import mobius, integer_nthroot
def A053289(n):
    if n==1: return 3
    def f(x): return int(n-2+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax)+1 >= kmax:
        kmax <<= 1
    rmin, rmax = 1, kmax
    while True:
        kmid = kmax+kmin>>1
        if f(kmid)+1 < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    while True:
        rmid = rmax+rmin>>1
        if f(rmid) < rmid:
            rmax = rmid
        else:
            rmin = rmid
        if rmax-rmin <= 1:
            break
    return kmax-rmax # Chai Wah Wu, Aug 13 2024

a(n) = A001597(n+1) - A001597(n). - Jonathan Vos Post, Feb 18 2008
From Amiram Eldar, Jun 30 2023: (Start)
Formulas from Jakimczuk (2016):
Lim sup_{n->oo} a(n)/(2*n) = 1.
Lim inf_{n->oo} a(n)/(2*n)^(2/3 + eps) = 0. (End)
Can be obtained by inserting 0 between 3 and 6 in A375702 and then adding 1 to all terms. In particular, for n > 2, a(n+1) - 1 = A375702(n). - Gus Wiseman, Sep 14 2024

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

A376559 Second differences of consecutive perfect powers (A001597). First differences of A053289.

Original entry on oeis.org

1, -3, 6, 2, -7, 3, -1, 9, 2, 2, 2, 2, -17, -1, 13, 9, 2, -7, -11, 9, -5, 20, 2, -16, -1, 21, 2, 2, -15, -11, 30, 2, 2, 2, 2, 2, 2, 2, -22, -15, 41, 2, 2, 2, -36, 3, 37, 2, 2, 2, -34, -11, 49, 2, 2, -66, 45, 3, -61, 2, 83, 2, 2, 2, 2, -63, 25, 42, 2, -9, -89

Offset: 1

Gus Wiseman, Sep 28 2024

sign

Perfect-powers A007916 are numbers with a proper integer root.

Does this sequence contain zero?

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, ...
with first differences (A376559):
  1, -3, 6, 2, -7, 3, -1, 9, 2, 2, 2, 2, -17, -1, 13, 9, 2, -7, -11, 9, -5, 20, ...

The version for A000002 is A376604, first differences of A054354.
For first differences we have A053289, union A023055, firsts A376268, A376519.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A112344 counts integer partitions into perfect-powers, factorizations A294068.
For perfect-powers: A053289 (first differences), A376560 (positive curvature), A376561 (negative curvature).
For second differences: A036263 (prime), A073445 (composite), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power), A376599 (non-prime-power).
Cf. A045542, A052410, A053707, A064113, A069623, A174965, A216765, A251092, A333254, A336416, A361102.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Differences[Select[Range[1000],perpowQ],2]

lista(nn) = my(v = concat (1, select(ispower, [1..nn])), w = vector(#v-1, i, v[i+1] - v[i])); vector(#w-1, i, w[i+1] - w[i]); \\ Michel Marcus, Oct 02 2024

from sympy import mobius, integer_nthroot
def A376559(n):
    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): return int(n-1+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    a = bisection(f,n,n)
    b = bisection(lambda x:f(x)+1,a,a)
    return a+bisection(lambda x:f(x)+2,b,b)-(b<<1) # Chai Wah Wu, Oct 02 2024

A102430 Triangle read by rows where T(n,k) is the number of integer partitions of n > 1 into powers of k > 1.

Original entry on oeis.org

2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 6, 3, 2, 2, 2, 6, 3, 2, 2, 2, 2, 10, 3, 3, 2, 2, 2, 2, 10, 5, 3, 2, 2, 2, 2, 2, 14, 5, 3, 3, 2, 2, 2, 2, 2, 14, 5, 3, 3, 2, 2, 2, 2, 2, 2, 20, 7, 4, 3, 3, 2, 2, 2, 2, 2, 2, 20, 7, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2, 26, 7, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2

Offset: 2

Marc LeBrun, Jan 08 2005

All entries above main diagonal are = 1.

			The T(9,3)=5 partitions of 9 into powers of 3: 111111111, 1111113, 11133, 333, 9.
From _Gus Wiseman_, Jun 07 2019: (Start)
Triangle begins:
   2
   2  2
   4  2  2
   4  2  2  2
   6  3  2  2  2
   6  3  2  2  2  2
  10  3  3  2  2  2  2
  10  5  3  2  2  2  2  2
  14  5  3  3  2  2  2  2  2
  14  5  3  3  2  2  2  2  2  2
  20  7  4  3  3  2  2  2  2  2  2
  20  7  4  3  3  2  2  2  2  2  2  2
  26  7  4  3  3  3  2  2  2  2  2  2  2
  26  9  4  4  3  3  2  2  2  2  2  2  2  2
  36  9  6  4  3  3  3  2  2  2  2  2  2  2  2
  36  9  6  4  3  3  3  2  2  2  2  2  2  2  2  2
  46 12  6  4  4  3  3  3  2  2  2  2  2  2  2  2  2
Row n = 8 counts the following partitions:
  8          3311       44         5111       611        71         8
  44         311111     41111      11111111   11111111   11111111   11111111
  422        11111111   11111111
  2222
  4211
  22211
  41111
  221111
  2111111
  11111111
(End)

Alois P. Heinz, Rows n = 2..500, flattened

Cf. A102431, A102432, A102433, A102434, A001700, A018819, A062051, A008645, A008650, A008652, A008648.
Same as A308558 except for the k = 1 column.
Row sums are A102431.
First column (k = 2) is A018819.
Second column (k = 3) is A062051.
Cf. A000961, A001597, A007916, A023894, A052410, A112344.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<0, 0,
      b(n, i-1, k)+(p-> `if`(p>n, 0, b(n-p, i, k)))(k^i)))
    end:
T:= (n, k)-> b(n, ilog[k](n), k):
seq(seq(T(n, k), k=2..n), n=2..20);  # Alois P. Heinz, Oct 12 2019

Table[Length[Select[IntegerPartitions[n],And@@(IntegerQ[Log[k,#]]&/@#)&]],{n,2,10},{k,2,n}] (* Gus Wiseman, Jun 07 2019 *)

T(1, k) = 1, T(n, 1) = choose(2n-1, n), T(n>1, k>1) = T(n-1, k) + (T(n/k, k) if k divides n, else 0)

Corrected and rewritten by Gus Wiseman, Jun 07 2019

A376561 Points of downward concavity in the sequence of perfect-powers (A001597).

Original entry on oeis.org

2, 5, 7, 13, 14, 18, 19, 21, 24, 25, 29, 30, 39, 40, 45, 51, 52, 56, 59, 66, 70, 71, 74, 87, 94, 101, 102, 108, 110, 112, 113, 119, 127, 135, 143, 144, 156, 157, 160, 161, 169, 178, 187, 196, 205, 206, 215, 224, 225, 234, 244, 263, 273, 283, 284, 293, 294, 304

Offset: 1

Gus Wiseman, Sep 30 2024

nonn

These are points at which the second differences are negative.
Perfect-powers (A001597) are numbers with a proper integer root.
Note that, for some sources, downward concavity is positive curvature.
From Robert Israel, Oct 31 2024: (Start)
The first case of two consecutive numbers in the sequence is a(4) = 13 and a(5) = 14.
The first case of three consecutive numbers is a(293) = 2735, a(294) = 2736, a(295) = 2737.
The first case of four consecutive numbers, if it exists, involves a(k) with k > 69755. (End)

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, 33, ...
with first differences (A376559):
  1, -3, 6, 2, -7, 3, -1, 9, 2, 2, 2, 2, -17, -1, 13, 9, 2, -7, -11, 9, -5, 20, 2, ...
with negative positions (A376561):
  2, 5, 7, 13, 14, 18, 19, 21, 24, 25, 29, 30, 39, 40, 45, 51, 52, 56, 59, 66, 70, ...

Robert Israel, Table of n, a(n) for n = 1..10000
Gus Wiseman, Points of downward concavity in the perfect-powers.

The version for A000002 is A025505, complement A022297. See also A054354, A376604.
For first differences we have A053289, union A023055, firsts A376268, A376519.
For primes instead of perfect-powers we have A258026.
For upward concavity we have A376560 (probably the complement).
A000961 lists the prime-powers inclusive, exclusive A246655.
A001597 lists the perfect-powers.
A007916 lists the non-perfect-powers.
A112344 counts partitions into perfect-powers, factorizations A294068.
A333254 gives run-lengths of differences between consecutive primes.
Second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power), A376599 (non-prime-power).
Cf. A025475, A045542, A052410, A054819, A064113, A069623, A174965, A216765, A251092, A336416, A336417, A375702.

N:= 10^6: # to use perfect powers <= N
P:= {seq(seq(i^m,i=2..floor(N^(1/m))), m=2 .. ilog2(N))}: nP:= nops(P):
P:= sort(convert(P,list)):
select(i -> 2*P[i] > P[i-1]+P[i+1], [$2..nP-1]); # Robert Israel, Oct 31 2024

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Join@@Position[Sign[Differences[Select[Range[1000],perpowQ],2]],-1]

A112345 Number of partitions of n into distinct perfect powers.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 1, 1, 3, 2, 1, 1, 3, 2, 1, 1, 3, 3, 1, 1, 3, 4, 1, 1, 4, 4, 1, 1, 4, 4, 2, 1, 4, 6, 2, 2, 4, 6, 2, 2, 4, 6, 3, 2, 5, 6, 4, 2, 6, 6, 4, 2, 6, 7, 4, 3, 6, 9, 4, 3, 7, 9, 5, 3, 7, 9, 6, 3, 7, 10, 6, 3, 8, 11, 6, 3, 8

Offset: 0

Reinhard Zumkeller, Sep 05 2005

nonn

			a(40) = #{36+4, 32+8, 27+9+4} = 3.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Perfect Power
Eric Weisstein's World of Mathematics, Partition

Cf. A001597, A000009, A033461, A039966, A112344, A284171.

G.f.: Product_{k>=2} (1 + x^A001597(k)). - Ilya Gutkovskiy, Mar 21 2017

a(0)=1 prepended by Ilya Gutkovskiy, Mar 21 2017

A284171 Number of partitions of n into distinct perfect powers (including 1).

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 3, 2, 1, 2, 3, 2, 1, 2, 3, 3, 2, 4, 5, 3, 2, 4, 5, 3, 2, 4, 6, 4, 2, 4, 7, 5, 2, 5, 8, 5, 2, 5, 8, 6, 3, 5, 10, 8, 4, 6, 10, 8, 4, 6, 10, 9, 5, 7, 11, 10, 6, 8, 12, 10, 6, 8, 13, 11, 7, 9, 15, 13, 7, 10, 16, 14, 8, 10, 16, 15, 9, 10, 17, 16, 9, 11

Offset: 0

Ilya Gutkovskiy, Mar 21 2017

nonn

Differs from the sequence A112345 which does not consider 1 as a perfect power.

			a(25) = 3 because we have [25], [16, 9] and [16, 8, 1].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Perfect Power
Index entries for related partition-counting sequences

Cf. A001597, A078635, A112344, A112345.

nmax = 100; CoefficientList[Series[(1 + x) Product[(1 + Boole[GCD @@ FactorInteger[k][[All, 2]] > 1] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Vec((1 + x) * prod(k=1, 100, 1 + (gcd(factorint(k)[,2])>1)*x^k) + O(x^101)) \\ Indranil Ghosh, Mar 21 2017

G.f.: Product_{k>=1} (1 + x^A001597(k)).

a(n) = A112345(n-1) + A112345(n).

A263717 Number of partitions of n into perfect odd powers (1 being excluded).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 1, 0, 2, 0, 3, 0, 0, 0, 1, 1, 0, 2, 0, 3, 0, 0, 0, 1, 1, 0, 2, 0, 3, 0, 1, 1, 2, 2, 0, 3, 0, 5, 0, 1, 1, 2, 2, 0, 3, 0, 5, 0, 1, 1, 2, 2, 0, 3, 1, 6

Offset: 1

Martin Y. Champel, Oct 24 2015

nonn

			a(97) = #{8*9+25, 5*9+25+27, 2*9+25+2*27} = 3.

Martin Y. Champel, Table of n, a(n) for n = 1...999

Cf. A075109, A112344.

Needs["Combinatorica`"]; Length@ Select[Combinatorica`Partitions@ #, AllTrue[#, And[PrimePowerQ@ #, ! PrimeQ@ #, OddQ@ #] &, 1] &] & /@ Range[2, 52] (* Michael De Vlieger, Nov 05 2015, Version 10 *)

# Python version 2.7
def a(n):
    base = sorted(list(set([a**b for b in range(2,int(log(n)/log(2))) for a in range(3,1+int(n**(1./b)),2)])))
    lb = len(base)
    if lb == 0:
        return 0
    sol = 0
    s = [n // base[0]]
    if lb == 1:
        if n % base[0]  == 0: return 1
        return 0
    while True:
        k = s.pop()
        while k < 0:
            if s ==(lb-1)*[0]:
                return sol
            k = s.pop() - 1
        s.append(k)
        x = n - sum([s[i]*base[i] for i in range(len(s))])
        ls = len(s)
        if ls == lb:
            continue
        a = x // base[ls]
        b = x % base[ls]
        if b == 0:
            s.append(a)
            sol +=1
            if len(s) == lb:
                s.pop()
                s.append(-1)
            r = s.pop() - 2
            s.append(r)
        else:
            s.append(a-1)
            if a!=0:
                if len(s) == lb: s[lb-1]=-1

A282500 Expansion of 1/(1 - Sum_{k = i^j, i>=1, j>=2} x^k).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 5, 8, 13, 19, 26, 37, 55, 81, 116, 167, 244, 358, 520, 752, 1091, 1589, 2311, 3354, 4870, 7081, 10298, 14963, 21734, 31580, 45900, 66704, 96919, 140827, 204654, 297413, 432180, 627996, 912565, 1326117, 1927054, 2800260, 4069160, 5913116, 8592675, 12486402, 18144506, 26366614

Offset: 0

Ilya Gutkovskiy, Feb 16 2017

nonn

Number of compositions (ordered partitions) into perfect powers (A001597).

			a(7) = 5 because we have  [4, 1, 1, 1], [1, 4, 1, 1], [1, 1, 4, 1], [1, 1, 1, 4] and [1, 1, 1, 1, 1, 1, 1].

Eric Weisstein's World of Mathematics, Perfect Powers
Index entries for sequences related to compositions

Cf. A001597, A078635, A112344.

nmax = 95; CoefficientList[Series[1/ (1 - x - Sum[Boole[GCD @@ FactorInteger[k][[All, 2]] > 1] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k = i^j, i>=1, j>=2} x^k).

a(n) ~ c / r^n, where r = 0.68816189979082638501485812136220175833447947220530020978433949588627... and c = 0.4267808681995359684192168334905096310027880655306734537865362460298... . - Vaclav Kotesovec, Feb 17 2017

A308558 Triangle read by rows where T(n,k) is the number of integer partitions of n > 0 into powers of k > 0.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 07 2019

			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)

Same as A102430 except for the k = 1 column.
Row sums are A102431(n) + 1.
Column k = 2 is A018819.
Column k = 3 is A062051.
Cf. A000961, A001597, A007916, A008284, A023894, A052410, A101417, A112344, A323053, A323054.

Table[If[k==1,1,Length[Select[IntegerPartitions[n],And@@(IntegerQ[Log[k,#]]&/@#)&]]],{n,10},{k,n}]

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

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A376559 Second differences of consecutive perfect powers (A001597). First differences of A053289.

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A102430 Triangle read by rows where T(n,k) is the number of integer partitions of n > 1 into powers of k > 1.

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A376561 Points of downward concavity in the sequence of perfect-powers (A001597).

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A112345 Number of partitions of n into distinct perfect powers.

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A284171 Number of partitions of n into distinct perfect powers (including 1).

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