A001597 -id:A001597 - cp's OEIS frontend

A075760 Nontrivial binomial coefficients which are perfect powers (A001597).

36, 1225, 19600, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025, 63955431761796, 2172602007770041, 73804512832419600

Offset: 1

Robert G. Wilson v, Oct 08 2002

Triangular-square numbers (A001110) are a subset, except for 0 and 1.

"For C(n,k) k>=4 and any l>=2 no solutions exist and this is what Erdos proved by an ingenious argument. ... C(50, 3) = 140^2 is the only solution for k = 3, l=2." page 13 of Aigner and Ziegler.

Martin Aigner and Gunter M. Ziegler, Proofs from THE BOOK, Second Edition, Springer-Verlag, Berlin, 2000, Chapter 3, "Binomial coefficients are (almost) never powers," pages 13-16.

Cf. A001110.

f[n_] := Apply[ GCD, Last[ Transpose[ FactorInteger[n]]]]; a = {}; Do[ If[ f[n(n - 1)/2] > 1, a = Append[a, Binomial[n, 2]]]; If[ f[n(n - 1)*(n - 2)/6] > 1, a = Append[a, Binomial[n, 3]]], {n, 5, 1500000}]

a(10)-a(12) from Sean A. Irvine, Mar 05 2025

A088413 A088259 indexed by A001597.

Original entry on oeis.org

0, 1, 2, 4, 7, 8, 12, 15, 18, 22, 27, 32, 34, 45, 50, 69, 72, 83, 92, 104, 112, 113, 117, 135, 142, 147, 151, 153, 158, 163, 176, 181, 187, 192, 203, 210, 214, 219, 241, 243, 248, 262, 269, 276, 280, 291, 298, 302, 307, 313, 325, 330, 347, 354, 363, 377, 392, 402

Offset: 1

Ray Chandler, Sep 29 2003

nonn

Cf. A001597, A088259.

a(n)=k such that A088259(n)=A001597(k+1).

Offset changed by Andrew Howroyd, Sep 22 2024

A284743 Positive numbers that are not the sum of (any number of) distinct perfect powers (A001597).

Original entry on oeis.org

2, 3, 6, 7, 11, 15, 19, 23

Offset: 1

Amiram Eldar, Apr 01 2017

Subsequence of A001422.

David Wells noted that 23 is the largest integer that is not the sum of distinct powers.

			22 is not in the sequence since 22 = 1 + 2^2 + 2^3 + 3^2.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin 1987, p. 101.

Cf. A001422, A001597.

PerfectPowerQ[n_] := n==1 || GCD@@FactorInteger[n][[All, 2]]>1; a=Select[Range[128], PerfectPowerQ[#] &]; nn = Dimensions[a][[1]]; t=Rest[CoefficientList[Series[Product[(1 + x^a[[k]]),{k,nn}],{x,0,a[[nn]]}], x]]; Flatten[Position[t, 0]]

A294076 Absolute difference between n-th stella octangula number (A007588) and the nearest perfect power (A001597).

Original entry on oeis.org

1, 0, 2, 2, 1, 2, 15, 3, 8, 5, 35, 50, 37, 25, 2, 11, 16, 8, 18, 10, 104, 5, 42, 25, 68, 104, 157, 35, 195, 92, 146, 15, 32, 17, 174, 134, 251, 145, 145, 263, 204, 160, 91, 230, 245, 124, 145, 337, 236, 24, 50, 26, 264, 415, 153, 234, 473, 552, 459, 182, 291

Offset: 0

Felix Fröhlich, Feb 07 2018

nonn

There are only two square stella octangula numbers, namely those corresponding to n = 1 and n = 169, so a(1) = 0 and a(169) = 0 (cf. Wikipedia link).

Wikipedia, Stella octangula number

Cf. A001597, A007588.

f[n_, i_: 1] := Block[{k = n, j = If[i == 1, 1, -1]}, While[Nor[k == 1, GCD @@ FactorInteger[k][[All, 2]] > 1], k = k + j]; k]; {1}~Join~Array[Min@ Abs@ {# - f[#], f[#, 0] - #} &[# (2 #^2 - 1)] &, 60] (* Michael De Vlieger, Feb 21 2018 *)

a007588(n) = n*(2*n^2-1)
is_a001597(n) = ispower(n) || n==1
nearestpower(n) = my(x=0); while(1, if(x < n, if(is_a001597(n-x), return(n-x), if(is_a001597(n+x), return(n+x))), if(is_a001597(n+x), return(n+x))); x++)
a(n) = abs(a007588(n)-nearestpower(a007588(n)))

A326119 a(n) is the absolute value of the alternating sum of the first n increasing perfect powers (A001597): 1, 1-4, 1-4+8, 1-4+8-9, ...

Original entry on oeis.org

1, 3, 5, 4, 12, 13, 14, 18, 18, 31, 33, 48, 52, 69, 56, 72, 72, 97, 99, 117, 108, 135, 121, 168, 156, 187, 174, 226, 215, 269, 243, 286, 290, 335, 341, 388, 396, 445, 455, 506, 494, 530, 559, 597, 628, 668, 663, 706, 738, 783, 817, 864, 864, 900, 949, 987, 1038

Offset: 1

Richard Locke Peterson, Sep 10 2019

nonn

			For n=8: a(8) = |1 - 4 + 8 - 9 + 16 - 25 + 27 - 32|.

Cf. A001597, A076408.

t = Select[Range@2400, # == 1 || GCD @@ Last /@ FactorInteger@# > 1 &]; Abs@ Accumulate[t (-1)^Range@ Length[t]] (* Giovanni Resta, Sep 11 2019 *)

seq(n)={my(v=vector(n), i=0, k=0, s=0); while(i<#v, k++; if(ispower(k)||k==1, s=k-s; i++; v[i]=abs(s))); v} \\ Andrew Howroyd, Sep 10 2019

a(n) = abs(Sum_{k=1..n} (-1)^k*A001597(k)). - Andrew Howroyd, Sep 10 2019

A332008 Numbers k such that phi(k) and phi(k+1) are perfect powers (A001597).

Original entry on oeis.org

1, 15, 16, 63, 101, 125, 255, 256, 272, 504, 512, 513, 629, 679, 1358, 1359, 1728, 1970, 2047, 2222, 2509, 2834, 3458, 3705, 4094, 4095, 4400, 4577, 4616, 4913, 5403, 6817, 7295, 7956, 8729, 11667, 11672, 16132, 16384, 16523, 17507, 23085, 24198, 24564, 24624, 25220, 25601, 27216, 27384, 28564, 29444

Offset: 1

Antonio Roldán, Feb 04 2020

nonn

			phi(101) = 10^2, and phi(102) = 2^5.
phi(3458) = 6^4, and phi(3459) = 48^2.

Cf. A000010, A001597, A166955, A281016.

[1] cat [k:k in [3..30000]|IsPower(EulerPhi(k))  and IsPower(EulerPhi(k+1))]; // Marius A. Burtea, Feb 05 2020

perfectPowerQ[1] = True; perfectPowerQ[n_] := GCD @@ FactorInteger[n][[;; , 2]] > 1; Select[Range[30000], And @@ perfectPowerQ /@ EulerPhi[# + {0, 1}] &] (* Amiram Eldar, Feb 04 2020 *)

v=[1]; for(i = 2, 30000, if(ispower(eulerphi(i)), if(ispower(eulerphi(i+1)), v = concat(v, i)))); v

A362426 Number of compositions (ordered partitions) of n into 2 perfect powers (A001597).

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 0, 1, 2, 2, 0, 2, 2, 0, 0, 1, 4, 1, 0, 2, 0, 0, 0, 2, 2, 2, 0, 2, 2, 0, 2, 1, 4, 2, 2, 4, 2, 0, 0, 4, 4, 0, 2, 2, 2, 0, 0, 2, 0, 3, 0, 4, 2, 1, 0, 0, 4, 2, 2, 0, 2, 0, 2, 1, 4, 0, 0, 4, 0, 0, 0, 3, 2, 2, 0, 2, 0, 0, 0, 2, 2, 2, 0, 0, 4, 0, 0, 0, 4, 2, 2

Offset: 0

Ilya Gutkovskiy, Apr 19 2023

nonn

Eric Weisstein's World of Mathematics, Perfect Power.

Cf. A001597.

perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[;; , 2]] > 1; a[n_] := Total[Multinomial @@ Tally[#][[;; , 2]] & /@ Select[IntegerPartitions[n, {2}], AllTrue[#, perfectPowerQ] &]]; Array[a, 100, 0] (* Amiram Eldar, May 05 2023 *)

A371223 Perfect powers (A001597) equal to the sum of a factorial number (A000142) and a Fibonacci number (A000045).

Original entry on oeis.org

1, 4, 8, 9, 25, 27, 32, 36, 121, 125, 128, 2704, 5041, 5184

Offset: 1

Gonzalo Martínez, Mar 23 2024

Listed terms are 1, 2^2, 2^3, 3^2, 5^2, 3^3, 2^5, 6^2, 11^2, 5^3, 2^7, 52^2, 71^2 and 72^2.
It is observed that 4, 8, 25, 121 and 5041 are also terms of A227644 (Perfect powers equal to the sum of two factorial numbers), where in turn 25, 121 and 5041 are terms of A085692 (Brocard's problem), while the first 4 terms and 36 are part of A272575 (Perfect powers that are the sum of two Fibonacci numbers).
On the other hand, 4, 8, 32 and 128 are terms of A000079.
The representation for each term is as follows.
= 1! + 0
= 1! + 3 = 2! + 2
= 3! + 2
= 1! + 8 = 3! + 3
= 4! + 1
= 3! + 21 = 4! + 3
= 4! + 8
= 2! + 34
= 5! + 1
= 5! + 5
= 5! + 8
= 5! + 2584
= 7! + 1
= 7! + 144

			128 is a term because 128 = 2^7 and 128 = 5! + 8, where 8 is a Fibonacci number.

Cf. A000045, A000142, A001597, A227644, A272575.

A380318 Product of the first n perfect powers (A001597).

Original entry on oeis.org

1, 1, 4, 32, 288, 4608, 115200, 3110400, 99532800, 3583180800, 175575859200, 11236854988800, 910185254092800, 91018525409280000, 11013241574522880000, 1376655196815360000000, 176211865192366080000000, 25374508587700715520000000, 4288291951321420922880000000, 840505222458998500884480000000, 181549128051143676191047680000000

Offset: 0

Ilya Gutkovskiy, Jan 20 2025

nonn

Eric Weisstein's World of Mathematics, Perfect Power.

Cf. A000442, A001044, A001597, A002110, A024923, A036691, A076408, A111059.

Join[{1},FoldList[Times,Join[{1},Select[Range[250],GCD@@FactorInteger[#][[All,2]]>1&]]]] (* Harvey P. Dale, May 03 2025 *)

A386242 a(n) is the least perfect power A001597 with binary weight n.

Original entry on oeis.org

1, 9, 25, 27, 121, 125, 1521, 2025, 5625, 24025, 42875, 59319, 32761, 393129, 851929, 1540081, 6275025, 15327225, 27258841, 41925625, 127893481, 243204025, 385611769, 268336125, 1979449081, 4823441401, 12870221809, 25698491351, 51354402813, 127506840561, 205822820329

Offset: 1

Hugo Pfoertner, Jul 23 2025

Cf. A000120, A001597, A231897.

upto = 10^11; L = Table[2 upto, {2 + Log2@ upto}]; Do[n = 1; While[(v = n^k) <= upto, nb = Plus @@ IntegerDigits[v, 2]; If[L[[nb]] > v, L[[nb]] = v]; n++], {k, 2, Log2[upto]}]; Take[L, Position[L, 2 upto][[1, 1]] - 1] (* Giovanni Resta, Jul 23 2025 *)

a(27)-a(31) from Giovanni Resta, Jul 23 2025

cp's OEIS Frontend

A075760 Nontrivial binomial coefficients which are perfect powers (A001597).

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A088413 A088259 indexed by A001597.

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A284743 Positive numbers that are not the sum of (any number of) distinct perfect powers (A001597).

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A294076 Absolute difference between n-th stella octangula number (A007588) and the nearest perfect power (A001597).

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A326119 a(n) is the absolute value of the alternating sum of the first n increasing perfect powers (A001597): 1, 1-4, 1-4+8, 1-4+8-9, ...

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A332008 Numbers k such that phi(k) and phi(k+1) are perfect powers (A001597).

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Magma

Mathematica

PARI

A362426 Number of compositions (ordered partitions) of n into 2 perfect powers (A001597).

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Mathematica

A371223 Perfect powers (A001597) equal to the sum of a factorial number (A000142) and a Fibonacci number (A000045).

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Examples

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A380318 Product of the first n perfect powers (A001597).

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