A001597 -id:A001597 - cp's OEIS frontend

A175052 Perfect powers (members of A001597) n where the next larger perfect power is congruent mod 2 to n.

4, 25, 32, 121, 128, 196, 225, 343, 484, 1000, 1331, 1728, 2048, 2187, 2197, 2704, 3025, 3375, 4913, 5776, 6859, 7744, 8000, 8100, 9261, 10648, 12167, 13824, 16641, 17424, 19683, 21904, 24389, 26896, 29791, 32768, 35721, 39204, 42849, 50625

Offset: 1

Leroy Quet, Dec 08 2009

nonn

			25 (25 = 5^2) are 27 (27 = 3^3) are consecutive perfect powers. Since both are odd, then 25 is in this sequence.
128 (128 = 2^7) and 144 (144 = 12^2) are consecutive perfect powers. Since both are even, then 128 is in this sequence.

Cf. A001597, A175053

Extended by Ray Chandler, Dec 10 2009

A285845 Powers (A001597) that are also cyclops numbers (A134808).

Original entry on oeis.org

11025, 19044, 21025, 24025, 32041, 38025, 42025, 47089, 51076, 58081, 59049, 65025, 66049, 67081, 74088, 75076, 87025, 93025, 1110916, 1140624, 1170724, 1190281, 1240996, 1270129, 1290496, 1340964, 1350244, 1380625, 1420864, 1430416, 1490841, 1510441

Offset: 1

Colin Barker, Apr 27 2017

The first term not in A160711 is 74088 = 42^3.

Intersection of A001597 and A134808. - Robert G. Wilson v, Apr 27 2017

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1156 terms from Robert G. Wilson v)

Cf. A001597, A134808, A160711.

Select[NestList[If[# == 1, 4, Min@ Table[(Floor[#^(1/k)] + 1)^k, {k, 2, 1 + Floor@ Log2@ #}]] &, 1, 1400], Function[n, And[OddQ@ Length@ #, #[[ Ceiling[Length[#]/2] ]] == 0, DigitCount[n, 10, 0] == 1] &@ IntegerDigits@ n]] (* Michael De Vlieger, Apr 27 2017, after Robert G. Wilson v at A001597 *)
cyclopsQ[n_Integer, b_: 10] := Module[{digitList = IntegerDigits[n, b], len, pos0s, flag}, len = Length[digitList]; pos0s = Select[Range[len], digitList[[#]] == 0 &]; flag = OddQ[len] && (Length[pos0s] == 1) && (pos0s == {(len + 1)/2}); Return[flag]]; (* from Alonso del Arte in A134808 *) min = 0; max = 1520000; t = Union@ Flatten@ Table[n^expo, {expo, Prime@ Range@ PrimePi@ Log2@ max}, {n, Floor[1 + min^(1/expo)], max^(1/expo)}]; Select[t, cyclopsQ] (* Robert G. Wilson v, Apr 27 2017 *)

is_cyclops(k) = {
  if(k==0, return(1));
  my(d=digits(k), j);
  if(#d%2==0 || d[#d\2+1]!=0, return(0));
  for(j=1, #d\2, if(d[j]==0, return(0)));
  for(j=#d\2+2, #d, if(d[j]==0, return(0)));
  return(1)}
L=List(); for(n=1, 100000, if(ispower(n) && is_cyclops(n), listput(L, n))); Vec(L)

A293190 a(n) = |{A001597(n) <= k <= A001597(n+1): 2*k^2-1 is prime}|.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 01 2017

Conjecture: (i) a(n) > 0 for all n > 0. In other words, for any perfect powers x^m and y^n with 0 < x^m < y^n, there is an integer z with x^m <= z <= y^n such that 2*z^2 - 1 is prime.
(ii) For any perfect powers x^m and y^n with 0 < x^m < y^n, there is an integer z with x^m <= z <= y^n such that 2*z + 3 (or 20*z^2 + 3) is prime.
(iii) For perfect powers x^m and y^n with 0 < x^m < y^n, there is a practical number q (cf. A005153) with x^m <= q <= y^n, unless x^m = 5^2 and y^n = 3^3, or x^m = 11^2 and y^n = 5^3, or x^m = 22434^2 and y^n = 55^5.
Compare this with the Redmond-Sun conjecture.

			a(1) = 3 since 2*2^2 - 1, 2*3^2-1 and 2*4^2-1 are all prime but 2*1^2 - 1 is not prime.
a(3) = 1 since A001597(3) = 8, A001597(4) = 9, 2*8^2 - 1 = 127 is prime but 2*9^2 - 1 is composite.
a(6) = 1 since A001597(6) = 25, A001597(7) = 27, 2*25^2 - 1 = 1249 is prime but 2*26^2 - 1 and 2*27^2 - 1 are composite.
a(14) = 1 since A001597(14) = 121, A001597(15) = 125, 2*125^2
- 1 = 31249 is prime but 2*k^2 - 1 is composite for every k = 121, 122, 123, 124.
a(361) = 1 since A001597(361) = 46^3 = 97336, A001597(362) = 312^2 = 97344, and k = 97342 is the only number among 97336,...,97344 with 2*k^2 - 1 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
Wikipedia, Redmond-Sun conjecture

Cf. A001597, A005153, A066049, A116086, A116455.

n=1;m=1;Do[Do[If[IntegerQ[k^(1/Prime[i])],Print[n," ",Sum[Boole[PrimeQ[2j^2-1]],{j,m,k}]];n=n+1;m=k;Goto[aa]],{i,1,PrimePi[Log[2,k]]}];Label[aa],{k,2,6561}]

A362422 Number of partitions of n into 2 perfect powers (A001597).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 19 2023

nonn

Eric Weisstein's World of Mathematics, Perfect Power.

Cf. A001597, A362423.

perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[;; , 2]] > 1; a[n_] := Count[IntegerPartitions[n, {2}], ?(AllTrue[#, perfectPowerQ] &)]; Array[a, 100, 0] (* _Amiram Eldar, May 05 2023 *)

A362423 Number of partitions of n into 3 perfect powers (A001597).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 1, 2, 1, 0, 2, 2, 2, 2, 1, 2, 1, 0, 2, 3, 3, 2, 2, 3, 2, 1, 3, 4, 3, 2, 4, 4, 1, 1, 3, 4, 2, 3, 3, 3, 3, 1, 3, 3, 4, 3, 3, 4, 3, 2, 2, 4, 5, 3, 4, 4, 2, 1, 4, 5, 2, 3, 3, 5, 1, 2, 4, 5, 4, 2, 4, 2, 4, 0, 3, 5, 5, 2

Offset: 0

Ilya Gutkovskiy, Apr 19 2023

nonn

Eric Weisstein's World of Mathematics, Perfect Power.

Cf. A001597, A362422.

perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[;; , 2]] > 1; a[n_] := Count[IntegerPartitions[n, {3}], ?(AllTrue[#, perfectPowerQ] &)]; Array[a, 100, 0] (* _Amiram Eldar, May 05 2023 *)

A362428 a(n) is the least positive integer that can be expressed as the sum of one or more consecutive perfect powers (A001597) in exactly n ways, or -1 if no such integer exists.

Original entry on oeis.org

1, 25, 441

Offset: 1

Ilya Gutkovskiy, Apr 19 2023

a(4) > 1.5 * 10^8.

No further terms < 2.8*10^14. - Michael S. Branicky, Apr 20 2023

			For n = 2: 25 = 25 = 9 + 16.
For n = 3: 441 = 441 = 216 + 225 = 128 + 144 + 169.

Eric Weisstein's World of Mathematics, Perfect Power

Cf. A001597.

A365295 a(n) is the least positive integer that can be expressed as the sum of two distinct perfect powers (A001597) in exactly n ways.

Original entry on oeis.org

1, 5, 17, 129, 468, 1025, 2628, 12025, 32045, 27625, 138125, 430625, 204425, 160225, 2010025, 2348125, 801125, 1743625, 2082925, 4978025, 4005625, 12325625, 30525625, 73046025, 5928325, 13287625, 46437625, 45177925, 35409725, 120737825, 52073125, 66438125, 29641625, 32846125, 956974625

Offset: 0

Ilya Gutkovskiy, Aug 31 2023

nonn

			For n = 2: a(2) = 17 = 1^2 + 2^4 = 2^3 + 3^2.
a(6) = 2628 via 3^3 + 51^2 = 2^7 + 50^2 = 18^2 + 48^2 = 21^2 + 3^7 = 2^9 + 46^2 = 30^2 + 12^3. - _David A. Corneth_, Sep 09 2023

Karl-Heinz Hofmann and Hugo Pfoertner, Table of n, a(n) for n = 0..77
Karl-Heinz Hofmann, Python program
Hugo Pfoertner, PARI program and results (terms < 2*10^9), Sep 10 2023.

Cf. A001597, A093195, A362424, A363040.

upto(n) = {n = (sqrtint(n) + 1)^2; my(v = vector(n), pows = List([1]), r = -1, res = []); for(j = 2, logint(n, 2), for(i = 2, sqrtnint(n, j), listput(pows, i^j))); pows = Set(pows); for(i = 1, #pows - 1, j = i+1; c = pows[i] + pows[j]; while(c <= n, v[c]++; j++; c = pows[i] + pows[j])); for(i = 1, #v, c = v[i]+1; if(c > #res, res = concat(res, vector(c - #res, j, oo))); if(i < res[c], res[c] = i)); res} \\ David A. Corneth, Sep 08 2023

PARI
```
\\ see link
```
Python
```
# see link
```

a(8)-a(10) from David Consiglio, Jr., Sep 08 2023

a(9) corrected and a(11)-a(34) from Hugo Pfoertner, Sep 10 2023

A373116 Fibonacci numbers whose digits product is a positive perfect power (A001597).

Original entry on oeis.org

1, 8, 55, 144, 4181, 17711, 196418, 1346269, 259695496911122585

Offset: 1

Gonzalo Martínez, May 25 2024

Since the product of the digits of Fibonacci(k) is required to be positive, Fibonacci(k) does not have zero as a digit. For this reason this list is probably finite, since it is conjectured that there are only finitely many Fibonacci numbers that do not contain the digit zero (see A076564). If the conjecture is true, the largest number possessing the property would be Fibonacci(85) = 259695496911122585 whose digit product is 194400^2.

Unlike A373049, here the product uses all the digits of Fibonacci(k).

			196418 is a term, because Fibonacci(27) = 196418 and the product of its digits is 1*9*6*4*1*8 = 12^3.

Cf. A000045, A001597, A076564, A246558, A370071, A373049.

powQ[n_] := n == 1 || GCD @@ FactorInteger[n][[;; , 2]] > 1; Select[Fibonacci[Range[2, 100]], powQ[Times @@ IntegerDigits[#]] &] (* Amiram Eldar, May 25 2024 *)

A380564 Triangle read by rows q(b1,b2) with 1A001597), and b1 is m^j, 0

Original entry on oeis.org

9, 0, 265, 5, 117032, 2123333591, 7, 44, 291720, 10757067, 57, 449, 16879, 18042, 19032324921, 0, 332930, 0, 2180306, 174631931663663360, 51981761666123, 9, 0, 93, 839, 407917265, 50732175, 197761284636128964, 10, 10, 133302001, 124343, 155133423353

Offset: 1

Dominic McCarty and Robert G. Wilson v, Feb 03 2025

a(703) has over 1000 digits, and so it is not included in the b-file.

			a(1) = floor(A380599);
a(3) = q(3,4) = 265_10 = 10021_4 = 100211_3;
a(5) = q(4,5) = 117032_10 = 12221112_5 = 12221112112_3;
a(29) = floor(A379651); etc.
Lower triangle of an array:
   \b1  2       3           4         5                   6               7
  b2\
   3|   9;
   4|   0,    265;
   5|   5, 117032, 2123333591;
   6|   7,     44,     291720, 10757067;
   7|  57,    449,      16879,    18042,        19032324921;
   8|   0, 332930,          0,  2180306, 174631931663663360, 51981761666123;
       ...

Dominic McCarty, Table of n, a(n) for n = 1..702 (Rows 1..37).
Dominic McCarty, Python program
Dominic McCarty, Illustrations of first 1830 terms

Cf. A181929, A379651, A380599.

A074321 Numbers n such that n-th perfect power + n is prime: n + A001597(n) is prime.

Original entry on oeis.org

1, 3, 4, 6, 10, 13, 24, 25, 34, 37, 49, 50, 52, 73, 74, 93, 106, 109, 127, 142, 160, 177, 178, 193, 208, 212, 228, 234, 235, 250, 253, 262, 275, 276, 279, 281, 282, 284, 298, 333, 336, 337, 339, 360, 385, 402, 405, 407, 417, 442, 468, 474, 478, 480, 487

Offset: 1

Zak Seidov, Oct 11 2002

Primes of the form n + A001597(n) in A075781.

			n=3: pp(3)+3=8+3=11 is prime.

Cf. A001597, A075781.

cp's OEIS Frontend

A175052 Perfect powers (members of A001597) n where the next larger perfect power is congruent mod 2 to n.

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A285845 Powers (A001597) that are also cyclops numbers (A134808).

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PARI

A293190 a(n) = |{A001597(n) <= k <= A001597(n+1): 2*k^2-1 is prime}|.

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A362422 Number of partitions of n into 2 perfect powers (A001597).

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A362423 Number of partitions of n into 3 perfect powers (A001597).

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A362428 a(n) is the least positive integer that can be expressed as the sum of one or more consecutive perfect powers (A001597) in exactly n ways, or -1 if no such integer exists.

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A365295 a(n) is the least positive integer that can be expressed as the sum of two distinct perfect powers (A001597) in exactly n ways.

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PARI

PARI

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A373116 Fibonacci numbers whose digits product is a positive perfect power (A001597).

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Mathematica

A380564 Triangle read by rows q(b1,b2) with 1A001597), and b1 is m^j, 0

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