cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 18 results. Next

A363190 Odd powerful numbers (A062739) k such that the next powerful number after k is also odd.

Original entry on oeis.org

25, 121, 225, 343, 1089, 1323, 2187, 2197, 3025, 3087, 3249, 5929, 6125, 6859, 7803, 8575, 9261, 10125, 11881, 11907, 14161, 15125, 16641, 16807, 19683, 19773, 21025, 22707, 25921, 27889, 29241, 29791, 30375, 33275, 36125, 41067, 42849, 44217, 45125, 45369, 49729
Offset: 1

Views

Author

Amiram Eldar, May 21 2023

Keywords

Comments

A076445 is a subsequence if there are no three consecutive integers that are powerful numbers (A001694).

Examples

			25 = 5^2 is a term since it is an odd powerful number and the next powerful number, 27 = 3^3, is also odd.

Links

Crossrefs

Cf. A001694, A062739, A076445, A363189, A363191, A363192.

Programs

  • Mathematica
    With[{pow = Select[Range[10^5], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]}, pow[[Select[Range[Length[pow] - 1], OddQ[pow[[#]]] && OddQ[pow[[#+1]]] &]]]]
  • PARI
    lista(kmax) = {my(c = 0); for(k = 1, kmax, if(ispowerful(k), c++; if(k%2, print1(c, ", ")))); }

A200049 Positions of squares of odd primes among odd powerful numbers A062739.

Original entry on oeis.org

2, 3, 5, 7, 9, 12, 14, 16, 20, 21, 27, 29, 30, 34, 37, 44, 45, 48, 52, 53, 58, 61, 65, 71, 75, 76, 78, 79, 84, 93, 97, 100, 101, 109, 111, 115, 119, 122, 128, 132, 133, 142, 144, 146, 147, 157, 165, 169, 170, 172, 178, 180, 188, 193, 198, 202, 203, 207, 211, 213, 221, 231, 233, 234, 238, 251, 254, 261, 263, 267, 271
Offset: 1

Views

Author

Zak Seidov, Nov 13 2011

Keywords

Links

Crossrefs

Cf. A000040, A062739.

Programs

  • PARI
    lista(nn) = select(x->(issquare(x) && isprime(sqrtint(x))), select(x->(ispowerful(x) && (x % 2)), vector(nn, k, k)), 1); \\ Michel Marcus, Jun 18 2017

Formula

A062739(a(n)) = (A000040(n+1))^2.

A018256 Divisors of 36.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 12, 18, 36
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

36 is a highly composite number: A002182(7)=36. - Reinhard Zumkeller, Jun 21 2010
Numbers with all prime indices and exponents <= 2. Reversing inequalities gives A062739, strict A353502. - Gus Wiseman, Jun 28 2022

Links

Crossrefs

Cf. A018253, A018261, A018266, A018293, A018321, A018350, A018412, A018609, A018676, A178877, A178878, A165412, A178858, A178859, A178860, A178861, A178862, A178863, A178864.
Cf. A003586, A004709, A018338, A018710, A062739, A353502.

Programs

Formula

Intersection of A003586 (3-smooth) and A004709 (cubefree). - Gus Wiseman, Jun 28 2022

A045967 a(1)=4; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+1}^{e_i+1}.

Original entry on oeis.org

4, 9, 25, 27, 49, 225, 121, 81, 125, 441, 169, 675, 289, 1089, 1225, 243, 361, 1125, 529, 1323, 3025, 1521, 841, 2025, 343, 2601, 625, 3267, 961, 11025, 1369, 729, 4225, 3249, 5929, 3375, 1681, 4761, 7225, 3969, 1849, 27225, 2209, 4563, 6125, 7569, 2809, 6075
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

If we had a(1) = 1 (instead of 4), then this would be multiplicative and a permutation of the odd powerful numbers (A062739). - Amiram Eldar, Aug 11 2022

References

Links

Crossrefs

Cf. A045966, A027748, A062739, A124010, A000040.
Cf. A151800.

Programs

  • Haskell
    a045967 1 = 4
a045967 n = product $ zipWith (^)
            (map a151800 $ a027748_row n) (map (+ 1) $ a124010_row n)
-- Reinhard Zumkeller, Jun 03 2013, Dec 23 2011
  • Mathematica
    a[1]=4; a[n_] := Thread[f = FactorInteger[n]; Times @@ Power[f[[All, 1]] // NextPrime , f[[All, 2]] + 1]]; Array[a, 50] (* Jean-François Alcover, Feb 03 2015 *)

Formula

Sum_{n>=1} 1/a(n) = 2*zeta(2)*zeta(3)/(3*zeta(6)) - 3/4. - Amiram Eldar, Aug 11 2022

Extensions

More terms from David W. Wilson

A335851 Numbers that are powerful in Gaussian integers.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 18, 25, 27, 32, 36, 49, 50, 54, 64, 72, 81, 98, 100, 108, 121, 125, 128, 144, 162, 169, 196, 200, 216, 225, 242, 243, 250, 256, 288, 289, 324, 338, 343, 361, 392, 400, 432, 441, 450, 484, 486, 500, 512, 529, 576, 578, 625, 648, 675, 676, 686
Offset: 1

Views

Author

Amiram Eldar, Jun 26 2020

Keywords

Comments

Numbers all of whose prime factors in Gaussian integers have multiplicity larger than 1.
The even powerful numbers divided by 4. - Amiram Eldar, May 28 2023

Examples

			2 is a term since 2 = -i * (1 + i)^2 in the ring of Gaussian integers. -i is a unit, and the multiplicity of its only Gaussian prime factor, 1 + i, is 2.

Links

Crossrefs

Disjoint union of A001694 and 2 * A062739.
Cf. A082695.

Programs

  • Mathematica
    gaussPowerQ[n_] := AllTrue[FactorInteger[n, GaussianIntegers -> True], Abs[First[#]] == 1 || Last[#] > 1 &]; Select[Range[1000], gaussPowerQ]

Formula

Sum_{n>=1} 1/a(n) = (4/3) * Sum_{n>=1} 1/A001694(n) = 4*zeta(2)*zeta(3)/(3*zeta(6)) = (4/3) * A082695 = 2.591461...

A363189 Indices of the odd terms in the sequence of powerful numbers (A001694).

Original entry on oeis.org

1, 4, 6, 7, 10, 13, 16, 17, 20, 24, 25, 28, 30, 31, 35, 39, 41, 43, 45, 48, 51, 56, 57, 60, 62, 63, 65, 68, 71, 75, 79, 82, 83, 84, 87, 90, 94, 97, 98, 99, 102, 103, 105, 107, 110, 114, 117, 120, 122, 125, 127, 129, 133, 138, 141, 142, 144, 145, 148, 151, 152
Offset: 1

Views

Author

Amiram Eldar, May 21 2023

Keywords

Comments

The asymptotic density of this sequence is (2-sqrt(2))/(3-sqrt(2)) = 0.369398... .
If A001694(k) is a term of A363190 then k and k+1 are consecutive integers in this sequence.

Examples

			The first 6 powerful numbers are 1, 4, 8, 9, 16 and 25. 1, 9 and 25 are odd and their positions in the sequence are 1, 4 and 6, respectively.

Links

Crossrefs

Cf. A001694, A062739, A363190, A363191, A363192.

Programs

  • Mathematica
    Position[Select[Range[7000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &], _?(OddQ[#] &)] // Flatten
  • PARI
    lista(kmax) = {my(c = 0); for(k = 1, kmax, if(ispowerful(k), c++; if(k%2, print1(c, ", ")))); }

A244623 Odd prime powers that are not primes.

Original entry on oeis.org

1, 9, 25, 27, 49, 81, 121, 125, 169, 243, 289, 343, 361, 529, 625, 729, 841, 961, 1331, 1369, 1681, 1849, 2187, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 14641, 15625, 16129, 16807, 17161, 18769, 19321, 19683
Offset: 1

Views

Author

Jani Melik, Jul 02 2014

Keywords

Comments

Intersection of A061345 and A014076.
A014076 set minus A061346.

Links

Crossrefs

Intersection of A005408 and A025475.
Cf. A061345 (odd prime powers), A061346 (odd neither prime nor prime power), A062739 (odd powerful), A075109 (perfect powers), A136141.

Programs

  • Mathematica
    Join[{1},Select[Range[1,20001,2],PrimePowerQ[#]&&(!PrimeQ[#])&]] (* Harvey P. Dale, Dec 11 2018 *)
  • PARI
    isok(p) = ((p%2) && !isprime(p) && isprimepower(p)) || (p==1); \\ Michel Marcus, Jul 06 2021
  • Sage
    def isA244623(n) :
   return(n % 2 == 1 and is_prime_power(n) == 1 and is_prime(n) == 0)
[n for n in (1..20000) if isA244623(n)]

Formula

a(n) = A079290(n) at least in the range n=3..94, and perhaps beyond. - R. J. Mathar, Aug 20 2014
Sum_{n>=1} 1/a(n) = 1/2 + Sum_{p prime} 1/(p*(p-1)) = 1/2 + A136141. - Amiram Eldar, Dec 21 2020

A363191 a(n) is the least start of a run of exactly n consecutive powerful numbers (A001694) that are even, or -1 if no such run exists.

Original entry on oeis.org

16, 4, 196, 968, 8712, 437400, 85730400, 5030690600, 264615012500, 5239012864, 550886816376, 2494017320776852
Offset: 1

Views

Author

Amiram Eldar, May 21 2023

Keywords

Comments

No more terms below 10^18.
At most one of the n even consecutive powerful numbers in the run is a perfect square. - David A. Corneth, May 21 2023

Examples

			a(1) = 16, since 16 = 2^4 is an even powerful number, preceded by an odd powerful number, 9 = 3^2, and followed by an odd powerful number, 25 = 5^2.
a(2) = 4, since 4 = 2^2 and 8 = 2^3 are two consecutive even powerful numbers, preceded by an odd powerful number, 1, and followed by an odd powerful number, 9 = 3^2.

Crossrefs

Cf. A001694, A062739, A349062, A363189, A363190, A363192.

Programs

  • Mathematica
    seq[lim_] := Module[{pow = Union[Flatten[Table[i^2*j^3, {j, 1, lim^(1/3)}, {i, 1, Sqrt[lim/j^3]}]]], s = {}, rem, ind}, rem = Mod[pow, 2]; Do[ind = SequencePosition[rem, Join[{1}, Table[0, {k}], {1}], 1]; If[ind == {}, Break[]]; AppendTo[s, pow[[ind[[1, 1]] + 1]]], {k, 1, Infinity}]; s]; seq[10^10]

A363192 a(n) is the least start of a run of exactly n consecutive powerful numbers (A001694) that are odd, or -1 if no such run exists.

Original entry on oeis.org

1, 25, 2187, 703125, 93096125, 10229709861, 197584409639, 32044275110699, 164029657560618375
Offset: 1

Views

Author

Amiram Eldar, May 21 2023

Keywords

Comments

No more terms below 10^18.
At most one of the n odd consecutive powerful numbers in the run is a perfect square. - David A. Corneth, May 21 2023

Examples

			a(1) = 1, since 1 is an odd powerful number, followed by an even powerful number, 4 = 2^2.
a(2) = 25, since 25 = 5^2 and 27 = 3^3 are two consecutive odd powerful numbers, preceded by an even powerful number, 16 = 2^4, and followed by an even powerful number, 32 = 2^5.

Crossrefs

Cf. A001694, A062739, A349062, A363189, A363190, A363191.

Programs

  • Mathematica
    seq[lim_] := Module[{pow = Union[Flatten[Table[i^2*j^3, {j, 1, lim^(1/3)}, {i, 1, Sqrt[lim/j^3]}]]], s = {}, rem, ind}, rem = Join[{0}, Mod[pow, 2]]; Do[ind = SequencePosition[rem, Join[{0}, Table[1, {k}], {0}], 1]; If[ind == {}, Break[]]; AppendTo[s, pow[[ind[[1, 1]]]]], {k, 1, Infinity}]; s]; seq[1.1*10^10]

A353502 Numbers with all prime indices and exponents > 2.

Original entry on oeis.org

1, 125, 343, 625, 1331, 2197, 2401, 3125, 4913, 6859, 12167, 14641, 15625, 16807, 24389, 28561, 29791, 42875, 50653, 68921, 78125, 79507, 83521, 103823, 117649, 130321, 148877, 161051, 166375, 205379, 214375, 226981, 274625, 279841, 300125, 300763, 357911
Offset: 1

Views

Author

Gus Wiseman, May 16 2022

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The initial terms together with their prime indices:
       1: {}
     125: {3,3,3}
     343: {4,4,4}
     625: {3,3,3,3}
    1331: {5,5,5}
    2197: {6,6,6}
    2401: {4,4,4,4}
    3125: {3,3,3,3,3}
    4913: {7,7,7}
    6859: {8,8,8}
   12167: {9,9,9}
   14641: {5,5,5,5}
   15625: {3,3,3,3,3,3}
   16807: {4,4,4,4,4}
   24389: {10,10,10}
   28561: {6,6,6,6}
   29791: {11,11,11}
   42875: {3,3,3,4,4,4}

Links

Crossrefs

The version for only parts is A007310, counted by A008483.
The version for <= 2 instead of > 2 is A018256, # of compositions A137200.
The version for only multiplicities is A036966, counted by A100405.
The version for indices and exponents prime (instead of > 2) is:
- listed by A346068
- counted by A351982
- only exponents: A056166, counted by A055923
- only parts: A076610, counted by A000607
The version for > 1 instead of > 2 is A062739, counted by A339222.
The version for compositions is counted by A353428, see A078012, A353400.
The partitions with these Heinz numbers are counted by A353501.
A000726 counts partitions with multiplicities <= 2, compositions A128695.
A001222 counts prime factors with multiplicity, distinct A001221.
A004250 counts partitions with some part > 2, compositions A008466.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A295341 counts partitions with some multiplicity > 2, compositions A335464.
Cf. A000720, A003963, A065483, A114901, A181819, A353503, A353508.

Programs

  • Mathematica
    Select[Range[10000],#==1||!MemberQ[FactorInteger[#],{?(#<5&),}|{,?(#<3&)}]&]

Formula

Sum_{n>=1} 1/a(n) = Product_{p prime > 3} (1 + 1/(p^2*(p-1))) = (72/95)*A065483 = 1.0154153584... . - Amiram Eldar, May 28 2022
Showing 1-10 of 18 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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