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 1-4 of 4 results.

A303277 If n = Product (p_j^k_j) then a(n) = (Sum (k_j))^(Sum (p_j)).

Original entry on oeis.org

1, 1, 1, 4, 1, 32, 1, 9, 8, 128, 1, 243, 1, 512, 256, 16, 1, 243, 1, 2187, 1024, 8192, 1, 1024, 32, 32768, 27, 19683, 1, 59049, 1, 25, 16384, 524288, 4096, 1024, 1, 2097152, 65536, 16384, 1, 531441, 1, 1594323, 6561, 33554432, 1, 3125, 128, 2187, 1048576, 14348907, 1, 1024, 65536
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 20 2018

Keywords

Examples

			a(48) = a(2^4 * 3^1) = (4 + 1)^(2 + 3) = 5^5 = 3125.

Links

Crossrefs

Cf. A000142, A000312, A001222, A002110, A007504, A008472, A008474, A008477, A022559, A034387, A039697, A088865, A263653, A285769.

Programs

  • Mathematica
    Join[{1}, Table[PrimeOmega[n]^DivisorSum[n, # &, PrimeQ[#] &], {n, 2, 55}]]
  • PARI
    a(n) = my(f=factor(n)); vecsum(f[,2])^vecsum(f[,1]); \\ Michel Marcus, Apr 21 2018

Formula

a(n) = bigomega(n)^sopf(n) = A001222(n)^A008472(n).
a(p^k) = k^p where p is a prime.
a(A000312(k)) = a(k)*k^A008472(k).
a(A000142(k)) = A022559(k)^A034387(k).
a(A002110(k)) = k^A007504(k).

A303278 If n = Product_j p_j^k_j where the p_j are distinct primes then a(n) = (Product_j k_j)^(Product_j p_j).

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 9, 8, 1, 1, 64, 1, 1, 1, 16, 1, 64, 1, 1024, 1, 1, 1, 729, 32, 1, 27, 16384, 1, 1, 1, 25, 1, 1, 1, 4096, 1, 1, 1, 59049, 1, 1, 1, 4194304, 32768, 1, 1, 4096, 128, 1024, 1, 67108864, 1, 729, 1, 4782969, 1, 1, 1, 1073741824, 1, 1, 2097152, 36, 1, 1, 1, 17179869184, 1, 1, 1, 46656, 1, 1, 32768
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 20 2018

Keywords

Comments

This is different from A008477, which is Product_j k_j^p_j. - N. J. A. Sloane, May 01 2021

Examples

			a(36) = a(2^2 * 3^2) = (2*2)^(2*3) = 4^6 = 4096.

Links

Crossrefs

Cf. A000026, A000142, A005117 (indices of ones), A005361, A007947, A008477, A034386, A039696, A135291, A285769, A303277.

Programs

  • Mathematica
    Table[Times@@Transpose[FactorInteger[n]][[2]]^Last[Select[Divisors[n], SquareFreeQ]], {n, 75}]
  • PARI
    a(n) = my(f=factor(n)); factorback(f[, 2])^factorback(f[, 1]); \\ Michel Marcus, Apr 21 2018

Formula

a(n) = tau(n/rad(n))^rad(n) = A005361(n)^A007947(n).
a(p^k) = k^p where p is a prime.
a(A000142(k)) = A135291(k)^A034386(k).

Extensions

Definition clarified by N. J. A. Sloane, May 01 2021

A306328 If n = Product (p_j^k_j) then a(n) = Sum (p_j)^Product (k_j).

Original entry on oeis.org

0, 2, 3, 4, 5, 5, 7, 8, 9, 7, 11, 25, 13, 9, 8, 16, 17, 25, 19, 49, 10, 13, 23, 125, 25, 15, 27, 81, 29, 10, 31, 32, 14, 19, 12, 625, 37, 21, 16, 343, 41, 12, 43, 169, 64, 25, 47, 625, 49, 49, 20, 225, 53, 125, 16, 729, 22, 31, 59, 100, 61, 33, 100, 64, 18, 16, 67, 361, 26, 14, 71, 15625, 73, 39, 64
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 07 2019

Keywords

Examples

			a(12) = a(2^2 * 3^1) = (2 + 3)^(2 * 1) = 25.

Crossrefs

Cf. A005361, A008472, A088865, A246655 (fixed points), A285769, A303277, A303278, A306329.

Programs

  • Mathematica
    Table[DivisorSum[n, # &, PrimeQ[#] &]^DivisorSigma[0, n/Last[Select[Divisors[n], SquareFreeQ]]], {n, 75}]

Formula

a(n) = sopf(n)^tau(n/rad(n)) = A008472(n)^A005361(n).

A306329 If n = Product (p_j^k_j) then a(n) = Product (p_j)^Sum (k_j).

Original entry on oeis.org

1, 2, 3, 4, 5, 36, 7, 8, 9, 100, 11, 216, 13, 196, 225, 16, 17, 216, 19, 1000, 441, 484, 23, 1296, 25, 676, 27, 2744, 29, 27000, 31, 32, 1089, 1156, 1225, 1296, 37, 1444, 1521, 10000, 41, 74088, 43, 10648, 3375, 2116, 47, 7776, 49, 1000, 2601, 17576, 53, 1296, 3025, 38416, 3249, 3364, 59, 810000
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 07 2019

Keywords

Examples

			a(12) = a(2^2 * 3^1) = (2 * 3)^(2 + 1) = 216.

Crossrefs

Cf. A000961 (fixed points), A001222, A007947, A088865, A285769, A303277, A303278, A306328.

Programs

  • Mathematica
    Table[Last[Select[Divisors[n], SquareFreeQ]]^PrimeOmega[n], {n, 60}]

Formula

a(n) = rad(n)^bigomega(n) = A007947(n)^A001222(n).
Showing 1-4 of 4 results.

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

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