A385454 - Contribution to The OEIS: Difference of the largest and smallest semiperimeters of an integral rectangle with area n.

 A385454

Difference of the largest and smallest semiperimeters of an integral rectangle with area n.

0

0, 0, 0, 1, 0, 2, 0, 3, 4, 4, 0, 6, 0, 6, 8, 9, 0, 10, 0, 12, 12, 10, 0, 15, 16, 12, 16, 18, 0, 20, 0, 21, 20, 16, 24, 25, 0, 18, 24, 28, 0, 30, 0, 30, 32, 22, 0, 35, 36, 36, 32, 36, 0, 40, 40, 42, 36, 28, 0, 45, 0, 30, 48, 49, 48, 50, 0, 48, 44, 54, 0, 56, 0

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OFFSET

1,6

COMMENTS

For all noncomposite n, a(n) = 0.

For each square k^2, a(k^2) = (k^2 + 1) - 2*k = (k-1)^2.

LINKS

James C. McMahon, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = 1 + n - A063655(n).

EXAMPLE

The largest semiperimeter of an integral rectangle with area 9 is 10 (1 x 9 rectangle); the smallest semiperimeter is 6 (3 x 3 rectangle). The difference, a(9) = 4.

MATHEMATICA

a[n_]:=1+n-2Median[Divisors[n]]; Array[a, 73]

CROSSREFS

Cf. A063655.

Sequence in context: A274441 A213859 A330492 * A350534 A101336 A300001

Adjacent sequences: A385442 A385443 A385444 * A385459 A385460 A385461

KEYWORD

nonn,new

AUTHOR

James C. McMahon, Jun 29 2025

STATUS

approved

Cotton Candy Nebula Photographed by the Vera C. Rubin Observatory

 

(Image credit: RubinObs/NOIRLab/SLAC/NSF/DOE/AURA)

The Vera C. Rubin Observatory recently came online and is producing wonderful photographs of space. The photo shown is the Cotton Candy nebula (Messier 20) located 5,000 light-years distant in the constellation Sagittarius.

Zoomable photo: https://noirlab.edu/public/images/noirlab2521ah/zoomable/

A385288 - Contribution to the OEIS: Numbers with a prime number of prime factors, counted with multiplicity, and whose prime factors are each raised to a prime exponent

 A385288

Numbers with a prime number of prime factors, counted with multiplicity, and whose prime factors are each raised to a prime exponent.

0

4, 8, 9, 25, 27, 32, 49, 72, 108, 121, 125, 128, 169, 200, 243, 288, 289, 343, 361, 392, 500, 529, 675, 800, 841, 961, 968, 972, 1125, 1323, 1331, 1352, 1369, 1372, 1568, 1681, 1800, 1849, 2048, 2187, 2197, 2209, 2312, 2700, 2809, 2888, 3087, 3125, 3267, 3481

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OFFSET

1,1

COMMENTS

a(n) = A114129(n) through n=25; then a(26) = 961 and A114129(26) = 864.

Subset of A056166.

Subset of A001694. - Michael De Vlieger, Jun 25 2025.

LINKS

James C. McMahon, Table of n, a(n) for n = 1..1000

EXAMPLE

200 = 2^3 * 5^2; 200 has a prime number of prime factors, counted with multiplicity (3 + 2 = 5), and exponents 3 and 2 are prime.

MATHEMATICA

Select[Range[10^4], AllTrue[Last/@FactorInteger[#], PrimeQ]&&PrimeQ[PrimeOmega[#]]&]

PROG

(PARI) isok(k) = my(f=factor(k)); isprime(bigomega(k)) && (sum(k=1, #f~, isprime(f[k, 2])) == omega(f)); \\ Michel Marcus, Jun 25 2025

CROSSREFS

Cf. A001248, A001694, A030078, A056166, A114129.

Sequence in context: A383211 A051676 A114129 * A383103 A053810 A076702

Adjacent sequences: A385283 A385284 A385286 * A385289 A385292 A385293

KEYWORD

nonn,new

AUTHOR

James C. McMahon, Jun 24 2025

STATUS

approved

Online Encyclopedia of Integer Sequences, OEIS.