A143513 -id:A143513 - cp's OEIS frontend

A143512 Numbers of the form 3^a * 5^b * 17^c * 257^d * 65537^e; products of Fermat primes.

1, 3, 5, 9, 15, 17, 25, 27, 45, 51, 75, 81, 85, 125, 135, 153, 225, 243, 255, 257, 289, 375, 405, 425, 459, 625, 675, 729, 765, 771, 867, 1125, 1215, 1275, 1285, 1377, 1445, 1875, 2025, 2125, 2187, 2295, 2313, 2601, 3125, 3375, 3645, 3825, 3855, 4131, 4335, 4369

Offset: 1

T. D. Noe, Aug 21 2008

nonn

Similar to A004729, which allows each Fermat prime to occur 0 or 1 times. Applying Euler's phi function to these numbers produces numbers in A143513.

If the well-known conjecture that there are only five prime Fermat numbers F_k = 2^(2^k) + 1, k=0,1,2,3,4, is true, then we have exactly Sum_{n>=1} 1/a(n) = Product_{k=0..4} F_k/(F_k-1) = 4294967295/2147483648 = 1.9999999995343387126922607421875. - Vladimir Shevelev and T. D. Noe, Dec 01 2010

T. D. Noe, Table of n, a(n) for n = 1..10000

nn=60; logs=Log[2.,{3,5,17,257,65537}]; lim=Floor[nn/logs]; t={}; Do[z={i,j,k,l,m}.logs; If[z

A368543 The number of divisors of n whose prime factors are all of the form 2^k + 1 (A092506).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 1, 4, 3, 4, 1, 6, 1, 2, 4, 5, 2, 6, 1, 6, 2, 2, 1, 8, 3, 2, 4, 3, 1, 8, 1, 6, 2, 4, 2, 9, 1, 2, 2, 8, 1, 4, 1, 3, 6, 2, 1, 10, 1, 6, 4, 3, 1, 8, 2, 4, 2, 2, 1, 12, 1, 2, 3, 7, 2, 4, 1, 6, 2, 4, 1, 12, 1, 2, 6, 3, 1, 4, 1, 10, 5, 2, 1, 6, 4, 2, 2, 4, 1, 12, 1, 3, 2, 2, 2, 12, 1, 2, 3, 9

Offset: 1

Amiram Eldar, Dec 29 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A019434, A092506, A143513.

q[n_] := AllTrue[FactorInteger[n][[;; , 1]], # - 1 == 2^IntegerExponent[# - 1, 2] &]; f[p_, e_] := If[q[p], e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f=factor(n)); prod(i=1, #f~, if((f[i,1]-1) >> valuation(f[i,1]-1, 2) == 1 , f[i,2] + 1, 1))};

Multiplicative with a(p^e) = e+1 if p is in A092506 (i.e., p is either 2 or a Fermat prime), and 1 otherwise.
a(n) >= 1, with equality if and only if all the prime factors of n are not of the form 2^k + 1.
a(n) <= A000005(n), with equality if and only if all the prime factors of n are in A092506 (n is in A143513 assuming that there are only 5 Fermat primes).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/Product_{k>=1} (1 - 1/A092506(k)) = 3.99999999906867742538... . This value is exactly 4294967295/1073741824 if there are only 5 Fermat primes.

A305759 Numbers that can be factored as a product of numbers of the form 2^k+1 (A000051).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 17, 18, 20, 24, 25, 27, 30, 32, 33, 34, 36, 40, 45, 48, 50, 51, 54, 60, 64, 65, 66, 68, 72, 75, 80, 81, 85, 90, 96, 99, 100, 102, 108, 120, 125, 128, 129, 130, 132, 135, 136, 144, 150, 153, 160, 162, 165, 170, 180, 192

Offset: 1

Nicholas Stearns, Jun 10 2018

nonn

If a(n) and a(m) are in the sequence, so is a(n)*a(m).

			a(11) = 15 = 3*5 = (2^1 + 1)*(2^2 + 1).

Cf. A000051, A143512, A143513.

up = 192; t = Complement[1+2^Range[0, Ceiling@Log2@up], {9}]; a = {}; ric[p_, w_] := Block[{q = p}, If[w == {}, AppendTo[a, p], While[q <= up, ric[q, Rest@w]; q *= w[[1]]]]]; ric[1, t]; Union[a] (* Giovanni Resta, Jun 14 2018 *)

More terms from Giovanni Resta, Jun 14 2018

cp's OEIS Frontend

A143512 Numbers of the form 3^a * 5^b * 17^c * 257^d * 65537^e; products of Fermat primes.

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A368543 The number of divisors of n whose prime factors are all of the form 2^k + 1 (A092506).

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A305759 Numbers that can be factored as a product of numbers of the form 2^k+1 (A000051).

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