A143512 -id:A143512 - cp's OEIS frontend

A143513 Numbers of the form 2^a * 3^b * 5^c * 17^d * 257^e * 65537^f; products of 2 and the Fermat primes.

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 17, 18, 20, 24, 25, 27, 30, 32, 34, 36, 40, 45, 48, 50, 51, 54, 60, 64, 68, 72, 75, 80, 81, 85, 90, 96, 100, 102, 108, 120, 125, 128, 135, 136, 144, 150, 153, 160, 162, 170, 180, 192, 200, 204, 216, 225, 240, 243, 250, 255, 256

Offset: 1

T. D. Noe, Aug 21 2008

nonn

Similar to A003401, which allows each Fermat prime to occur 0 or 1 times. Euler's phi function maps this sequence into itself. The odd terms of this sequence are in A143512.

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

nn=34; logs=Log[2.,{2,3,5,17,257,65537}]; lim=Floor[nn/logs]; t={}; Do[z={i,j,k,l,m,n}.logs; If[z

Sum_{a(n) is odd} 1/a(n) = Sum_{a(n) is even} 1/a(n). If there are only five Fermat primes: 3,5,17,257,65537 (this is a well-known conjecture), then we have exactly Sum_{n>=1} 1/a(n) = 4294967295/1073741824 = 3.999999999068677425384521484375, which is twice the sum of the reciprocals of A143512. - Vladimir Shevelev and T. D. Noe, Dec 01 2010

A143510 Numbers m such that the equation phi(x) = m has even but no odd solutions.

Original entry on oeis.org

16842752, 33685504, 67371008, 134742016, 269484032, 538968064, 1077936128, 2155872256, 4294967296

Offset: 1

T. D. Noe, Aug 21 2008

In the unlikely event that Carmichael's conjecture is proved false, the counterexamples will be in this sequence. The number a(1) = 16842752 = 257*2^16 is mentioned in problem E3361. If there are only five Fermat primes, then 2^k is in this sequence for all k>31. It appears that for every product d of Fermat primes (A143512), the number 2^k * d is in this sequence for some k. The link to "Numbers Like 16842752" lists examples for various d.

Conjecture: if the least solution to phi(x) = m is even, then m is in this sequence. - Jianing Song, Nov 07 2022

R. K. Guy, Unsolved problems in number theory, B39.

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
T. D. Noe, Numbers Like 16842752
William P. Wardlaw, L. L. Foster and R. J. Simpson, Problem E3361, Amer. Math. Monthly, Vol. 98, No. 5 (May, 1991), 443-444.
Eric Weisstein's World of Mathematics, Carmichael's Totient Function Conjecture

Cf. A143511 (least k such that phi(k)=m).

isok(k) =  numinvphi(k) && select(x->((x%2) == 1), invphi(k)) == 0; \\ using invphi from PARI scripts link; Michel Marcus, Oct 09 2023; corrected by Max Alekseyev, Oct 14 2023

Definition corrected by Max Alekseyev, Oct 14 2023

A174688 All different products of not necessarily distinct terms of A001317.

Original entry on oeis.org

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

Vladimir Shevelev, Dec 01 2010

nonn

Sequence differs from A143512 beginning with a(970).

			9 = 3^2 is a term since 3 is in A001317.

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

Cf. A001317, A000215, A143512.

f[n_] := FromDigits[Table[Mod[Binomial[n, k], 2], {k, 0, n}], 2]; n = 13; v = Array[f, n, 0]; vmax = v[[-1]]; s = {1}; Do[v1 = v[[k]]; rmax = Floor[Log[v1, vmax]]; s1 = v1^Range[0, rmax]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= vmax &]; s = Union[s, s2], {k, 2, n}]; s (* Amiram Eldar, Sep 27 2020 *)

Sum_{n>=1} 1/a(n) = 2.
Let m_a(n) = (-1)^A010060(n), if n is squarefree, and 0, otherwise (a-analog of Möbius function). Then Sum_{n>=1} m_a(n)/a(n) = 1/2.
A generalization: Sum_{n>=1} 1/(a(n))^s = Product_{Fermat numbers F} (1-F^(-s))^(-1), where s>0 (an analog of Euler identity for primes, where, for real s, s>1).

Offset corrected and more terms added by Amiram Eldar, Sep 27 2020

A339345 Lexicographically earliest sequence of odd numbers such that the asymptotic density of the numbers which are coprime to all the terms of this sequence is 1/2.

Original entry on oeis.org

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

Amiram Eldar, Nov 30 2020

nonn

Numbers whose prime divisors are all in A339344.
Closed under multiplication.
First differs from A143512 and A174688 at n = 970.

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

Cf. A143512, A174688, A339344.

seq[m_] := Module[{v = {1}, r = 1, p = 3, k, n = m + 1, s = {1}, v1, s1, s2, rmax}, Do[AppendTo[v, p]; r *= 1 - 1/p; p = NextPrime[r/(r - 1/2)], {m}]; vmax = v[[-1]]; Do[v1 = v[[k]]; rmax = Floor[Log[vmax]/Log[v1]]; s1 = v1^Range[0, rmax]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= vmax &]; s = Union[s, s2], {k, 2, n}]; s]; seq[5]

Sum_{n>=1} 1/a(n) = 2.

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

A332334 Let a(1) = a(2) = 1, and for n > 2 let a(n) = p where p is the largest prime such that p# divides phi(n), where phi is Euler's totient function and # is the primorial.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 3, 3, 2, 3, 2, 2, 2, 2, 3, 3, 3, 2, 2, 5, 2, 2, 2, 3, 3, 3, 3, 3, 2, 2, 3, 3, 2, 3, 2, 2, 2, 3, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 2, 5, 5, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 3, 3, 2, 3, 5, 3, 3, 2, 3, 2, 2, 3, 2, 3, 2

Offset: 1

Charles R Greathouse IV, Feb 09 2020

nonn

Pollack and Pomerance show that the normal order of a(n) is log log n/log log log n. The maximal order is log n (for primorial primes A018239, by the prime number theorem) and the minimal order, for n > 2, is 2 (for products of Fermat primes A143512, apart from 1).

Paul Pollack and Carl Pomerance, Phi, primorials, and Poisson, arXiv:2001.06727 [math.NT], 2020.

Cf. A018239, A143512.

a(n)=my(ph=eulerphi(n)); my(p=1); forprime(q=2,, if(ph%q, return(p), p=q))

cp's OEIS Frontend

A143513 Numbers of the form 2^a * 3^b * 5^c * 17^d * 257^e * 65537^f; products of 2 and the Fermat primes.

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A143510 Numbers m such that the equation phi(x) = m has even but no odd solutions.

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A174688 All different products of not necessarily distinct terms of A001317.

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A339345 Lexicographically earliest sequence of odd numbers such that the asymptotic density of the numbers which are coprime to all the terms of this sequence is 1/2.

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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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A332334 Let a(1) = a(2) = 1, and for n > 2 let a(n) = p where p is the largest prime such that p# divides phi(n), where phi is Euler's totient function and # is the primorial.

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