A122254 -id:A122254 - cp's OEIS frontend

A048135 Tomahawk-constructible n-gons.

3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 26, 27, 28, 30, 32, 34, 35, 36, 37, 38, 39, 40, 42, 45, 48, 51, 52, 54, 56, 57, 60, 63, 64, 65, 68, 70, 72, 73, 74, 76, 78, 80, 81, 84, 85, 90, 91, 95, 96, 97, 102, 104, 105, 108, 109, 111, 112

Offset: 1

David W. Wilson

nonn

Andrew M. Gleason, Angle Trisection, the Heptagon and the Triskaidecagon, American Mathematical Monthly, 95 (1988), 185-194.

Cf. A005109, A048136, A122254, A122255.

from itertools import count, islice
from sympy import primefactors, totient
def A048135_gen(): # generator of terms
    yield from filter(lambda n: set(primefactors(totient(n))) <= {2,3}, count(3))
A048135_list = list(islice(A048135_gen(),66)) # Chai Wah Wu, Apr 02 2025

a(n) = A122254(n+2); A122255(a(n)) = 1. - Reinhard Zumkeller, Aug 29 2006

A048136 Tomahawk-nonconstructible n-gons.

Original entry on oeis.org

11, 22, 23, 25, 29, 31, 33, 41, 43, 44, 46, 47, 49, 50, 53, 55, 58, 59, 61, 62, 66, 67, 69, 71, 75, 77, 79, 82, 83, 86, 87, 88, 89, 92, 93, 94, 98, 99, 100, 101, 103, 106, 107, 110, 113, 115, 116, 118, 121, 122, 123, 124, 125, 127, 129, 131, 132, 134

Offset: 1

David W. Wilson

nonn

Andrew M. Gleason, Angle Trisection, the Heptagon and the Triskaidecagon, American Mathematical Monthly, 95 (1988), 185-194.

Cf. A005109, A048135, A122254, A122255.

from itertools import count, islice
from sympy import primefactors, totient
def A048136_gen(): # generator of terms
    yield from filter(lambda n: not set(primefactors(totient(n))) <= {2,3}, count(3))
A048136_list = list(islice(A048136_gen(),58)) # Chai Wah Wu, Apr 02 2025

A122255(a(n)) = 0: complement of A122254. - Reinhard Zumkeller, Aug 29 2006

A122255 Characteristic function of numbers with 3-smooth Euler's totient (A000010).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1

Offset: 1

Reinhard Zumkeller, Aug 29 2006

Multiplicative because A000010 is. - Andrew Howroyd, Aug 01 2018

			For n = 25, phi(25) = 20 = 2^2 * 5^1, and this is not 3-smooth, thus a(25) = 0.
For n = 26, phi(26) = 12 = 2^4 * 3^1, and here there are no larger prime factors than 3 (12 is 3-smooth), thus a(26) = 1. - _Antti Karttunen_, Aug 22 2017

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for characteristic functions

Cf. A000010, A006530, A065333, A122261, A122256 (partial sums).

Characteristic function of A122254.

a[n_] := Boole[FactorInteger[EulerPhi[n]][[-1, 1]] <= 3];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 20 2019 *)

a(n)=n=eulerphi(n); n>>=valuation(n, 2); n==3^valuation(n, 3) \\ Charles R Greathouse IV, Feb 21 2013

a(n) = if A006530(A000010(n)) <= 3 then 1 else 0.
a(A122254(n)) = a(A048135(n)) = 1; a(A048136(n)) = 0.
a(n) = if n=1 then 0 else A122256(n) - A122256(n-1).
a(n) = A122261(n) for n < 25.
a(n) = A065333(A000010(n)). - Antti Karttunen, Aug 22 2017
Multiplicative with a(p^e) = 1 for e = 1 and A006530(p-1) <= 3 or p <= 3; otherwise 0. - Andrew Howroyd, Aug 01 2018

A122260 Multiplicative closure of Pierpont primes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 28, 30, 32, 34, 35, 36, 37, 38, 39, 40, 42, 45, 48, 49, 50, 51, 52, 54, 56, 57, 60, 63, 64, 65, 68, 70, 72, 73, 74, 75, 76, 78, 80, 81, 84, 85, 90, 91, 95, 96, 97, 98, 100, 102, 104, 105, 108

Offset: 1

Reinhard Zumkeller, Aug 29 2006

If u and v are terms then also u*v is a term; A005109 is the generating subsequence;
A122261(a(n)) = 1;
A122254 is a subsequence: a(n) = A122254(n) = A048737(n) for n < 22.

			15 = 3 * 5 is a term since both 3 and 5 are Pierpont primes.

Ivan Neretin, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pierpont Prime.

Cf. A005109, A048737, A122254, A122261.

mx = 108; Select[Range@mx, Complement[FactorInteger[#][[All, 1]], Select[Prime@Range@mx, Max[FactorInteger[# - 1][[All, 1]]] < 5 &], {1}] == {} &] (* Ivan Neretin, Aug 13 2015 *)

sm3(n)=n>>=valuation(n,2);n==3^valuation(n,3)
is(n)=my(f=factor(n)[,1]);for(i=1,#f,if(!sm3(f[i]),return(0)));1 \\ Charles R Greathouse IV, Feb 21 2013

Sum_{n>=1} 1/a(n) = Product_{p in A005109} p/(p-1) = 5.80109266072985445612... - Amiram Eldar, Sep 27 2020

A048737 Numbers whose prime divisors consist of primes p such that 2^p-1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 28, 30, 31, 32, 34, 35, 36, 38, 39, 40, 42, 45, 48, 49, 50, 51, 52, 54, 56, 57, 60, 61, 62, 63, 64, 65, 68, 70, 72, 75, 76, 78, 80, 81, 84, 85, 89, 90, 91, 93, 95, 96, 98, 100, 102, 104, 105

Offset: 1

J. Lowell

nonn

Multiplicative closure of A000043. - Charles R Greathouse IV, Feb 21 2013

			10 = 2 * 5 is a term since both 2 and 5 are Mersenne exponents (A000043).

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000043, A122254, A122260.

p = Join[{1}, MersennePrimeExponent @ Range[12]]; Select[Range[p[[-1]]], AllTrue[FactorInteger[#][[;; , 1]], MemberQ[p, #] &] &] (* Amiram Eldar, Sep 27 2020 *)

a(n) = A122254(n) = A122260(n) for n < 22. - Reinhard Zumkeller, Aug 29 2006

Sum_{n>=1} 1/a(n) = Product_{p in A000043} p/(p-1) = 5.7838... - Amiram Eldar, Sep 27 2020

More terms from James Sellers, Apr 22 2000

A354356 Numbers k such that sigma(k) is 3-smooth (has no larger prime factors than 3).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 17, 21, 22, 23, 30, 31, 33, 34, 35, 42, 46, 47, 51, 53, 55, 62, 66, 69, 70, 71, 77, 85, 93, 94, 102, 105, 106, 107, 110, 115, 119, 127, 138, 141, 142, 154, 155, 159, 161, 165, 170, 186, 187, 191, 210, 213, 214, 217, 230, 231, 235, 238, 253, 254, 255, 265, 282, 310, 318, 321, 322, 329

Offset: 1

Antti Karttunen, May 24 2022

nonn

The prime terms in this sequence are in A005105. - Amiram Eldar, May 25 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n)

Cf. A000203, A065333, A354355 (characteristic function).
Cf. A005105, A046528, A354357 (subsequences).
Cf. also A122254.

Select[Range[330], Max@ FactorInteger[DivisorSigma[1, #]][[All, 1]] <= 3 &] (* Michael De Vlieger, May 24 2022 *)

A065333(n) = ((3^valuation(n, 3)<A065333
A354355(n) = A065333(sigma(n));
isA354356(n) = A354355(n);

A348867 Numbers whose numerator and denominator of the harmonic mean of their divisors are both 3-smooth numbers.

Original entry on oeis.org

1, 2, 3, 6, 28, 40, 84, 120, 135, 224, 270, 672, 819, 1638, 3780, 10880, 13392, 30240, 32640, 32760, 167400, 950976, 1303533, 2178540, 2607066, 3138345, 4713984, 6276690, 8910720, 14705145, 17428320, 29410290, 45532800, 52141320, 179734464, 301953024, 311323824

Offset: 1

Amiram Eldar, Nov 02 2021

nonn

The terms that are also harmonic numbers (A001599) are those whose harmonic mean of divisors (A001600) is a 3-smooth number. Of the 937 harmonic numbers below 10^14, 38 are terms in this sequence.
If a term is not a harmonic number, then its numerator and denominator of the harmonic mean of its divisors are powers of 2 and 3, or vice versa.
If k1 and k2 are coprime terms, then k1*k2 is also a term. In particular, if k is an odd term, then 2*k is also a term.

			2 is a term since the harmonic mean of its divisors is 4/3 = 2^2/3.
3 is a term since the harmonic mean of its divisors is 3/2.
40 is a term since the harmonic mean of its divisors is 32/9 = 2^5/3^2.

Amiram Eldar, Table of n, a(n) for n = 1..47

Cf. A000079, A000244, A001599, A001600, A003586, A099377, A099378.
Subsequence of A348868.
Similar sequences: A074266, A122254, A348658, A348659.

smQ[n_] := n == 2^IntegerExponent[n, 2] * 3^IntegerExponent[n, 3]; h[n_] := DivisorSigma[0, n]/DivisorSigma[-1, n]; q[n_] := smQ[Numerator[(hn = h[n])]] && smQ[Denominator[hn]]; Select[Range[10^5], q]

A379259 a(n) is the number of iterations that n requires to reach a 3-smooth number under the map x -> phi(x).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 2, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 2, 3, 0, 2, 1, 0, 1, 2, 1, 2, 0, 2, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 4, 0, 2, 2, 1, 1, 2, 0, 2, 1, 1, 2, 3, 1, 2, 2, 1, 0, 1, 2, 3, 1, 3, 1, 2, 0, 1, 1, 2, 1, 2, 1, 2, 1, 0, 2, 3, 1, 1, 2, 2

Offset: 1

Amiram Eldar, Dec 19 2024

If k is a 3-smooth number then phi(k) is also a 3-smooth number. Therefore, a(n) counts the numbers that are not 3-smooth numbers in the trajectory from n to a 3-smooth number (including n if it is not a 3-smooth number).

The indices of records, 1, 5, 11, 23, 47, ..., seem to be A246491 except for the first term (checked up to A246491(15)).

			a(1) = a(2) = a(3) = a(4) = 0 because 1, 2, 3 and 4 are already 3-smooth numbers.
a(5) = 1 because after one iteration 5 -> phi(5) = 4, a 3-smooth number, 4, is reached.
a(23) = 3 because after 3 iterations 23 -> 22 -> 10 -> 4, a 3-smooth number, 4, is reached.

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

Cf. A000010, A003434, A003586, A086420, A122254, A246491.

smQ[n_] := n == Times @@ ({2, 3}^IntegerExponent[n, {2, 3}]); a[n_] := -1 + Length@ NestWhileList[EulerPhi, n, ! smQ[#] &]; Array[a, 100]

issm(n) = my(m = n >> valuation(n, 2)); m == 3^valuation(m, 3);
a(n) = {my(c = 0); while(!issm(n), c++; n = eulerphi(n)); c;}

a(A003586(n)) = 0.

cp's OEIS Frontend

A048135 Tomahawk-constructible n-gons.

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A048136 Tomahawk-nonconstructible n-gons.

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A122255 Characteristic function of numbers with 3-smooth Euler's totient (A000010).

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A122260 Multiplicative closure of Pierpont primes.

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A048737 Numbers whose prime divisors consist of primes p such that 2^p-1 is prime.

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A354356 Numbers k such that sigma(k) is 3-smooth (has no larger prime factors than 3).

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A348867 Numbers whose numerator and denominator of the harmonic mean of their divisors are both 3-smooth numbers.

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A379259 a(n) is the number of iterations that n requires to reach a 3-smooth number under the map x -> phi(x).

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