A005109 -id:A005109 - cp's OEIS frontend

A081428 Class 9- primes.

34549, 86371, 103613, 120919, 138059, 149519, 172583, 172741, 224563, 276293, 282059, 282143, 293659, 299417, 316691, 352399, 368513, 379903, 397303, 403061, 414577, 451499, 483179, 486527, 489431, 500947, 506537, 517747, 518047, 541799

Offset: 1

Robert G. Wilson v, Mar 20 2003

nonn

R. K. Guy, Unsolved Problems in Number Theory, A18.

R. J. Mathar, Table of n, a(n) for n = 1..6505

Cf. A005113, A056637, A005109, A005110, A005111, A005112, A081424, A081425, A081426, A081427, A081429, A081430.

PrimeFactors[n_Integer] := Flatten[Table[ #[[1]], {1}] & /@ FactorInteger[n]]; f[n_Integer] := Block[{m = n}, If[m == 0, m = 1, While[ IntegerQ[m/2], m /= 2]; While[ IntegerQ[m/3], m /= 3]]; Apply[Times, PrimeFactors[m] - 1]]; ClassMinusNbr[n_] := Length[NestWhileList[f, n, UnsameQ, All]] - 3; Prime[ Select[ Range[50000], ClassMinusNbr[ Prime[ # ]] == 9 &]]

A113432 Pierpont semiprimes: semiprimes of the form (2^K)*(3^L)+1.

Original entry on oeis.org

4, 9, 10, 25, 33, 49, 55, 65, 82, 129, 145, 217, 289, 649, 865, 973, 1537, 1945, 2049, 2305, 3073, 4097, 4609, 5833, 6145, 6913, 8193, 8749, 9217, 11665, 13123, 15553, 20737, 23329, 24577, 27649, 31105, 34993, 41473, 62209, 69985, 73729, 78733

Offset: 1

Jonathan Vos Post, Nov 01 2005

			a(1) = 4 = (2^0)*(3^1)+1 = 2^2 hence the semiprime A001358(1).
a(2) = 9 = (2^3)*(3^0)+1 = 3^2 hence the semiprime A001358(3).
a(3) = 10 = (2^0)*(3^2)+1 = 2 * 5 hence the semiprime A001358(4).
a(4) = 25 = (2^3)*(3^1)+1 = 5^2 hence the semiprime A001358(9).
a(5) = 33 = (2^5)*(3^0)+1 = 3 * 11 hence the semiprime A001358(11).
a(6) = 49 = (2^4)*(3^1)+1 = 7^2 hence the semiprime A001358(17).
a(7) = 55 = (2^1)*(3^3)+1 = 5 * 11 hence the semiprime A001358(19).

Vincenzo Librandi, Table of n, a(n) for n = 1..89
Chris Caldwell, "Pierpont primes." primeform posting, Oct 25, 2005.
Chris Caldwell, "Pierpont primes." primeform posting, Oct 25, 2005. [Cached copy]
Eric Weisstein's World of Mathematics, Pierpont Prime
Eric Weisstein's World of Mathematics, Semiprime

Cf. A001358, A003586, A005109, A055600, A111153, A111206, A113433, A113434.

Select[Range[10^5], Plus @@ Last /@ FactorInteger[ # ] == 2 && Max @@ First /@ FactorInteger[ # - 1] < 5 &] (* Ray Chandler, Jan 24 2006 *)

{a(n)} = Intersection of {(2^K)*(3^L)+1} A055600 and semiprimes A001358. a(n) is in this sequence iff there exist nonnegative integers K and L such that Omega((2^K)*(3^L)+1) = 2.

A122254 Numbers with 3-smooth Euler's totient (A000010).

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, 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, 114

Offset: 1

Reinhard Zumkeller, Aug 29 2006

An integer n>=3 belongs to this sequence if and only if a regular n-gon can be constructed using straightedge and conic sections (details in Gibbins and Smolinsky, see below). - Austin Shapiro, Nov 14 2021

Products of 3-smooth numbers (A003586) and squarefree numbers whose prime factors are all Pierpont primes (A005109). - Amiram Eldar, Dec 03 2022

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Aliska Gibbins and Lawrence Smolinsky, Geometric Constructions with Ellipses, The Mathematical Intelligencer 31(1) (2009), 57-62.

Cf. A000010, A003586 (3-smooth), A005109.

Subsequence of A122260.

Select[Range@115, Max[FactorInteger[EulerPhi[#]][[All, 1]]] < 5 &] (* Ivan Neretin, Jul 28 2015 *)

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

list(lim)=my(v=List(),u,t);for(i=0,log(lim--+1.5)\log(3),t=3^i;while(t<=lim,if(isprime(t+1),listput(v,t+1));t<<=1));v=vecsort(Vec(v));u=List([1]);for(i=3,#v,for(j=1,#u,t=v[i]*u[j];if(t>lim,next(2));listput(u,t)));u=vecsort(Vec(u));v=List(u);for(i=1,#u,t=u[i];while((t*=3)<=lim,listput(v,t)));u=Vec(v);v=List(u);for(i=1,#u,t=u[i];while((t<<=1)<=lim,listput(v,t)));vecsort(Vec(v)) \\ Charles R Greathouse IV, Feb 22 2013

from itertools import count, islice
from sympy import multiplicity, factorint
def A065333(n): return int(3**(multiplicity(3,m:=n>>(~n&n-1).bit_length()))==m)
def A122254_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(p<=3 or (e==1 and A065333(p-1)) for p,e in factorint(n).items()), count(max(startvalue,1)))
A122254_list = list(islice(A122254_gen(),40)) # Chai Wah Wu, Dec 20 2024

a(n) = A048135(n-2) for n>2.
a(n) = A122260(n) = A048737(n) for n < 22.
Sum_{n>=1} 1/a(n) = 3 * Product_{p > 3 in A005109} (1 + 1/p) = 5.38288865867495675807... . - Amiram Eldar, Dec 03 2022

A111344 Pierpont 4-almost primes: numbers with exactly 4 prime divisors, not necessarily distinct, of the form 2^K*3^L + 1.

Original entry on oeis.org

513, 13825, 32769, 59050, 110593, 157465, 177148, 186625, 262145, 279937, 497665, 1259713, 1327105, 2097153, 2125765, 2519425, 4718593, 4782970, 5668705, 6718465, 17915905, 18874369, 22674817, 33554433, 38263753, 56623105

Offset: 1

Jonathan Vos Post, Nov 08 2005

nonn

			a(1) = 513 = (2^9)*(3^0)+1 = 3 * 3 * 3 * 19.
a(2) = 13825 = (2^9)*(3^3)+1 = 5 * 5 * 7 * 79.
a(3) = 32769 = (2^15)*(3^0)+1 = 3 * 3 * 11 * 331.
a(4) = 59050 = (2^0)*(3^10)+1 = 2 * 5 * 5 * 1181.
a(10) = 279937 = (2^7)*(3^7)+1 = 7 * 7 * 29 * 197 (lots of sevens).
a(24) = 33554433 = (2^25)*(3^0) = 3 * 11 * 251 * 4051.
a(60) = 31381059610 = (2^0)*(3^22)+1 = 2 * 5 * 5501 * 570461.
a(168) = 16677181699666570 = (2^0)*(3^34)+1 = 2 * 5 * 956353 * 1743831169.

Charles R Greathouse IV, Table of n, a(n) for n = 1..2500
Eric Weisstein's World of Mathematics, Pierpont Prime
Eric Weisstein's World of Mathematics, Almost Prime

Intersection of A014613 and A055600.
A005109 gives the Pierpont primes, which are primes of the form (2^K)*(3^L)+1.
A113432 gives the Pierpont semiprimes, 2-almost primes of the form (2^K)*(3^L)+1.
A112797 gives the Pierpont 3-almost primes, of the form (2^K)*(3^L)+1.
A111345 gives the Pierpont 5-almost primes, of the form (2^K)*(3^L)+1.
A111346 gives the Pierpont 6-almost primes, of the form (2^K)*(3^L)+1.
A113739 gives the Pierpont 7-almost primes, of the form (2^K)*(3^L)+1.
A113740 gives the Pierpont 8-almost primes, of the form (2^K)*(3^L)+1.
A113741 gives the Pierpont 9-almost primes, of the form (2^K)*(3^L)+1.

is(n)=bigomega(n)==4 && n-1 == 2^valuation(n-1,2)*3^valuation(n-1,3) \\ Charles R Greathouse IV, Feb 01 2017

list(lim)=my(v=List(),L=lim\1-1); for(e=0,logint(L,3), my(t=3^e); while(t<=L, if(bigomega(t+1)==4, listput(v, t+1)); t*=2)); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

Extended by Ray Chandler, Nov 08 2005

Name edited by Charles R Greathouse IV, Feb 01 2017

A113739 Pierpont 7-almost primes. 7-almost primes of form (2^K)*(3^L)+1.

Original entry on oeis.org

339738625, 10460353204, 83682825625, 669462604993, 2641807540225, 3761479876609, 7625597484988, 18075490334785, 35184372088833, 481469424205825, 488038239039169, 570630428688385, 1125899906842625

Offset: 1

Jonathan Vos Post, Nov 08 2005

nonn

			a(1) = 339738625 = (2^22)*(3^4)+1 = 5 * 5 * 5 * 17 * 29 * 37 * 149.
a(2) = 10460353204 = (2^0)*(3^21)+1 = 2 * 2 * 7 * 7 * 43 * 547 * 2269.
a(3) = 83682825625 = (2^3)*(3^21)+1 = 5 * 5 * 5 * 5 * 7 * 631 * 30313.
a(4) = 669462604993 = (2^6)*(3^21)+1 = 7 * 13 * 19 * 31 * 67 * 277 * 673.
a(7) = 7625597484988 = (2^0)*(3^27)+1 = 2 * 2 * 7 * 19 * 37 * 19441 * 19927.
a(9) = 35184372088833 = (2^45)*(3^0)+1 = 3 * 3 * 3 * 11 * 19 * 331 * 18837001.
a(13) = 1125899906842625 = (2^50)*(3^0)+1 = 5 * 5 * 5 * 41 * 101 * 8101 * 268501.
a(16) = 5559060566555524 = (2^0)*(3^33)+1 = 2 * 2 * 7 * 67 * 661 * 25411 * 176419.
a(28) = 9223372036854775809 = (2^63)*(3^0)+1 = 3 * 3 * 3 * 19 * 43 * 5419 * 77158673929.

Charles R Greathouse IV, Table of n, a(n) for n = 1..716
Eric Weisstein's World of Mathematics, Pierpont Prime
Eric Weisstein's World of Mathematics, Almost Prime

Intersection of A046308 and A055600.
A005109 gives the Pierpont primes, which are primes of the form (2^K)*(3^L)+1.
A113432 gives the Pierpont semiprimes, 2-almost primes of the form (2^K)*(3^L)+1.
A112797 gives the Pierpont 3-almost primes, of the form (2^K)*(3^L)+1.
A111344 gives the Pierpont 4-almost primes, of the form (2^K)*(3^L)+1.
A111345 gives the Pierpont 5-almost primes, of the form (2^K)*(3^L)+1.
A111346 gives the Pierpont 6-almost primes, of the form (2^K)*(3^L)+1.
A113740 gives the Pierpont 8-almost primes, of the form (2^K)*(3^L)+1.
A113741 gives the Pierpont 9-almost primes, of the form (2^K)*(3^L)+1.

list(lim)=my(v=List(), L=lim\1-1); for(e=0, logint(L, 3), my(t=3^e); while(t<=L, if(bigomega(t+1)==7, listput(v, t+1)); t*=2)); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

a(n) is in this sequence iff there exist nonnegative integers K and L such that Omega((2^K)*(3^L)+1) = 7.

Extended by Ray Chandler, Nov 08 2005

A113740 Pierpont 8-almost primes. 8-almost primes of form (2^K)*(3^L)+1.

Original entry on oeis.org

1999004627104432129, 4052555153018976268, 8754997675608244225, 9606056659007943745, 11832592569282330625, 22769912080611422209, 68309736241834266625, 354577405862133891073, 12449449430074295092225

Offset: 1

Jonathan Vos Post, Nov 08 2005

nonn

			a(1) = 1999004627104432129 = (2^18)*(3^27)+1 = 7 * 13 * 19 * 109 * 127 * 181 * 6949 * 66403.
a(2) = 4052555153018976268 = (2^0)*(3^39)+1 = 2 * 2 * 7 * 79 * 157 * 2887 * 10141 * 398581.
a(3) = 8754997675608244225 = (2^55)*(3^5)+1 = 5 * 5 * 11 * 11 * 1201 * 1229 * 16451 * 119191.
a(4) = 9606056659007943745 = (2^6)*(3^36)+1 = 5 * 13 * 17 * 89 * 109 * 281 * 18793 * 169693.
a(13) = 717897987691852588770250 = (2^0)*(3^50)+1 = 2 * 5 * 5 * 5 * 101 * 1181 * 394201 * 61070817601.
a(29) = 1570042899082081611640534564 = (2^0)*(3^57)+1 = 2 * 2 * 7 * 2851 * 3079 * 53923 * 101917 * 1162320517.

Charles R Greathouse IV, Table of n, a(n) for n = 1..993
Eric Weisstein's World of Mathematics, Pierpont Prime
Eric Weisstein's World of Mathematics, Almost Prime

Intersection of A046310 and A055600.
A005109 gives the Pierpont primes, which are primes of the form (2^K)*(3^L)+1.
A113432 gives the Pierpont semiprimes, 2-almost primes of the form (2^K)*(3^L)+1.
A112797 gives the Pierpont 3-almost primes, of the form (2^K)*(3^L)+1.
A111344 gives the Pierpont 4-almost primes, of the form (2^K)*(3^L)+1.
A111345 gives the Pierpont 5-almost primes, of the form (2^K)*(3^L)+1.
A111346 gives the Pierpont 6-almost primes, of the form (2^K)*(3^L)+1.
A113739 gives the Pierpont 7-almost primes, of the form (2^K)*(3^L)+1.
A113741 gives the Pierpont 9-almost primes, of the form (2^K)*(3^L)+1.

list(lim)=my(v=List(), L=lim\1-1); for(e=0, logint(L, 3), my(t=3^e); while(t<=L, if(bigomega(t+1)==8, listput(v, t+1)); t*=2)); Set(v) \\ Charles R Greathouse IV, Feb 06 2017

a(n) is in this sequence iff there exist nonnegative integers K and L such that Omega((2^K)*(3^L)+1) = 8.

Extended by Ray Chandler, Nov 08 2005

A113741 Pierpont 9-almost primes. 9-almost primes of form (2^K)*(3^L)+1.

Original entry on oeis.org

1601009443167990625, 1897492673384285185, 39346408075296537575425, 46005119909369701466113, 221073919720733357899777, 2153693963075557766310748, 3925770232266214525108225

Offset: 1

Jonathan Vos Post, Nov 08 2005

			a(1) = 1601009443167990625 = (2^5)*(3^35)+1 = 5 * 5 * 5 * 5 * 5 * 7 * 11 * 241 * 27608073601.
a(2) = 1897492673384285185 = (2^10)*(3^32)+1 = 5 * 13 * 13 * 13 * 41 * 41 * 373 * 2357 * 116881.

Charles R Greathouse IV, Table of n, a(n) for n = 1..434
Eric Weisstein's World of Mathematics, Pierpont Prime
Eric Weisstein's World of Mathematics, Almost Prime

Intersection of A046312 and A055600.
A005109 gives the Pierpont primes, which are primes of the form (2^K)*(3^L)+1.
A113432 gives the Pierpont semiprimes, 2-almost primes of the form (2^K)*(3^L)+1.
A112797 gives the Pierpont 3-almost primes, of the form (2^K)*(3^L)+1.
A111344 gives the Pierpont 4-almost primes, of the form (2^K)*(3^L)+1.
A111345 gives the Pierpont 5-almost primes, of the form (2^K)*(3^L)+1.
A111346 gives the Pierpont 6-almost primes, of the form (2^K)*(3^L)+1.
A113739 gives the Pierpont 7-almost primes, of the form (2^K)*(3^L)+1.
A113740 gives the Pierpont 8-almost primes, of the form (2^K)*(3^L)+1.

list(lim)=my(v=List(), L=lim\1-1); for(e=0, logint(L, 3), my(t=3^e); while(t<=L, if(bigomega(t+1)==9, listput(v, t+1)); t*=2)); Set(v) \\ Charles R Greathouse IV, Feb 02 2017

a(n) is in this sequence iff there exist nonnegative integers K and L such that Omega((2^K)*(3^L)+1) = 9.

Extended by Ray Chandler, Nov 08 2005

A048135 Tomahawk-constructible n-gons.

Original entry on oeis.org

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

A077498 Primes of the form 2^r*7^s + 1.

Original entry on oeis.org

2, 3, 5, 17, 29, 113, 197, 257, 449, 1373, 3137, 50177, 65537, 114689, 268913, 470597, 614657, 1075649, 3294173, 7340033, 9834497, 210827009, 275365889, 359661569, 469762049, 1129900997, 1438646273, 1927561217, 7909306973, 52613349377

Offset: 1

Amarnath Murthy, Nov 07 2002

nonn

			197 = 2^2*7^2 + 1 is a member.

Cf. A005109, A077497, A077499, A077500.

Corrected and extended by Ray Chandler, Aug 02 2003

cp's OEIS Frontend

A081428 Class 9- primes.

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A113432 Pierpont semiprimes: semiprimes of the form (2^K)*(3^L)+1.

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A122254 Numbers with 3-smooth Euler's totient (A000010).

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A111344 Pierpont 4-almost primes: numbers with exactly 4 prime divisors, not necessarily distinct, of the form 2^K*3^L + 1.

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A113739 Pierpont 7-almost primes. 7-almost primes of form (2^K)*(3^L)+1.

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A113740 Pierpont 8-almost primes. 8-almost primes of form (2^K)*(3^L)+1.

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A113741 Pierpont 9-almost primes. 9-almost primes of form (2^K)*(3^L)+1.

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A048135 Tomahawk-constructible n-gons.

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