A005109 -id:A005109 - cp's OEIS frontend

A077500 Primes of the form 2^r*p^s + 1, where p is an odd prime.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 47, 53, 59, 73, 83, 89, 97, 101, 107, 109, 113, 137, 149, 163, 167, 173, 179, 193, 197, 227, 233, 251, 257, 263, 269, 293, 317, 347, 353, 359, 383, 389, 401, 433, 449, 467, 479, 487, 503, 509, 557, 563, 569, 577, 587

Offset: 1

Amarnath Murthy, Nov 07 2002

nonn

Primes p such that p-1 has at most one odd prime divisor.

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

Cf. A005109, A077497, A077498, A077499.

Select[Prime[Range[110]],Length[Select[FactorInteger[#-1] [[All, 1]], OddQ]]<2&] (* Harvey P. Dale, Oct 09 2017 *)

Corrected and extended by Sascha Kurz, Jan 04 2003

A111345 Pierpont 5-almost primes. 5-almost primes of form (2^K)*(3^L)+1.

Original entry on oeis.org

4375, 19684, 7077889, 7962625, 34012225, 100663297, 129140164, 452984833, 459165025, 544195585, 644972545, 918330049, 5159780353, 7346640385, 8589934593, 13947137605, 14495514625, 23219011585, 27518828545, 28991029249

Offset: 1

Jonathan Vos Post, Nov 08 2005

			a(1) = 4375 = (2^1)*(3^7)+1 = 5 * 5 * 5 * 5 * 7.
a(2) = 19684 = (2^0)*(3^9)+1 = 2 * 2 * 7 * 19 * 37.
a(3) = 7077889 = (2^18)*(3^3)+1 = 7 * 13 * 13 * 31 * 193 (prime factors each have all odd digits).
a(4) = 7962625 = (2^15)*(3^5)+1 = 5 * 5 * 5 * 11 * 5791 (again, coincidentally, prime factors each have all odd
digits).
a(7) = 129140164 = (2^0)*(3^17)+1 = 2 * 2 * 103 * 307 * 1021.
a(15) = 8589934593 = (2^33)*(3^0)+1 = 3 * 3 * 67 * 683 * 20857.
a(21) = 34359738369 = (2^35)*(3^0)+1 = 3 * 11 * 43 * 281 * 86171.
a(30) = 793437161473 = (2^11)*(3^18)+1 = 11 * 11 * 11 * 43 * 13863281.
a(32) = 847288609444 = (2^0)*(3^25)+1 = 2 * 2 * 61 * 151 * 22996651.
a(47) = 68630377364884 = (2^0)*(3^29)+1 = 2 * 2 * 523 * 6091 * 5385997.
a(48) = 70368744177665 = (2^46)*(3^0)+1 = 5 * 277 * 1013 * 1657 * 30269.
a(81) = 50031545098999708 = (2^0)*(3^35)+1 = 2 * 2 * 61 * 547 * 374857981681.
a(89) = 144115188075855873 = (2^57)*(3^0)+1 = 3 * 3 * 571 * 174763 * 160465489.
a(99) = 450283905890997364 = (2^0)*(3^37)+1 = 2 * 2 * 18427 * 107671 * 56737873.
a(113) = 4611686018427387905 = (2^62)*(3^0)+1 = 5 * 5581 * 8681 * 49477 * 384773.

Eric Weisstein's World of Mathematics, Pierpont Prime
Eric Weisstein's World of Mathematics, Almost Prime

Intersection of A014614 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.
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.

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)==5, 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) = 5.

Extended by Ray Chandler, Nov 08 2005

A111346 Pierpont 6-almost primes. 6-almost primes of form (2^K)*(3^L)+1.

Original entry on oeis.org

14348908, 134217729, 1073741825, 139314069505, 231928233985, 264479053825, 282429536482, 618475290625, 705277476865, 3570467226625, 4398046511105, 8349416423425, 21134460321793, 35664401793025, 91507169819845

Offset: 1

Jonathan Vos Post, Nov 08 2005

			a(1) = 14348908 = (2^0)*(3^15)+1 = 2 * 2 * 7 * 31 * 61 * 271.
a(2) = 134217729 = (2^27)*(3^0)+1 = 3 * 3 * 3 * 3 * 19 * 87211.
a(3) = 1073741825 = (2^30)*(3^0)+1 = 5 * 5 * 13 * 41 * 61 * 1321.
a(4) = 139314069505 = (2^18)*(3^12)+1 = 5 * 13 * 17 * 61 * 337 * 6133.
a(100) = 151115727451828646838273 = (2^77)*(3^0)+1 = 3 * 43 * 617 * 683 * 78233 * 35532364099.
a(127) = 9671406556917033397649409 = (2^83)*(3^0)+1 = 3 * 499 * 1163 * 2657 * 155377 * 13455809771.
a(153) = 523347633027360537213511522 = (2^0)*(3^56)+1 = 2 * 17 * 113 * 193 * 19489 * 36214795668330833.
a(169) = 2475880078570760549798248449 = (2^91)*(3^0)+1 = 3 * 43 * 2731 * 224771 * 1210483 * 25829691707.

Eric Weisstein's World of Mathematics, Pierpont Prime
Eric Weisstein's World of Mathematics, Almost Prime

Intersection of A046306 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.
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.

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)==6, 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) = 6.

Extended by Ray Chandler, Nov 08 2005

A112797 Pierpont 3-almost primes. 3-almost primes of form (2^K)*(3^L)+1.

Original entry on oeis.org

28, 244, 325, 385, 730, 1025, 1729, 2188, 5185, 6562, 7777, 16385, 26245, 36865, 46657, 49153, 55297, 82945, 93313, 221185, 354295, 419905, 531442, 559873, 589825, 663553, 708589, 884737, 1119745, 1572865, 1594324, 1889569, 2985985

Offset: 1

Jonathan Vos Post, Nov 08 2005

			a(1) = 28 = (2^0)*(3^3)+1 = 2 * 2 * 7.
a(2) = 244 = (2^0)*(3^5)+1 = 2 * 2 * 61.
a(3) = 325 = (2^2)*(3^4)+1 = 5 * 5 * 13.
a(4) = 385 = (2^7)*(3^1)+1 = 5 * 7 * 11.
a(11) = 7777 = (2^5)*(3^5)+1 = 7 * 11 * 101.
a(115) = 94143178828 = (2^0)*(3^23)+1 = 2 * 2 * 23535794707.
a(119) = 137438953473 = (2^37)*(3^0)+1 = 3 * 1777 * 25781083.
a(196) = 281474976710657 = (2^48)*(3^0)+1 = 193 * 65537 * 22253377.

Eric Weisstein's World of Mathematics, Pierpont Prime
Eric Weisstein's World of Mathematics, Almost Prime

Intersection of A014612 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.
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.
A113741 gives the Pierpont 9-almost primes, of the form (2^K)*(3^L)+1.

Take[Select[2^#[[1]] 3^#[[2]] + 1 & /@ Tuples[Range[0, 20], 2],
PrimeOmega[ #]  ==  3 &] // Union, 40] (* Harvey P. Dale, Jan 02 2021 *)

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)==3, 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) = 3.

Extended by Ray Chandler, Nov 08 2005

A002200 Primes of the form 2^q3^r5^s + 1.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 61, 73, 97, 101, 109, 151, 163, 181, 193, 241, 251, 257, 271, 401, 433, 487, 541, 577, 601, 641, 751, 769, 811, 1153, 1201, 1297, 1459, 1601, 1621, 1801, 2161, 2251, 2593, 2917, 3001, 3457, 3889, 4001, 4051, 4801, 4861

Offset: 1

N. J. A. Sloane

nonn

M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 53.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 200 terms from Harry J. Smith)

Cf. A174144, A005109, A077497.

K:=10^7;; # to get all terms <= K.
A:=Filtered([1..K],IsPrime);;
B:=List(A,i->Factors(i-1));;
C:=[];;  for i in B do if Elements(i)=[2] or Elements(i)=[2,3]  or Elements(i)=[2,5] or Elements(i)=[2,3,5]  then Add(C,Position(B,i)); fi; od;
A002200:=Concatenation([2],List(C,i->A[i])); # Muniru A Asiru, Sep 10 2017

[p: p in PrimesUpTo(5000) | forall{d: d in PrimeDivisors(p-1) | d le 5}]; // Bruno Berselli, Sep 24 2012

up=10^6; a=1; Sort[Reap[While[ aGiovanni Resta, Jul 18 2017 *)

{ default(primelimit, 16600000); n=0; forprime (p=2, 16600000, m=p-1; p2=p3=p5=0; s=m; r=0; while(r==0, q=s\2; r=s-2*q; s=q; if(r==0, p2++)); s=m; r=0; while(r==0, q=s\3; r=s-3*q; s=q; if(r==0, p3++)); s=m; r=0; while(r==0, q=s\5; r=s-5*q; s=q; if(r==0, p5++)); if (m == 2^p2*3^p3*5^p5, n++; write("b002200.txt", n, " ", p)); if (n >= 200, break); ); } \\ Harry J. Smith, May 25 2009

{ n=5000; cache=10^5; v=vector(cache); x2=2; x3=3; x5=5; i=j=k=1; v[1]=1; for(m=2,cache,v[m]=t=min(x2,min(x3,x5)); if(x2==t,x2=2*v[i++]); if(x3==t,x3=3*v[j++]); if(x5==t,x5=5*v[k++]);); i=0; c=0; while(cJean-Marie Madiot, Jul 17 2017

Better description and more terms from Vladeta Jovovic, May 08 2003

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

A122261 Characteristic function of numbers having only factors that are Pierpont primes.

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

Offset: 1

Reinhard Zumkeller, Aug 29 2006

			For n = 11 = 11^1, 11 is not a Pierpoint prime because 11-1 = 10 = 2*5 has a prime factor larger than 3, thus a(11) = 0.
For n = 25 = 5^2, 5 is a Pierpoint prime as 5-1 = 4 = 2^2 does not have any prime factors larger than 3, thus a(25) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..12289
Eric Weisstein's World of Mathematics, Pierpont Prime
Index entries for characteristic functions

Cf. A005109, A065333, A122255, A122262 (partial sums).

Characteristic function of A122260.

Block[{nn = 105, s}, s = Select[Sort@ Flatten@ Table[2^i*3^j + 1, {i, 0, Log2@ nn}, {j, 0, Log[3, nn/2^i]}] , PrimeQ]; Table[Boole[n == 1] + Boole@ AllTrue[FactorInteger[n][[All, 1]], MemberQ[s, #] &], {n, nn}]] (* Michael De Vlieger, Aug 23 2017, after Robert G. Wilson v at A005109 *)

A065333(n) = ((3^valuation(n, 3)<Charles R Greathouse IV, Aug 21 2011
A122261(n) = factorback(apply(p -> A065333(p-1), (factor(n)[, 1]))); \\ Antti Karttunen, Aug 22 2017

Multiplicative with a(p) = A065333(p-1), for p prime.
a(n) = if n=1 then 0 else A122262(n) - A122262(n-1).
a(A122260(n)) = 1.
a(n) = A122255(n) for n < 25.

An unnecessary part removed from the formula and the Example section added by Antti Karttunen, Aug 22 2017

A051913 Numbers k such that phi(k)/phi(phi(k)) = 3.

Original entry on oeis.org

7, 9, 13, 14, 18, 19, 21, 26, 27, 28, 35, 36, 37, 38, 39, 42, 45, 52, 54, 56, 57, 63, 65, 70, 72, 73, 74, 76, 78, 81, 84, 90, 91, 95, 97, 104, 105, 108, 109, 111, 112, 114, 117, 119, 126, 130, 133, 135, 140, 144, 146, 148, 152, 153, 156, 162, 163, 168, 171, 180, 182

Offset: 1

J. H. Conway and Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Dec 17 1999

Also numbers k such that phi(k) = 2^a*3^b with a, b > 0.
Also numbers k such that a regular k-gon can be constructed using conics but not with merely a compass and straightedge.
"Constructed using conics" means that one can draw any conic, once its focus, its vertex and a point on its directrix are constructed. Points at intersections are thereby constructed. (Videla's definition is slightly more complicated, but equivalent.) One can use parabolas to take cube roots; hyperbolas yield trisected angles. - Don Reble, Apr 23 2007

			Phi(999) = Phi(3*3*3*37) = 648 = 8*81.

George E. Martin, Geometric Constructions, Springer, 1997, p. 140.

T. D. Noe, Table of n, a(n) for n=1..1000
C. R. Videla, On points constructible from conics, Mathematical Intelligencer, 19, No. 2, pp. 53-57 (1997).

Cf. A000010, A003401, A003586, A058383.

[n: n in [1..200] | EulerPhi(n)/EulerPhi(EulerPhi(n)) eq 3]; // Vincenzo Librandi, Apr 17 2015

lf[x_] := Length[FactorInteger[x]] eu[x_] := EulerPhi[x] Do[s=lf[eu[n]]; If[Equal[s, 2]&&Equal[Mod[eu[n], 6], 0], Print[n]], {n, 1, 1000}] (* Labos Elemer, Dec 28 2001 *)
f[n_] := Block[{m = n}, If[m > 0, While[ IntegerQ[m/2], m /= 2]; While[ IntegerQ[m/3], m /= 3]]; m == 1]; fQ[n_] := Block[{pff = Select[ FactorInteger[n], #[[1]] > 3 &]}, pf = Flatten[{2, Table[ #[[1]], {1}] & /@ pff}]; pfe = Union[ Flatten[{1, Table[ #[[2]], {1}] & /@ pff}]]; If[ Union[f /@ (pf - 1)] == {True} && pfe == {1} && !IntegerQ[ Log[2, EulerPhi[ n]]], True, False]]; Select[ Range[184], fQ[ # ] &] (* Robert G. Wilson v, Apr 05 2005 *)
Select[Range[200],EulerPhi[#]/EulerPhi[EulerPhi[#]]==3&] (* Harvey P. Dale, Jul 11 2025 *)

from itertools import count, islice
from sympy import primefactors, totient
def A051913_gen(): # generator of terms
    yield from filter(lambda n: primefactors(totient(n)) == [2,3], count(1))
A051913_list = list(islice(A051913_gen(),30)) # Chai Wah Wu, Apr 02 2025

Numbers k of the form 2^a*3^b*p*q*r*..., where p, q, r, ... are distinct primes of the form 2^x*3^y + 1 (i.e., belong to A005109) and phi(k) is not a power of 2 [Videla]. - Robert G. Wilson v, Apr 05 2005

Additional comments from Labos Elemer, Dec 28 2001
Additional comments from Benoit Cloitre, Jan 26 2002
Edited by N. J. A. Sloane, Apr 21 2007
Entries checked by Don Reble, Apr 23 2007

A077499 Primes of the form 2^r*11^s + 1.

Original entry on oeis.org

2, 3, 5, 17, 23, 89, 257, 353, 1409, 2663, 30977, 65537, 170369, 495617, 5767169, 23068673, 59969537, 82458113, 453519617, 3429742097, 4715895383, 15352201217, 39909726209, 1857616347137, 45732811767809, 96757023244289

Offset: 1

Amarnath Murthy, Nov 07 2002

nonn

Primes p such that p-1 has at most one odd prime divisor 11.

Cf. A005109, A077497, A077498, A077500.

More terms from Ray Chandler, Aug 02 2003

A122259 Primes p such that p - 1 is not 3-smooth.

Original entry on oeis.org

11, 23, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 79, 83, 89, 101, 103, 107, 113, 127, 131, 137, 139, 149, 151, 157, 167, 173, 179, 181, 191, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347

Offset: 1

Reinhard Zumkeller, Aug 29 2006

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pierpont Prime

Cf. A005109, A006530, A122257.

smooth3Q[n_] := n == 2^IntegerExponent[n, 2]*3^IntegerExponent[n, 3];
Select[Prime[Range[100]], !smooth3Q[#-1]&] (* Jean-François Alcover, Oct 17 2021 *)

is(n)=my(q=n-1); q>>=valuation(q,2); q/=3^valuation(q,3); q>1 && isprime(n) \\ Charles R Greathouse IV, Oct 29 2018

A122257(a(n)) = 0;
A006530(a(n) - 1) > 3.
a(n) ~ n log n. - Charles R Greathouse IV, Oct 29 2018

cp's OEIS Frontend

A077500 Primes of the form 2^r*p^s + 1, where p is an odd prime.

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A111345 Pierpont 5-almost primes. 5-almost primes of form (2^K)*(3^L)+1.

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A111346 Pierpont 6-almost primes. 6-almost primes of form (2^K)*(3^L)+1.

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A112797 Pierpont 3-almost primes. 3-almost primes of form (2^K)*(3^L)+1.

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A002200 Primes of the form 2^q*3^r*5^s + 1.

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A122260 Multiplicative closure of Pierpont primes.

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A122261 Characteristic function of numbers having only factors that are Pierpont primes.

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A002200 Primes of the form 2^q3^r5^s + 1.