A058383 -id:A058383 - cp's OEIS frontend

A336772 Sums s of positive exponents such that no prime of the form 2^j*3^k + 1 with j + k = s exists.

12, 24, 33, 46, 48, 60, 72, 74, 80, 96, 102, 111, 118, 120, 130, 132, 141, 142, 144, 147, 159, 162, 165, 166, 168, 186, 200, 216, 234, 240, 242, 252, 258, 288, 306, 309, 312, 318, 358, 370, 374, 375, 384, 399, 405, 408, 414, 420, 432, 435, 462, 464, 468, 478

Offset: 1

Hugo Pfoertner, based on a suggestion from Rainer Rosenthal, Aug 24 2020

nonn

			a(1) = 12, because none of the 11 numbers {2^1*3^11+1, 2^2*3^10+1, ..., 2^11*3^1+1} = {354295, 236197, 157465, 104977, 69985, 46657, 31105, 20737, 13825, 9217, 6145} is prime,
a(2) = 24: none of the 23 numbers {2^1*3^23+1, 2^2*3^22+1, ..., 2^23*3^1+1} = {188286357655, 125524238437, 83682825625, 55788550417, ..., 56623105, 37748737, 25165825} is prime.

Hugo Pfoertner, Table of n, a(n) for n = 1..500

Cf. A033845, A058383, A336773.

for(s=2,500, my(t=1); for(j=1,s-1, my(k=s-j); if(isprime(2^j*3^k+1),t=0;break)); if(t,print1(s,", ")))

A061599 Primes p such that the greatest prime divisor of p-1 is 5.

Original entry on oeis.org

11, 31, 41, 61, 101, 151, 181, 241, 251, 271, 401, 541, 601, 641, 751, 811, 1201, 1601, 1621, 1801, 2161, 2251, 3001, 4001, 4051, 4801, 4861, 6481, 7681, 8101, 8641, 9001, 9601, 9721, 11251, 14401, 15361, 16001, 19441, 21601, 21871, 22501, 23041, 24001

Offset: 1

Labos Elemer, Jun 13 2001

nonn

Prime numbers n for which cos(2Pi/n) is an algebraic number of 5th degree. - Artur Jasinski, Dec 13 2006

The least significant digit of each term is one. - Harvey P. Dale, Jul 07 2024

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

The 3rd in a family of sequences after A019434(=Fermat-primes) and A058383.
Cf. A019434, A058383, A023503, A034694, A006530, A006093, A035095, A000040.
Cf. A004729, A058383, A125867-A125875, A024899.

Do[If[Take[FactorInteger[EulerPhi[2n + 1]][[ -1]],1] == {5} && PrimeQ[2n + 1], Print[2n + 1]], {n, 1, 10000}] (* Artur Jasinski, Dec 13 2006 *)
Select[Prime[Range[3000]],Max[FactorInteger[#-1][[;;,1]]]==5&] (* Harvey P. Dale, Jul 07 2024 *)

{ default(primelimit, 167772161); n=0; forprime (p=3, 167772161, f=factor(p - 1)~; if (f[1, length(f)]==5, write("b061599.txt", n++, " ", p)) ) } \\ Harry J. Smith, Jul 25 2009

list(lim)=my(v=List(), s, t); lim\=1; lim--; for(i=1, logint(lim\2, 5), t=2*5^i; for(j=0, logint(lim\t, 3), s=t*3^j; while(s<=lim, if(isprime(s+1), listput(v, s+1)); s<<=1))); Set(v) \\ Charles R Greathouse IV, Oct 29 2018

Primes of the form 2^a*3^b*5^c + 1 with a and c > 0.

A125867 Numbers k such that p=6k+1 is prime and cos(2*Pi/p) is an algebraic number of a 3-smooth degree, but not 2-smooth.

Original entry on oeis.org

1, 2, 3, 6, 12, 16, 18, 27, 32, 72, 81, 96, 128, 192, 216, 243, 432, 486, 576, 648, 1728, 2048, 2916, 3072, 6561, 8748, 23328, 24576, 34992, 55296, 78732, 104976, 124416, 131072, 139968, 165888, 196608, 248832, 294912, 331776, 442368, 839808

Offset: 1

Artur Jasinski, Dec 13 2006

nonn

Numbers k such that p=6k+1 is prime and the greatest prime divisor of p-1 is 3.

Cf. A024899, A058383, A125866-A125878.

Do[If[Take[FactorInteger[EulerPhi[6n+1]][[ -1]], 1]=={3} && PrimeQ[6n+1],Print[n]],{n,1,100000}]

Edited by Don Reble, Apr 24 2007

A266268 Numbers n such that phi(n) = 3*phi(n-1).

Original entry on oeis.org

7, 13, 19, 37, 73, 91, 97, 109, 163, 193, 433, 487, 577, 703, 769, 793, 925, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 4699, 5551, 6697, 7999, 8701, 10369, 10591, 11803, 12289, 16471, 17497, 18433, 33251, 39367, 52489, 56791, 79249, 124357, 127927, 137899

Offset: 1

Jaroslav Krizek, Dec 26 2015

nonn

Prime terms are in A058383.
See A266276(n) = the smallest numbers k such that phi(k) = n * phi(k-1) for n >=1: 2, 3, 7, 1261, 11242771, ...
Number of terms < 10^k: 1, 7, 17, 29, 41, 86, 205, 446, 1001, 2295, ..., . - Robert G. Wilson v, Jan 24 2016
All terms are == +-1 (mod 6) but mostly 1 (> 95%). - Robert G. Wilson v, Jan 24 2016

			19 is in the sequence because phi(19) = 18 = 3*phi(18) = 3*6.

Robert G. Wilson v, Table of n, a(n) for n = 1..2467 (first 405 terms from G. C. Greubel)

Cf. A000010, A058383, A171271 (numbers n such that phi(n) = 2*phi(n-1)), A266276.

[n: n in [2..2*10^5] | EulerPhi(n) eq 3*EulerPhi(n-1)]; // Vincenzo Librandi, Dec 26 2015

Select[Range[5000], EulerPhi[ # ]==3*EulerPhi[ #-1]&] (* G. C. Greubel, Dec 26 2015 *)

isok(n) = eulerphi(n) == 3*eulerphi(n-1); \\ Michel Marcus, Dec 27 2015

lista(nn) = for(n=1, nn, if(eulerphi(n) == 3*eulerphi(n-1), print1(n, ", "))); \\ Altug Alkan, Jan 24 2016

a(n) = A067143(n) + 1.

A217035 Generalized cuban primes (A007645) which are also Class 1- (or Pierpont) primes (A005109).

Original entry on oeis.org

3, 7, 13, 19, 37, 73, 97, 109, 163, 193, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489, 139969, 147457, 209953, 331777, 472393, 629857, 746497, 786433, 839809, 995329, 1179649, 1492993, 1769473

Offset: 1

Jonathan Vos Post, Sep 24 2012

Is this the union of A058383 and {3}? - R. J. Mathar, Sep 28 2012
Yes, it is, because the only Fermat prime == 0 or 1 mod 3 is 3. - Robert Israel, Mar 02 2018
Generalized cuban primes are primes of the form x^2 + xy + y^2; or: primes of form x^2 + 3*y^2; or: primes == 0 or 1 mod 3. Class 1- (or Pierpont) primes: primes of the form 2^t*3^u + 1.

Ray Chandler, Table of n, a(n) for n = 1..8379 (terms < 10^1000)

Cf. A007645, A005109, A058383.

nn = 100000; t1 = Join[{3}, Select[Prime[Range[nn]], MemberQ[{1}, Mod[#, 3]] &]]; t2 = Select[Prime[Range[nn]], Max @@ First /@ FactorInteger[# - 1] < 5 &]; Intersection[t1, t2] (* T. D. Noe, Sep 26 2012 *)

A007645 INTERSECTION A005109.

A284037 Primes p such that p-1 and p+1 have two distinct prime factors.

Original entry on oeis.org

11, 13, 19, 23, 37, 47, 53, 73, 97, 107, 163, 193, 383, 487, 577, 863, 1153, 2593, 2917, 4373, 8747, 995327, 1492993, 1990657, 5308417, 28311553, 86093443, 6879707137, 1761205026817, 2348273369087, 5566277615617, 7421703487487, 21422803359743, 79164837199873

Offset: 1

Giuseppe Coppoletta, Mar 28 2017

nonn

Either p-1 or p+1 must be of the form 2^i * 3^j, since among three consecutive numbers exactly one is a multiple of 3. - Giovanni Resta, Mar 29 2017

Subsequence of A219528. See the previous comment. - Jason Yuen, Mar 08 2025

			7 is not a term because n + 1 = 8 has only one prime factor.
23 is a term because it is prime and n - 1 = 22 has two distinct prime factors (2, 11) and n + 1 = 24 has two distinct prime factors (2, 3).
43 is not a term because n - 1 = 42 has three distinct prime factors (2, 3, 7).

Robert Israel, Table of n, a(n) for n = 1..51 (all terms < 10^75)

Cf. A000668, A001221, A005105, A005109, A033845, A058383, A067386, A092506, A215504, A219528, A275598.

N:= 10^20: # To get all terms <= N
Res:= {}:
for i from 1 to ilog2(N) do
  for j from 1 to floor(log[3](N/2^i)) do
    q:= 2^i*3^j;
    if isprime(q-1) and nops(numtheory:-factorset((q-2)/2^padic:-ordp(q-2,2)))=1 then Res:= Res union {q-1} fi;
    if isprime(q+1) and nops(numtheory:-factorset((q+2)/2^padic:-ordp(q+2,2)))=1 then Res:= Res union {q+1} fi
od od:
sort(convert(Res,list)); # Robert Israel, Apr 16 2017

mx = 10^30; ok[t_] := PrimeQ[t] && PrimeNu[t-1]==2==PrimeNu[t+1]; Sort@ Reap[Do[ w = 2^i 3^j; Sow /@ Select[ w+ {1,-1}, ok], {i, Log2@ mx}, {j, 1, Log[3, mx/2^i]}]][[2, 1]] (* terms up to mx, Giovanni Resta, Mar 29 2017 *)

isok(n) = isprime(n) && (omega(n-1)==2) && (omega(n+1)==2); \\ Michel Marcus, Apr 17 2017

omega=sloane.A001221; [n for n in prime_range(10^6) if 2==omega(n-1)==omega(n+1)]

sorted([2^i*3^j+k for i in (1..40) for j in (1..20) for k in (-1,1) if is_prime(2^i*3^j+k) and sloane.A001221(2^i*3^j+2*k)==2])

A001221(a(n)) = 1 and A001221(a(n) - 1) = A001221(a(n) + 1) = 2.

a(33)-a(34) from Giovanni Resta, Mar 29 2017

A061638 Primes p such that the greatest prime divisor of p-1 is 7.

Original entry on oeis.org

29, 43, 71, 113, 127, 197, 211, 281, 337, 379, 421, 449, 491, 631, 673, 701, 757, 883, 1009, 1051, 1373, 1471, 2017, 2269, 2521, 2647, 2689, 2801, 3137, 3361, 3529, 4201, 4481, 5881, 6301, 7001, 7057, 7351, 7561, 7841, 8233, 8821, 10501, 10753, 12097

Offset: 1

Labos Elemer, Jun 13 2001

nonn

Prime numbers n for which cos(2*Pi/n) is an algebraic number of 7th degree. - Artur Jasinski, Dec 13 2006

			For n = {4, 8, 9, 12}, a(n)-1 = {70, 210, 280, 420} = 7*{10, 30, 40, 60}.

Harry J. Smith and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 500 terms from Smith)

The 4th in a family of sequences after A019434(=Fermat-primes), A058383, A061599.
Cf. A019434, A058383, A023503, A034694, A006530, A006093, A035095, A061599.
Cf. A004729, A058383, A125867-A125875, A024899.

Select[Prime[Range[2000]],FactorInteger[#-1][[-1,1]] ==7&]  (* Harvey P. Dale, Mar 12 2011 *)

default(primelimit, 108864001); n=0; forprime (p=3, 108864001, f=factor(p - 1)~; if (f[1, length(f)]==7, write("b061638.txt", n++, " ", p))) \\ Harry J. Smith, Jul 25 2009

list(lim)=my(v=List(), t, t5, t7); lim\=1; lim--; for(a=1, logint(lim\2, 7), t7=2*7^a; for(b=0, logint(lim\t7, 5), t5=5^b*t7; for(c=0, logint(lim\t5, 3), t=3^c*t5; while(t<=lim, if(isprime(t+1), listput(v, t+1)); t<<=1)))); Set(v) \\ Charles R Greathouse IV, Oct 29 2018

Primes of form 2^a*3^b*5^c*7^d + 1 with a and d > 1.

A277173 Numbers m such that b^sigma(m) == b^phi(m) == b^numdiv(m) == b^m (mod m) for every integer b.

Original entry on oeis.org

1, 2, 6, 12, 24, 60, 120, 126, 240, 420, 480, 504, 672, 780, 1248, 1260, 2340, 2520, 3360, 4680, 5040, 5460, 6240, 6552, 8160, 8736, 9360, 10080, 11424, 16380, 21216, 26208, 27360, 32760, 38304, 43680, 57120, 65520, 71136, 74592, 106080, 131040, 147168, 148512, 171360, 191520, 202464, 325920, 355680, 372960

Offset: 1

David A. Corneth and Altug Alkan, Oct 02 2016

nonn

Are terms products products of primes of the form 2^i*3^j + 1, A058383, for some nonnegative i and j? This is true for all terms up to 7.6*10^6. 7600320 is divisible by 29, which isn't of the form 2^j*3^i+1. Up to 10^8, all of the terms are divisible by only 16 distinct prime factors. That is: omega(lcm(all terms up to 10^8)) = 16.

Subsequence of A124240.

			6 is a term because for the primes up to 6, (2, 3 and 5), b^sigma(6) == b^phi(6) == b^numdiv(6) == b^6 (mod 6). This is sufficient to prove for all values b up to 6.

Robert G. Wilson v, Table of n, a(n) for n = 1..510

Cf. A124240.

fQ[n_] := Block[{b = 2, s = DivisorSigma[1, n], e = EulerPhi[n], d = DivisorSigma[0, n]}, While[b < n && PowerMod[b, s, n] == PowerMod[b, e, n] == PowerMod[b, d, n] == PowerMod[b, n, n], b = NextPrime@ b]; b >= n]; lst = {1}; k = 2; While[k < 400000, If[ fQ@ k, AppendTo[lst, k]]; k ++]; lst (* Robert G. Wilson v, Nov 04 2016 *)

isk(n, k) = {Mod(k, n)^sigma(n)==Mod(k, n)^n && Mod(k, n)^eulerphi(n)==Mod(k, n)^n && Mod(k, n)^numdiv(n)==Mod(k, n)^n}
is(n) = my(i);forprime(i=2, n, if(isk(n, i)==0,return(0))) ; 1
upto(lim) = my(l=List());for(n=1, lim, if(is(n), listput(l,n))); l

A379445 a(n) = gpf(prime(n)-1)*gpf(prime(n)+1), where gpf is A006530.

Original entry on oeis.org

4, 6, 6, 15, 21, 6, 15, 33, 35, 10, 57, 35, 77, 69, 39, 145, 155, 187, 21, 111, 65, 287, 55, 21, 85, 221, 159, 33, 133, 14, 143, 391, 161, 185, 95, 1027, 123, 581, 1247, 445, 65, 57, 291, 77, 55, 371, 259, 2147, 437, 377, 85, 55, 35, 86, 1441, 335, 85, 3197, 329, 3337

Offset: 2

Hugo Pfoertner, Dec 28 2024

nonn

Observation: Even terms of A006881 not occurring in this sequence are, e.g., 22, 34, 38, 46, ..., due to the sparseness of Mersenne primes (A000668) and Fermat primes (A000215). Also missing are many multiples of 3, e.g., 3*{31, 67, 79, 83, 101, 103, 113, ...}, as a consequence of the gaps of A058383 and A268640 and the size distribution of prime factors, i.e., the rareness of smooth numbers.

			a(43390) = 146 because 2^19-1 = A000668(5) is the 43390th prime and the greatest prime factor of 2^19-2 is 73.

Hugo Pfoertner, Table of n, a(n) for n = 2..10000
Hugo Pfoertner, 1 million terms of A379445, 7z compressed file (5 MB) (2025).

Cf. A006530, A023503, A023509, A087713.
Each term > 4 is element of A006881.
Cf. A000215, A000668, A058383, A268640.

Table[Times @@ Map[FactorInteger[#][[-1, 1]] &, Prime[n] + {-1, 1}], {n, 2, 61}] (* Michael De Vlieger, Jan 20 2025 *)

a379445(n) = my (p=prime(n), fm=factor(p-1), fp=factor(p+1)); fm[#fm~,1]*fp[#fp~,1]

a(n) = A023503(n)*A023509(n). - Michel Marcus, Jan 21 2025

A125868 Odd numbers k such that cos(2*Pi/k) is an algebraic number of a 5-smooth degree, but not 3-smooth.

Original entry on oeis.org

11, 25, 31, 33, 41, 55, 61, 75, 77, 93, 99, 101, 123, 125, 143, 151, 155, 165, 175, 181, 183, 187, 205, 209, 217, 225, 231, 241, 251, 271, 275, 279, 287, 297, 303, 305, 325, 341, 369, 375, 385, 401, 403, 407, 425, 427, 429, 451, 453, 465, 475, 495, 505, 525

Offset: 1

Artur Jasinski, Dec 13 2006

nonn

A regular polygon of a(n) sides can be constructed if one also has an angle trisector and 5-sector.

Cf. A058383, A061599, A125866-A125878.

Do[If[Take[FactorInteger[EulerPhi[2n+1]][[ -1]],1]=={5},Print[2n+1]],{n,1,1000}]

Edited by Don Reble, Apr 24 2007

cp's OEIS Frontend

A336772 Sums s of positive exponents such that no prime of the form 2^j*3^k + 1 with j + k = s exists.

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A061599 Primes p such that the greatest prime divisor of p-1 is 5.

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A125867 Numbers k such that p=6k+1 is prime and cos(2*Pi/p) is an algebraic number of a 3-smooth degree, but not 2-smooth.

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A266268 Numbers n such that phi(n) = 3*phi(n-1).

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Magma

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A217035 Generalized cuban primes (A007645) which are also Class 1- (or Pierpont) primes (A005109).

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A284037 Primes p such that p-1 and p+1 have two distinct prime factors.

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Maple

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Sage

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A061638 Primes p such that the greatest prime divisor of p-1 is 7.

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