A005105 -id:A005105 - cp's OEIS frontend

A081640 a(n) = n-th prime of class 12- according to the Erdős-Selfridge classification.

14920303, 18224639, 24867247, 26532953, 34548443, 38003011, 39800743, 41319599, 41443483, 45604771, 46432667, 47247763, 49734341, 49734493, 49749439, 51591833, 53014667, 55257977, 59681383, 59700749, 60804817

Offset: 1

Robert G. Wilson v, Mar 23 2003

nonn

The first 184 resp. 300 terms of A081430 allow us to deduce 44 resp. 84 consecutive terms of this sequence. - M. F. Hasler, Apr 05 2007

			a(1) = 14920303 = 1+2*A081430(3)*3 is the smallest 12- prime

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

M. F. Hasler, Table of n, a(n) for n=1..282

Cf. A005113, A005105, A005106, A005107, A005108, A081633, A081633, A081635, A081636, A081637, A081638.

Cf. A056637, A081430, A081641.

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[3610000], ClassMinusNbr[ Prime[ # ]] == 12 &]]

nextclassminus( a, p=1, n=[] )={ while( p, n=concat(n,p); p=0; for( i=1,#a, if( p & 2*a[i] >= p-1, break); for( k=ceil(n[ #n]/2/a[i]),a[ #a]/a[i], if( p & 2*k*a[i] >= p-1, break); if( isprime(2*k*a[i]+1), p=2*k*a[i]+1; break(1+(k==1)); ))));vecextract(n,"^1")}; A081640 = nextclassminus(A081430) \\ M. F. Hasler, Apr 05 2007

{ a(n) } = { p = 2*m*A081430(k)+1 | k=1,2,...,oo and m=1,2,... such that p is prime and m has no factor of class > 11- } - M. F. Hasler, Apr 05 2007

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 21 2007

A081641 a(n) = n-th prime of class 13- according to the Erdős-Selfridge classification.

Original entry on oeis.org

36449279, 53065907, 59681213, 69096887, 132756479, 135388367, 164255999, 179043637, 188991053, 207290663, 241560239, 279709259, 309550999, 364492781, 372993983, 377982103, 398007431, 406165099, 425633717, 445901987, 447609067, 516737983

Offset: 1

Robert G. Wilson v, Mar 23 2003

nonn

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

M. F. Hasler, Table of n, a(n) for n=1..1641

Cf. A005113, A005105, A005106, A005107, A005108, A081633, A081633, A081635, A081636, A081637, A081638.

Cf. A056637, A081640, A129248.

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[23733333], ClassMinusNbr[ Prime[ # ]] == 12 &]]

A081641 = nextclassminus(A081640) /* cf. A081640 - M. F. Hasler, Apr 05 2007 */

Edited by N. J. A. Sloane, May 14 2008 at the suggestion of R. J. Mathar.

A129472 Primes p of Erdos-Selfridge class 4+ with largest prime factor of p+1 not of class 3+.

Original entry on oeis.org

3181, 4513, 4957, 6067, 7177, 8731, 9397, 10433, 13171, 14947, 15761, 17389, 19387, 19609, 22051, 22273, 22453, 22717, 23531, 23753, 24197, 26161, 27823, 28711, 37369, 37591, 38183, 38923, 39293, 40993, 41143, 42697, 43067, 44621, 44843

Offset: 1

M. F. Hasler, Apr 17 2007

See A129470

			a(1) = 3181 = -1+2*37*43 is a prime of class 4+ since 37 is of class 3+, but the largest divisor of 3181+1 is 43 which is only of class 2+.

Subsequence of A129470; see also A129471, A129473, A129477-A129478, A129469, A005113, A005105-A005107.

class(n,s=1)={n=factor(n+s)[,1];if(n[ #n]<=3,1,for(i=2,#n,n[1]=max(class(n[i],s)+1,n[1]));n[1])}; A129472(n=100,p=1,a=[])={ local(f); while( #a 3 & 3 > class(f[ #f]), f=factor(1+p=nextprime(p+1))[,1]); forstep( i=#f-1,2,-1, if( 4 < f[1] = max( f[1],1+class( f[i] )), next(2))); if( f[1] == 4, a=concat(a,p); /*print(#a," ",p)*/)); a}

A173062 Primes of the form 2^r * 13^s - 1.

Original entry on oeis.org

3, 7, 31, 103, 127, 337, 1663, 5407, 8191, 131071, 346111, 524287, 2970343, 3655807, 22151167, 109051903, 617831551, 1631461441, 2007952543, 2147483647, 32127240703, 194664464383, 275716983697, 958348132351, 1357375919743, 1670616516607, 49834102882303, 57349132609183

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Feb 09 2010

nonn

s = 0 is "trivial" case of Mersenne primes: 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ...
Mersenne prime exponents r: 2, 3, 5, 7, 13, 17, 19, 31, ...
Necessarily r odd as for r = 2*k and p a prime of form 6*n+1: 2^(2*k) * p^j - 1 a multiple of 3.
Proof by induction with 2^2 * p^1 - 1 = 4*(6*n+1) - 1 = 3*(8*n+1), 2^2(k+1) * p^j - 1 = 4* (2^k * p^j - 1) + 3.
No prime in case i = j = k (k>1) as a^k - 1 has divisor a - 1.

			2^2*13^0 - 1 = 3 = prime(2) => a(1).
2^3*13^1 - 1 = 103 = prime(27) => a(4).
2^7*13^9 - 1 = 1357375919743 = prime(50467169414) => a(25).
list of (r,s) pairs: (2,0), (3,0), (5,0), (3,1), (7,0), (1,2), (7,1), (5,2), (13,0), (17,0), (11,2), (19,0), (3,5), (7,4), (17,2), (23,1), (7,6), (1,8), (5,7), (31,0), (9,7), (19,5), (1,10), (25,4), (7,9), (11,8), (27,5), (5,11), (25,6), (19,8), (13,10), (3,13), (29,7), (5,14), (39,5), (15,13), (5,16), ...

Peter Bundschuh, Einfuehrung in die Zahlentheorie, Springer-Verlag GmbH Berlin, 2002.
Leonard E. Dickson, History of the Theory of numbers, vol. I, Dover Publications, 2005.
Paulo Ribenboim, Wilfrid Keller, and Joerg Richstein, Die Welt der Primzahlen, Springer-Verlag GmbH Berlin, 2006.

Jinyuan Wang, Table of n, a(n) for n = 1..950

Cf. A000668, A005105, A077313, A077314, A077315.

lista(nn) = {my(q=1/2, p, w=List([])); for(r=0, logint(nn, 2), q=2*q; p=q/13; for(s=0, logint(nn\q, 13), p=13*p; if(ispseudoprime(p-1), listput(w, p-1)))); Set(w); } \\ Jinyuan Wang, Feb 23 2020

from itertools import count, islice
from sympy import integer_log, isprime
def A173062_gen(): # generator of terms
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(n):
        def f(x): return n+x-sum(((x+1)//13**i).bit_length() for i in range(integer_log(x+1,13)[0]+1))
        return bisection(f,n-1,n-1)
    return filter(lambda n:isprime(n), map(g,count(1)))
A173062_list = list(islice(A173062_gen(),30)) # Chai Wah Wu, Mar 31 2025

a(26)-a(28) from Jinyuan Wang, Feb 23 2020

A173236 Primes of the form 2^r * 13^s + 1.

Original entry on oeis.org

2, 3, 5, 17, 53, 257, 677, 3329, 13313, 35153, 65537, 2768897, 13631489, 2303721473, 3489660929, 4942652417, 11341398017, 10859007357953, 1594691292233729, 31403151600910337, 310144109150467073, 578220423796228097

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Feb 13 2010

nonn

Necessarily r is even (elementary proof by induction).
s=0 is (trivial) case of 2 and the known five Fermat primes: 2, 3, 5, 17, 257, 65537 (A092506).
Fermat prime exponents r are 0, 1, 2, 4, 8, 16.

			2^0*13^0 + 1 = 2 = prime(1) => a(1).
2^10*13^1 + 1 = 13313 = prime(1581) => a(9).
list of (r,s): (0,0), (1,0), (2,0), (4,0), (2,1), (8,0), (2,2), (8,1), (10,1), (4,3), (16,0), (14,2), (20,1), (20,3), (28,1), (10,6), (26,2), (10,9), (32,5), (40,4), (10,13), (22,10), (32,8), (48,4), (20,13), (2,18), (28,11), (50,6).

Emil Artin, Galoissche Theorie, Verlag Harri Deutsch, Zürich, 1973.
Leonard E. Dickson, History of the Theory of numbers, vol. I, Dover Publications, 2005.
Paulo Ribenboim, Wilfrid Keller, and Joerg Richstein, Die Welt der Primzahlen, Springer-Verlag GmbH Berlin, 2006.

Cf. A092506, A005105, A092506, A173062.

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)=[2,13] then Add(C,Position(B,i)); fi; od;
A173236:=Concatenation([2],List(C,i->A[i])); # Muniru A Asiru, Sep 10 2017

from itertools import count, islice
from sympy import isprime, integer_log
def A173236_gen(): # generator of terms
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(n):
        def f(x): return n+x-sum(((x-1)//13**i).bit_length() for i in range(integer_log(x-1,13)[0]+1))
        return bisection(f,n+1,n+1)
    return filter(lambda n:isprime(n), map(g,count(1)))
A173236_list = list(islice(A173236_gen(),30)) # Chai Wah Wu, Mar 31 2025

A333646 Numbers k divisible by the largest prime factor of the sum of divisors of k; a(1) = 1.

Original entry on oeis.org

1, 6, 15, 28, 30, 33, 40, 42, 51, 66, 69, 84, 91, 95, 102, 105, 117, 120, 135, 138, 140, 141, 145, 159, 165, 182, 186, 190, 210, 213, 224, 231, 234, 255, 270, 273, 280, 282, 285, 287, 290, 295, 308, 318, 321, 330, 345, 357, 364, 395, 420, 426, 435, 440, 445, 455

Offset: 1

Amiram Eldar, Jun 05 2020

nonn

Pomerance (1973) proved that all the harmonic numbers (A001599) are in this sequence.
If m is a product of distinct Mersenne primes (A046528), m > 1 and 3 | m, then 2*m is a term.
If p is a term of A005105 then, 6*p is a term for p > 3, and 3*p is a term if p is not a Mersenne prime (A000668).

			15 is a term since sigma(15) = 24, 3 is the largest prime factor of 24, and 15 is divisible by 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Carl Pomerance, On a Problem of Ore: Harmonic Numbers, unpublished manuscript, 1973; abstract *709-A5, Notices of the American Mathematical Society, Vol. 20, 1973, page A-648, entire volume.

A001599 and A105402 are subsequences.

Cf. A000203, A000668, A005105, A006530, A046528, A071190.

Select[Range[500], Divisible[#, FactorInteger[DivisorSigma[1, #]][[-1, 1]]] &]

Numbers k such that A071190(k) | k.

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);

A376699 Positions of primes in the sequence of numbers of the form 2^i * 3^j - 1 (A069353).

Original entry on oeis.org

3, 4, 5, 6, 8, 10, 11, 13, 15, 16, 18, 21, 22, 25, 31, 32, 36, 39, 40, 42, 51, 57, 61, 63, 65, 66, 71, 73, 79, 82, 94, 97, 106, 107, 110, 120, 121, 127, 128, 129, 130, 138, 142, 144, 161, 192, 204, 205, 212, 216, 232, 234, 244, 259, 264, 265, 308, 329, 346, 348

Offset: 1

Amiram Eldar, Oct 02 2024

nonn

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

Cf. A003586, A005105, A069353, A375906, A376700, A376701.

With[{lim = 10^10}, Position[Sort@ Flatten@ Table[2^i*3^j - 1, {i, 0, Log2[lim]}, {j, 0, Log[3, lim/2^i]}], _?PrimeQ] // Flatten]

lista(lim) = {my(s = List()); for(i = 0, logint(lim, 2), for(j = 0, logint(lim >> i, 3), listput(s, 2^i * 3^j - 1))); s = Set(s); for(i = 1, #s, if(isprime(s[i]), print1(i, ", ")));}

from itertools import count, islice
from sympy import isprime, integer_log
def A069353(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(((x+1)//3**i).bit_length() for i in range(integer_log(x+1,3)[0]+1))
    return bisection(f,n-1,n-1)
def A376699_gen(): # generator of terms
    return filter(lambda n:isprime(A069353(n)), count(1))
A376699_list = list(islice(A376699_gen(),30)) # Chai Wah Wu, Mar 31 2025

A069353(a(n)) = A003586(a(n)) - 1 = A005105(n).

A101253 a(n) = n-th prime of Erdős-Selfridge classification n+.

Original entry on oeis.org

2, 19, 113, 617, 1877, 8753, 52517, 255043, 1532173, 9287521, 48499459, 353653063, 2136716521, 18171787987, 111795382441

Offset: 1

Jonathan Vos Post, Dec 16 2004

nonn

Diagonalization of the Erdős-Selfridge classification of primes n+. See A101231 for diagonalization of the Erdős-Selfridge classification of primes n-.

			a(1) = 2 because 2 is the first element of A005105.
a(2) = 19 because 19 is the 2nd element of A005106.
a(3) = 113 because 113 is the 3rd element of A005107.
a(4) = 617 because 617 is the 4th element of A005108.
a(5) = 1877 because 1877 is the 5th element of A081633.
a(6) = 8753 because 8753 is the 6th element of A081634.

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

Cf. A101231, A005105, A005106, A005107, A005108, A081633, A081634, A081635, A081636, A081637, A081638, A081639, A005113, A019268.

Cf. A084071, A090468, A129474, A129475.

More terms from David Wasserman, Mar 26 2008

A268640 Primes of the form 2^i * 3^j - 1 for positive i, j.

Original entry on oeis.org

5, 11, 17, 23, 47, 53, 71, 107, 191, 383, 431, 647, 863, 971, 1151, 2591, 4373, 6143, 6911, 8747, 13121, 15551, 23327, 27647, 62207, 73727, 139967, 165887, 294911, 314927, 442367, 472391, 497663, 786431, 995327, 1062881, 2519423, 10616831, 17915903, 18874367

Offset: 1

Muniru A Asiru, Oct 15 2017

nonn

a(n) is congruent to 5 (mod 6).

			a(1) = 5 = 2^1 * 3^1 - 1.
a(2) = 11 = 2^2 * 3^1 - 1.
a(3) = 17 = 2^1 * 3^2 - 1.
a(4) = 23 = 2^3 * 3^1 - 1.
a(5) = 47 = 2^4 * 3^1 - 1.
List of (i, j): (1, 1), (2, 1), (1, 2), (3, 1), (4, 1), (1, 3), (3, 2), (2, 3), (6, 1), (7, 1), (4, 3), (3, 4), (5, 3), (2, 5), (7, 2), (5, 4), ...

Ray Chandler, Table of n, a(n) for n = 1..7151 (terms < 10^1000)

Cf. A000040, A000668, A005105.

K:=10^7+1;; # to get all terms <= K.
A:=Filtered([1..K],IsPrime);;
A268640:=List(Positions(List(A,i->Elements(Factors(i+1))),[2,3]),i->A[i]);

N:= 10^10: # to get all terms <= N
R:= {}:
for b from 1 to floor(log[3]((N+1)/2)) do
     R:= R union select(isprime, {seq(2^a*3^b-1,
          a=1..ilog2((N+1)/3^b))})
od:
sort(convert(R,list)); # Robert Israel, Oct 15 2017

{ A005105 } \ { 2 } \ { A000668 }.

cp's OEIS Frontend

A081640 a(n) = n-th prime of class 12- according to the Erdős-Selfridge classification.

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A081641 a(n) = n-th prime of class 13- according to the Erdős-Selfridge classification.

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A129472 Primes p of Erdos-Selfridge class 4+ with largest prime factor of p+1 not of class 3+.

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A173062 Primes of the form 2^r * 13^s - 1.

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A173236 Primes of the form 2^r * 13^s + 1.

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A333646 Numbers k divisible by the largest prime factor of the sum of divisors of k; a(1) = 1.

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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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