A005105 -id:A005105 - cp's OEIS frontend

A090468 Class 13+ primes.

545587687, 852480757, 1048438561, 1150849009, 1323457987, 1745980517, 1756123441, 1785398401, 1798736161, 1892507347, 1937020021, 2002155601, 2136716521, 2150905573, 2229004913, 2548101601, 2671514917, 2838761021

Offset: 1

Robert G. Wilson v, Nov 26 2003

nonn

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

M. F. Hasler, Table of n, a(n) for n=1..850

Cf. A005113, A005105 - A005108, A081633 - A081639, A084071, A129474.

For Maple program see Mathar link in A005105.

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]]; ClassPlusNbr[n_] := Length[ NestWhileList[f, n, UnsameQ, All]] - 3; Prime[ Select[ Range[195000000], ClassPlusNbr[ Prime[ # ]] == 13 &]]

A129470 Primes p such that the largest prime factor of p+1 has Erdős-Selfridge class+ < N-1 if p is of class N+.

Original entry on oeis.org

883, 1747, 2417, 2621, 3181, 3301, 3533, 3571, 3691, 3853, 4027, 4133, 4513, 4783, 4861, 4957, 5303, 5381, 5393, 5563, 5641, 5821, 6067, 6577, 6991, 7177, 7253, 7331, 8059, 8093, 8377, 8731, 8839, 8929, 8969, 9221, 9281, 9397, 9613, 9931

Offset: 1

M. F. Hasler, Apr 16 2007

In practice the class+ of a prime p is most often given by 1 + the class of the largest prime factor of p+1; terms of this sequence are counterexamples to this "rule". Terms of this sequence are at least of class 3+, since primes of class 1+ and 2+ have all prime factors of p+1 of class 1+. Terms a(k) of this sequence are >= -1 + 2*A005113(N-1) * nextprime(A005113(N-1)), where N is the class of a(k).

			a(3) = 883 = -1 + 2*13*17 is a prime of class 3+ since 13 is of class 2+, but the largest divisor of 883+1 is 17 which is only of class 1+.

Cf. A005105, A005106, A005107, A005108, A005113, A081633, A081634, A081635, A081636, A081637, A081638, A081639, A084071, A090468, A129469, A129471, A129472, A129473, A129474, A129475, A129477, A129478.

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])}; A129470(n=100,p=1,a=[])={ local(f); while( #a 3, f=factor(1+p=nextprime(p+1))[,1]); forstep( i=#f,2,-1, f[i]=class( f[i] ); if( f[i] > f[ #f], a=concat(a,p); /*print(#a," ",p);*/ break))); a}

A129474 Primes of Erdos-Selfridge class 14+.

Original entry on oeis.org

1704961513, 7281416041, 7638227617, 9462536833, 11934730597, 13237911481, 13282423003, 13522629793, 13942983841, 14185279861, 16029089501, 16221987853, 17434233041, 18171787987, 19639505461, 20717555041

Offset: 1

M. F. Hasler, Apr 16 2007

nonn

Primes of class r (or r+) are by definition the primes p for which p + 1 has all factors of a lower class < r, but at least one factor of class r - 1. See A005113 for more information.

a(1..149) calculated using A090468 up to 37.5e9, which gives A129474(150) > 75e9.

			a(1) = A005113[14] = 1704961513 = -1+2*852480757, where 852480757 = A090468[2]

M. F. Hasler, Table of n, a(n) for n = 1..149

Cf. A005113, A005105-A005108, A081633-A081639, A084071, A090468, A129475.

class(n, s=1) = { if(!isprime(n),0, if(!(n=factor(n+s)[,1]) || n[ #n]<=3,1, for(i=2,#n,n[1]=max(class(n[i],s)+1,n[1]));n[1]))};
nextclass(a,s=1,p,n=[])={if(!p,p=nextprime(a[ #a]+1)); print("producing primes of class ",1+class(a[1],s),["+","-"][1+(s<0)]," up to 2*",p); for(i=1,#a,for(k=1,p/a[i],if(isprime(2*k*a[i]-s),n=concat(n,2*k*a[i]-s))));vecsort(n)};
A129474=nextclass(A090468,1)

{ a(n) } = { p = 2*m*A090468(k)-1 | k=1,2,3... and m=1,2,3... such that p is prime and m has no factor of class > 13+ }

A129471 Primes p of Erdos-Selfridge class 3+ with largest prime factor of p+1 not of class 2+.

Original entry on oeis.org

883, 1747, 2417, 2621, 3301, 3533, 3571, 3691, 3853, 4027, 4133, 4783, 4861, 5303, 5381, 5393, 5563, 5641, 5821, 6577, 6991, 7253, 7331, 8059, 8093, 8377, 8839, 8929, 8969, 9221, 9281, 9613, 9931, 10069, 10477, 10487, 10601, 10607, 10903, 11491

Offset: 1

M. F. Hasler, Apr 17 2007

See A129470

			a(1) = 883 = -1+2*13*17 is a prime of class 3+ since 13 is of class 2+, but the largest divisor of 883+1 is 17 which is only of class 1+.

Subsequence of A129470; see also A129472-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])}; A129471(n=100,p=1,a=[])={ local(f); while( #a 3 & 2 > class(f[ #f]), f=factor(1+p=nextprime(p+1))[,1]); forstep( i=#f-1,2,-1, if( 3 < f[1] = max( f[1],1+class( f[i] )), next(2))); if( f[1] == 3, a=concat(a,p); /*print(#a," ",p)*/)); a}

A129473 Primes p of Erdos-Selfridge class 5+ with largest prime factor of p+1 not of class 4+.

Original entry on oeis.org

15913, 18541, 22921, 36353, 47741, 49201, 52267, 55333, 60589, 64969, 66137, 66721, 69203, 72707, 73291, 74167, 75773, 78401, 79861, 80737, 82051, 84533, 90227, 90373, 95191, 95483, 95629, 97673, 99133, 101323, 103951, 104681, 104827

Offset: 1

M. F. Hasler, Apr 17 2007

See A129470

			a(1) = 15913 = -1+2*73*109 is a prime of class 5+ since 73 is of class 4+, but the largest divisor of 15913+1 is 109 which is only of class 2+.

Subsequence of A129470; see also A129471-A129472, A129477-A129478, A129469, A005113, A005105-A005108, A081633.

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])}; A129473(n=100,p=1,a=[])={ local(f); while( #a 3 & 4 > class(f[ #f]), f=factor(1+p=nextprime(p+1))[,1]); forstep( i=#f-1,2,-1, if( 5 < f[1] = max( f[1],1+class( f[i] )), next(2))); if( f[1] == 5, a=concat(a,p); /*print(#a," ",p)*/)); a}

A129477 Primes p of Erdos-Selfridge class 6+ with largest prime factor of p+1 not of class 5+.

Original entry on oeis.org

2146141, 2182897, 2954773, 3199813, 3224317, 3285577, 3383593, 3505933, 3555121, 3567373, 3653137, 3775417, 3864037, 4250977, 4298533, 4328053, 4493773, 4504651, 4519981, 4572037, 4647277, 4692637, 4719061, 4726537

Offset: 1

M. F. Hasler, Apr 17 2007

See A129470

			a(1) = 2146141 = -1+2*1021*1051 = A129469[6] is a prime of class 6+ since 2146141+1 has prime factor 1021=A081633[1]=A005113[5] of class 5+, but the largest prime factor of 2146141+1 is 1051=A005107[65] of class 3+.

Subsequence of A129470; see also A129471-A129473, A129478, A129469, A005113, A005105-A005108, A081633-A081634.

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])}; a129477(n=100,p=1,a=[])={local(f,a5=A005113[5]);p=max(p,a5*nextprime(a5+1)*2-1); while( #a2 & f[ #f-1] >= a5 & 5 > class(f[ #f]), f=factor(1+p=nextprime(p+1))[,1]); forstep( i=#f-1,2,-1, if( 6 < f[1] = max( f[1],1+class( f[i] )), next(2))); if( f[1] == 6, a=concat(a,p); print(#a," ",p))); a}

A129478 Primes p of Erdos-Selfridge class 7+ with largest prime factor of p+1 not of class 6+.

Original entry on oeis.org

17227801, 18207913, 18592957, 19433053, 19608073, 19678081, 20028121, 20518177, 20658193, 20833213, 21043237, 21218257, 21533293, 21743317, 22128361, 22303381, 23668537, 25068697, 25418737, 25453741

Offset: 1

M. F. Hasler, Apr 17 2007

nonn

See A129470

			a(1) = 17227801 = -1+2*2917*2953 = A129469[7] is a prime of class 7+ since 17227801+1 has prime factor 2917 = A081634[1] = A005113[6] of class 6+, but the largest prime factor of 17227801+1 is 2953 = A005107[175] of class 3+.

Subsequence of A129470; see also A129471-A129473, A129477, A129469, A005113, A005105-A005108, A081633-A081635.

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])}; a129478(n=100,p=1,a=[])={local(f,a6=A005113[6]);p=max(p,a6*nextprime(a6+1)*2-2); while( #a2 & f[ #f-1] >= a6 & 6 > class(f[ #f]), f=factor(1+p=nextprime(p+1))[,1]); forstep( i=#f-1,2,-1, if( 7 < f[1] = max( f[1],1+class( f[i] )), next(2))); if( f[1] == 7, a=concat(a,p); print(#a," ",p))); a}

A293194 Primes of the form 2^q * 3^r * 5^s - 1.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 19, 23, 29, 31, 47, 53, 59, 71, 79, 89, 107, 127, 149, 179, 191, 199, 239, 269, 359, 383, 431, 449, 479, 499, 599, 647, 719, 809, 863, 971, 1151, 1249, 1279, 1439, 1499, 1619, 1999, 2399, 2591, 2699, 2879, 2999, 4049, 4373, 4799, 4999, 5119, 5399, 6143, 6911

Offset: 1

Muniru A Asiru, Oct 02 2017

nonn

Mersenne primes A000668 occur when (q, r, s) = (q, 0, 0) with q > 0.
a(2) = 3 is a Mersenne prime but a(3) = 7 is not.
For n > 2, all terms = {1, 5} mod 6.

			3 is a member because 3 is a prime number and 2^2 * 3^0 * 5^0 - 1 = 3.
89 is a member because 89 is a prime number and 2^1 * 3^2 * 5^1 - 1 = 89.
list of (q, r, s): (0, 1, 0), (2, 0, 0), (1, 1, 0), (3, 0, 0), (2, 1, 0), (1, 2, 0), (2, 0, 1), (3, 1, 0),(1, 1, 1), (5, 0, 0), (4, 1, 0), (1, 3, 0), (2, 1, 1), ...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000668, A005105, A293425.

K := 10^5 + 1;; # to get all terms less than or equal to K
A := Filtered([1 .. K], IsPrime);; I := [3, 5];;
B := List(A, i -> Elements(Factors(i + 1)));;
C := List([0 .. Length(I)], j -> List(Combinations(I, j), i -> Concatenation([2], i)));
A293194 := Concatenation([2], List(Set(Flat(List([1 .. Length(C)], i -> List([1 .. Length(C[i])], j -> Positions(B, C[i][j]))))), i -> A[i]));

N:= 10^6: # to get all terms <= N
R:= {}:
for c from 0 to floor(log[5]((N+1))) do
  for b from 0 to floor(log[3]((N+1)/5^c)) do
     R:= R union select(isprime, {seq(2^a*3^b*5^c-1,
         a=0..ilog2((N+1)/(3^b*5^c)))})
od od:
sort(convert(R,list)); # Robert Israel, Oct 15 2017

With[{n = 7000}, Sort@ Select[Flatten@ Table[2^q * 3^r * 5^s - 1, {q, 0, Log[2, n/(1)]}, {r, 0, Log[3, n/(2^q)]}, {s, 0, Log[5, n/(2^q * 3^r)]}], PrimeQ]] (* Michael De Vlieger, Oct 02 2017 *)

lista(nn) = {forprime(p=2,nn, if (vecmax(factor(p+1)[,1]) <= 5, print1(p, ", ")););} \\ Michel Marcus, Oct 06 2017

from itertools import count, islice
from sympy import integer_log, isprime
def A293194_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 f(x):
        c = x
        for i in range(integer_log(x,5)[0]+1):
            for j in range(integer_log(m:=x//5**i,3)[0]+1):
                c -= (m//3**j).bit_length()
        return c
    yield from filter(isprime,(bisection(lambda k:n+f(k),n,n)-1 for n in count(1)))
A293194_list = list(islice(A293194_gen(),30)) # Chai Wah Wu, Mar 31 2025

A048252 Largest number whose sum of divisors is 6^n.

Original entry on oeis.org

1, 5, 22, 187, 1219, 7597, 46117, 278857, 1676377, 10067797, 60450517, 362758177, 2176626817, 13060193977, 78363525817, 470183516857, 2820894903487, 16926601754197, 101559860054047, 609359671998037, 3656158318966357

Offset: 0

Labos Elemer

nonn

Terms of this sequence are products of distinct terms in A005105. - Ray Chandler, Sep 01 2010

Ray Chandler, Table of n, a(n) for n = 0..1000

Cf. A006532, A020477, A019422, A019423, A018427, A048251-A048256.

a(n) = {sn = 6^n; forstep(x=sn, 1, -1, if (sigma(x) == sn, return (x)););} \\ Michel Marcus, Dec 15 2013

a(9)-a(14) from Donovan Johnson, Sep 02 2008
a(15)-a(20) from Donovan Johnson, Nov 22 2008
Edited and extended by Ray Chandler, Sep 01 2010

A069356 Primes of the form 2^i*3^j - (i+j) with i, j >= 0.

Original entry on oeis.org

2, 5, 7, 43, 67, 103, 157, 281, 503, 641, 1451, 3061, 4597, 6553, 8179, 10357, 15541, 34981, 78721, 209939, 524269, 1062869, 2097131, 13436909, 25509151, 28311529, 63700969, 113246183, 153054989, 516560633, 573308903, 774840959, 805306339

Offset: 1

Reinhard Zumkeller, Mar 18 2002

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..300 [Corrected by Sean A. Irvine]

Cf. A069355, A005105, A069358, A069346.

Take[ Select[ Union[ Flatten[ Table[2^i*3^j - (i + j), {i, 0, 28}, {j, 0, 18}]]], PrimeQ[ # ] &], 33]

Edited and extended by Robert G. Wilson v, May 09 2003

cp's OEIS Frontend

A090468 Class 13+ primes.

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A129470 Primes p such that the largest prime factor of p+1 has Erdős-Selfridge class+ < N-1 if p is of class N+.

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A129474 Primes of Erdos-Selfridge class 14+.

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A129471 Primes p of Erdos-Selfridge class 3+ with largest prime factor of p+1 not of class 2+.

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A129473 Primes p of Erdos-Selfridge class 5+ with largest prime factor of p+1 not of class 4+.

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A129477 Primes p of Erdos-Selfridge class 6+ with largest prime factor of p+1 not of class 5+.

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A129478 Primes p of Erdos-Selfridge class 7+ with largest prime factor of p+1 not of class 6+.

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A293194 Primes of the form 2^q * 3^r * 5^s - 1.

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