A056637 -id:A056637 - cp's OEIS frontend

A081430 Class 11- primes.

1266767, 1520159, 2486717, 3316619, 4144541, 4512947, 4836779, 5389519, 5638379, 6218827, 6448979, 6633457, 6771419, 6907247, 7460149, 7462639, 7600597, 7739033, 7874627, 8153567, 8291573, 9110639, 9112319, 9121003

Offset: 1

Robert G. Wilson v, Mar 20 2003

nonn

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

M. F. Hasler, Table of n, a(n) for n=1..1000

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

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[300000, 1000000], ClassMinusNbr[ Prime[ # ]] == 1 &]]

A027763 Smallest k such that 2^^n is not congruent to 2^^(n-1) mod k, where 2^^n denotes the power tower 2^2^...^2 (in which 2 appears n times).

Original entry on oeis.org

2, 3, 5, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 531441, 1594323, 4782969, 14348907, 43046721, 86093443, 344373773, 688747547, 3486784401

Offset: 1

R. K. Guy

This sequence shares many terms with A056637, the least prime of class n-. Note that 3^(n-1) is an upper bound for each term and the upper bound is reached for n=13 and n=14. Are all subsequent terms 3^(n-1)? The Mathematica code uses the TowerMod function in the CNT package, which is described in the book by Bressoud and Wagon. - T. D. Noe, Mar 13 2009
For n=15, n=16, and n=17, the terms are also of the form 3^(n-1), but for n=18 and n=19, the terms are prime. - Wayne VanWeerthuizen, Aug 26 2014
A185816(a(n)) = n. - Reinhard Zumkeller, Sep 02 2014
Prime terms seen up to n=20 are in eleven instances of the form j*a(n-1)+1, for j=2, 4, 6, or 12. Note, though, that a(2)=5 and a(8)=719 are exceptions to this pattern. - Wayne VanWeerthuizen, Sep 06 2014

			2^^2=2^2=4 and 2^^3=2^2^2=16. We find 4 = 16 (mod k) until k=5. So a(3)=5. - _T. D. Noe_, Mar 13 2009

David Bressoud and Stan Wagon, A Course in Computational Number Theory, Key College Pub., 2000, p. 96.
Stan Wagon, posting to Problem of the Week mailing list, Dec 15 1997.

D. Bressoud, CNT.m Computational Number Theory Mathematica package.
Stan Wagon, Putnam Problem Notes
Eric W. Weisstein, MathWorld: Power Tower

Cf. A056637, A173927, A185816, A246491.

Needs["CNT`"]; k=1; Table[While[TowerMod[2,n,k]==TowerMod[2,n-1,k], k++ ]; k, {n,10}] (* T. D. Noe, Mar 13 2009 *)

Improved the name and changed the offset because I just prepended a term. - T. D. Noe, Mar 13 2009
Corrected and extended by T. D. Noe, Mar 13 2009
Terms a(15)-a(19) from Wayne VanWeerthuizen, Aug 26 2014
Terms a(20)-a(21) from Wayne VanWeerthuizen, Sep 06 2014

A081428 Class 9- primes.

Original entry on oeis.org

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 &]]

A246491 Smallest k such that 3^^n is not congruent to 3^^(n-1) mod k, where 3^^n denotes the power tower 3^3^...^3 (in which 3 appears n times).

Original entry on oeis.org

3, 5, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 1266767, 14920303, 36449279, 377982107

Offset: 1

Wayne VanWeerthuizen, Aug 27 2014

This sequence is a member of an interesting class of sequences defined by the rule, "Smallest k such that b^^n is not congruent to b^^(n-1) mod k, where b^^n denotes the power tower b^b^...^b (in which b appears n times)," for some constant b. Different choices for b, with b>=2, determine a sequence of this class. The first sequence of this class is A027763, which uses b=2. This is the sequence for b=3. Adjacent sequences follow for b=4 through b=9.
Sequences of this class can contain only terms that are either prime numbers or powers of primes (A000961).
Powers of three (A000244) are commonplace as terms in sequences of this class, occurring significantly more often than powers of other primes.
For successive values of b, the sequence of first terms of all sequences of this class is A007978.
It appears that the sequence for b=x contains the term y, if and only if the sequence for b=x+A003418(y) also contains the term y, where A003418(y) is the least common multiple of all the integers from 1 to y. Can this be proved?
Sequences of this class generally share many terms with other sequences of this class, although each appears to be unique. These individual sequences are very similar to each other, so much so that among the first 7000 of them, only 120 distinct values less than 50000 occur (compare that to the 5217 available primes or powers of primes that are less than 50000.)
Sequences of this class tend to share many terms with A056637, the least prime of class n-.

			(3^^3) mod 11 = 9, (3^^2) mod 11 = 5. 11 is the least whole number for which this is true, thus a(3)=11.

Wayne VanWeerthuizen, Sage/Cython source code for all sequences of this class.
Wayne VanWeerthuizen, Initial terms of sequences of this class for b=2..10001

Cf. A027763, A246492, A246493, A246494, A246495, A246496, A246497

Large overlap with A056637.

A246496 Smallest k such that 8^^n is not congruent to 8^^(n-1) mod k, where 8^^n denotes the power tower 8^8^...^8 (in which 8 appears n times).

Original entry on oeis.org

2, 3, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 1266767, 14920303

Offset: 1

Wayne VanWeerthuizen, Aug 27 2014

For general remarks regarding sequences of this type, see A246491.

Cf. A027763, A056637, A246491 - A246497

Large overlap with A056637.

A082449 Let f(p) = greatest prime divisor of p-1. Sequence gives smallest prime which takes at least n steps to reach 2 when f is iterated.

Original entry on oeis.org

2, 3, 7, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 1266767, 14619833, 36449279, 377982107, 1432349099, 22111003847

Offset: 0

N. J. A. Sloane, Apr 25 2003

There is a remarkable and unexplained agreement: if 3 and 7 are replaced by 11 and 14619833 is replaced by 14920303, the result is sequence A056637 (least prime of class n-, according to the Erdős-Selfridge classification of primes).
From David A. Corneth, Oct 18 2016: (Start):
If a(n) * k + 1 is prime then a(n + 1) <= a(n) * k + 1.
a(18), a(19), ..., a(23) <= 309554053859, 619108107719, 19811459447009, 433142367554861, 866284735109723, 22523403112852799 respectively. (End)
Conjecture: a(n) is the smallest prime p such that b(p) = n, where f(2) = 0 and for an odd prime p, f(p) = 1 + max{q|(p-1), q prime} f(q). In other words, a(n) is the smallest prime p such that A364332(primepi(p)) = n. Verified for n <= 13. - Jianing Song, Apr 28 2024

			a(2) = 7 since 7 -> 3 -> 2 takes two steps, and smaller primes require less than 2 steps.
For p = 2879, 8 steps are needed (2879 -> 1439 -> 719 -> 359 -> 179 -> 89 -> 11 -> 5 -> 2), so a(8) = 2879, since smaller primes require less than 8 steps.

Steven G. Johnson, Postings to Number Theory List, Apr 23 and Apr 25, 2003.

Cf. A006530, A023503, A083647, A056637, A364332.

(* Assuming a(n) > 2 a(n-1) if n>1 *) Clear[a, f]; f[p_] := FactorInteger[p - 1][[-1, 1]]; f[2] = 2; a[n_] := a[n] = For[p = NextPrime[2 a[n-1]], True, p = NextPrime[p], k = 0; If[Length[FixedPointList[f, p]] == n+2, Return[p]]]; a[0]=2; a[1]=3; Table[Print[a[n]]; a[n], {n, 0, 16}] (* Jean-François Alcover, Oct 18 2016 *)

Edited by Klaus Brockhaus, May 01 2003

a(16) from Donovan Johnson, Nov 17 2008

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

Original entry on oeis.org

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

A129248 a(n) = n-th prime of class 14- according to the Erdős-Selfridge classification.

Original entry on oeis.org

377982107, 437391349, 716174549, 742922699, 1385934359, 1603768277, 1780127639, 1790436371, 1895437139, 1968261067, 2066951933, 2109424013, 2148523669, 2150787839, 2238778847, 2299583987, 2334899909, 2368121663

Offset: 1

M. F. Hasler, Apr 05 2007, Apr 21 2007

nonn

a[1..2] calculated using A081641[1..11]; a[3] <= 716174549.

M. F. Hasler, Table of n, a(n) for n=1..339

Cf. A056637, A081640, A081641, A129249, A129250.

A129248 = nextclassminus(A081641) /* cf. A081640 */

{ a(n) } = { p = 2*m*A081641(k)+1 | k=1,2,...,oo and m=1,2,... such that p is prime and m has no factor of class > 13- }

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.

A126805 "Class-" (or "class-minus") number of prime(n) according to the Erdős-Selfridge classification of primes.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 2, 2, 4, 2, 3, 2, 3, 2, 1, 2, 3, 3, 1, 2, 2, 3, 1, 2, 2, 2, 2, 4, 2, 2, 2, 1, 4, 3, 4, 2, 2, 1, 2, 3, 2, 2, 3, 2, 3, 2, 2, 2, 1, 3, 4, 2, 4, 2, 5, 2, 2, 3, 2, 3, 3, 2, 4, 3, 3, 5, 3, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 1, 2, 2, 2, 2, 4, 3, 4, 3, 1, 2, 4, 3, 3, 2, 3, 2, 2, 5, 3, 3, 2

Offset: 1

R. J. Mathar, Feb 23 2007

This gives the "class-" number as opposed to the "class+" number. Not to be confused with the "class-number" of quadratic form theory.

a(n)=1 if A000040(n) is in A005109, a(n)=2 if A000040(n) is in A005110, a(n)=3 if A000040(n) is in A005111 etc.

Cf. A056637.

A126805 := proc(n)
    option remember;
    local p, pe, a;
    if isprime(n) then
        a := 1;
        for pe in ifactors(n-1)[2] do
            p := op(1, pe);
            if p > 3 then
                a := max(a, procname(p)+1);
            end if;
        end do;
        a ;
    else
        -1;
    end if;
end proc:
seq(A126805(ithprime(n)),n=1..100) ;

a [n_] := a[n] = Module[{p, pf, e, res}, If[PrimeQ[n], pf = FactorInteger[n-1]; res = 1; For[e = 1, e <= Length[pf], e++, p = pf[[e, 1]]; If[p > 3, res = Max[res, a[p]+1]]]; Return[res], -1]]; Table[a[Prime[n]], {n, 1, 105}] (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

A126805(n) = { if( n>0, n=-prime(n)); if(( n=factor(-1-n)[,1] ) & n[ #n]>3, vecsort( vector( #n, i, A126805(-n[i]) ))[ #n]+1, 1) } \\ M. F. Hasler, Apr 16 2007

a(n) = max { a(p)+1 ; prime(p) is > 3 and divides prime(n)-1 } union { 1 } - M. F. Hasler, Apr 16 2007

cp's OEIS Frontend

A081430 Class 11- primes.

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A027763 Smallest k such that 2^^n is not congruent to 2^^(n-1) mod k, where 2^^n denotes the power tower 2^2^...^2 (in which 2 appears n times).

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A081428 Class 9- primes.

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A246491 Smallest k such that 3^^n is not congruent to 3^^(n-1) mod k, where 3^^n denotes the power tower 3^3^...^3 (in which 3 appears n times).

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A246496 Smallest k such that 8^^n is not congruent to 8^^(n-1) mod k, where 8^^n denotes the power tower 8^8^...^8 (in which 8 appears n times).

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A082449 Let f(p) = greatest prime divisor of p-1. Sequence gives smallest prime which takes at least n steps to reach 2 when f is iterated.

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

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A129248 a(n) = n-th prime of class 14- according to the Erdős-Selfridge classification.

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