A005105 -id:A005105 - cp's OEIS frontend

A283418 Numbers n such that n and n+1 are primitive abundant.

82004, 158235, 326864, 442035, 516704, 1102724, 1606275, 2151435, 2697435, 2912084, 2921535, 2979675, 3002804, 3241755, 3647475, 4322835, 5801984, 5905844, 6069195, 7251075, 7387604, 7553924, 8272124, 8788724, 9292724, 9909584

Offset: 1

Emmanuel Vantieghem, May 02 2017

nonn

Intersection of A091191 and -1 + A091191.

			82004 is in the sequence because it is abundant (sum divisors = 164640, > 2*82004) and 82005 is also abundant (sum divisors = 165888, > 2*82005).

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A091191, A005105, A096399.

fQ[m_] := DivisorSigma[1, m] > 2 m;
gQ[m_] := fQ[m] && Union[fQ /@ Rest[Most[Divisors[m]]]] == {False};
V = Select[Range[10^7], gQ]; Intersection[V, V - 1]

A081635 Class 7+ primes.

Original entry on oeis.org

15013, 16333, 22093, 24841, 43321, 49003, 52517, 54721, 62533, 63761, 69061, 69073, 70061, 74597, 75781, 75793, 75913, 82561, 83233, 84673, 87433, 87509, 88793, 91081, 92761, 94321, 98737, 99367, 101641, 105097, 110881, 111973, 114343

Offset: 1

Robert G. Wilson v, Mar 20 2003

nonn

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

T. D. Noe, Table of n, a(n) for n=1..10000

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

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[10820], ClassPlusNbr[ Prime[ # ]] == 7 &]]

A081638 Class 10+ primes.

Original entry on oeis.org

1065601, 2424973, 5114881, 7222561, 8124481, 8524091, 8647411, 8650321, 9190681, 9287521, 9590417, 10617601, 10929817, 11996161, 12349093, 12508081, 12786181, 12971117, 13570681, 14113027, 14308123, 14312743, 14476807

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..360

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

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[150000], ClassPlusNbr[ Prime[ # ]] == 10 &]]

A081634 Class 6+ primes.

Original entry on oeis.org

2917, 4933, 5413, 7507, 8167, 8753, 10567, 10627, 11047, 11261, 11677, 12073, 12251, 12421, 12433, 12553, 12721, 14293, 15017, 17041, 18181, 18493, 19267, 19333, 20023, 21193, 21313, 21661, 22397, 24481, 25933, 26261, 26437, 27361

Offset: 1

Robert G. Wilson v, Mar 20 2003

nonn

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

T. D. Noe, Table of n, a(n) for n=1..10000

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

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[3000], ClassPlusNbr[ Prime[ # ]] == 6 &]]

A081636 Class 8+ primes.

Original entry on oeis.org

49681, 109441, 120103, 151561, 198733, 210193, 246241, 255043, 266401, 280243, 295873, 326659, 326701, 347773, 355171, 360421, 368881, 397633, 397673, 423001, 441877, 447137, 471241, 480541, 489989, 499397, 508037, 511507, 532757, 539401

Offset: 1

Robert G. Wilson v, Mar 20 2003

nonn

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

T. D. Noe, Table of n, a(n) for n=1..10000

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

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[44535], ClassPlusNbr[ Prime[ # ]] == 8 &]]

A081637 Class 9+ primes.

Original entry on oeis.org

532801, 710341, 720617, 1212487, 1261157, 1372081, 1457293, 1490429, 1532173, 1657801, 1788547, 1789093, 1809601, 1829293, 1887877, 1944181, 1960141, 1997587, 2121853, 2161853, 2474413, 2484049, 2557441, 2578801, 2613607

Offset: 1

Robert G. Wilson v, Mar 20 2003

nonn

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

T. D. Noe, Table of n, a(n) for n=1..10000

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

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[196000], ClassPlusNbr[ Prime[ # ]] == 9 &]]

A069353 Numbers of form 2^i*3^j - 1 with i, j >= 0.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 8, 11, 15, 17, 23, 26, 31, 35, 47, 53, 63, 71, 80, 95, 107, 127, 143, 161, 191, 215, 242, 255, 287, 323, 383, 431, 485, 511, 575, 647, 728, 767, 863, 971, 1023, 1151, 1295, 1457, 1535, 1727, 1943, 2047, 2186, 2303, 2591, 2915, 3071, 3455, 3887

Offset: 1

Reinhard Zumkeller, Mar 18 2002

nonn

Are there infinitely many primes in this sequence? See A005105.

If m is a term then also 2*m + 1 and 3*m + 2.

Graham Everest, Peter Rogers, and Thomas Ward, A higher-rank Mersenne problem, Algorithmic Number Theory: 5th International Symposium, ANTS-V Sydney, Australia, July 7-12, 2002 Proceedings 5, Lect. Notes Computer Sci. 2369, Springer Berlin Heidelberg, 2002, pp. 95-107.

Cf. A003586, A055600, A069355, A005105.

With[{max = 4000}, Sort[Flatten[Table[2^i*3^j - 1, {i, 0, Log2[max]}, {j, 0, Log[3, max/2^i]}]]]] (* Amiram Eldar, Jul 13 2023 *)

from sympy import 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) # Chai Wah Wu, Mar 31 2025

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

A129469 Least prime of Erdos-Selfridge class n+ in A129470.

Original entry on oeis.org

883, 3181, 15913, 2146141, 17227801, 456185017, 4960846573, 568124640697, 2273325467773, 145351829612377, 9302101084613641, 595332797734595317, 5813792718345189961, 1139502378775815768313, 166245781044286357673761

Offset: 3

M. F. Hasler, Apr 16 2007

nonn

The sequence starts at offset 3, since primes of class 1+ and 2+ have all prime factors (of p+1) of class 1+. Definitions imply that a(n) >= -1+2*A005113(n-1)*nextprime(1+A005113(n-1)). We have a(n) = -1+2*A005113(n-1)*p for all n<18, with p prime for n>3. This holds probably for all n.

			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 2+.
a(4) = 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+.

Cf. A129470, A005113, A005105 - A005108, A081633 - A081639, A084071, A090468, A129474 - A129475.

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])}; A129469={vector(#A005113-1,i,t=A005113[i+1]; t=[t,nextprime(t+1)-1,0];until( isprime( t[3] = -1+2*t[1]*t[2] ) & (f=factor( 1+t[3] )[,1]) & class(f[ #f],1)= i+1, print("Warning, crossed a prime of class >= ",i+1,"+, p=", t[2]); ); ); print(i+2," ",t[3]); t[3])}

A084071 Class 12+ primes.

Original entry on oeis.org

68198461, 115084901, 138358573, 156811273, 213397621, 220576331, 234432217, 260050573, 282261961, 290996753, 330864497, 353653063, 371500819, 383616341, 406915273, 426240379, 445800983, 446707201, 449558323, 460339577, 472782553

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..51

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

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[25000000], ClassPlusNbr[ Prime[ # ]] == 12 &]]

A126433 Class+ number of prime(n) according to the Erdős-Selfridge classification of primes.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Mar 23 2007

nonn

a(n)=1 if A000040(n) is in A005105. a(n)=2 if A000040(n) is in A005106, a(n)=3 if in A005107 etc. The locations of records are implicit in A005113.

Cf. A101253.

A126433 := proc(n)
    option remember;
    local p, pf, e, a;
    if isprime(n) then
        pf := ifactors(n+1)[2];
        a := 1;
        for e from 1 to nops(pf) do
            p := op(1, op(e, pf));
            if p > 3 then
                a := max(a, procname(p)+1);
            end if;
        end do;
        a ;
    else
        -1;
    end if;
end proc:
seq(A126433(ithprime(n)),n=1..100) ;
A126433 := n -> if n>0 then A126433(-ithprime(n)) else numtheory[factorset](1-n); if % subset{2,3} then 1 else 1+max(seq(A126433(-i),i=%)) fi fi; map(%,[$1..999]); # M. F. Hasler, Apr 02 2007

classPlus[p_] := classPlus[p] = If[f = FactorInteger[p + 1][[All, 1]]; q = Last[f]; q == 2 || q == 3, 1, Max[classPlus /@ f] + 1]; classPlus /@ Prime /@ Range[105] (* Jean-François Alcover, Jun 24 2013 *)

A126433(n) = { if( n>0, n=-prime(n)); n=factor(1-n)[,1]; if( n[ #n]>3, vecsort( vector( #n, i, A126433(-n[i]) ))[ #n]+1, 1) }; vector(999,i,A126433(i))

cp's OEIS Frontend

A283418 Numbers n such that n and n+1 are primitive abundant.

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A081635 Class 7+ primes.

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A081638 Class 10+ primes.

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A081634 Class 6+ primes.

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A081636 Class 8+ primes.

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A081637 Class 9+ primes.

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Maple

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A069353 Numbers of form 2^i*3^j - 1 with i, j >= 0.

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A129469 Least prime of Erdos-Selfridge class n+ in A129470.

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