A005109 -id:A005109 - cp's OEIS frontend

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

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

A374577 Pierpont primes are primes of the form 2^t*3^u + 1; this sequence gives the t's in order.

Original entry on oeis.org

0, 1, 2, 1, 2, 4, 1, 2, 3, 5, 2, 1, 6, 8, 4, 1, 6, 8, 7, 4, 1, 5, 2, 7, 4, 7, 12, 3, 11, 1, 3, 16, 6, 14, 5, 12, 3, 5, 10, 18, 7, 12, 17, 11, 16, 13, 15, 8, 16, 4, 6, 19, 2, 20, 2, 18, 15, 1, 6, 22, 11, 21, 1, 13, 12, 11, 26, 25, 30, 19, 24, 20, 27, 16, 23, 11

Offset: 1

William C. Laursen, Jul 11 2024

nonn

This sequence gives the exponents of 2's in the Pierpont primes, A374578 gives the exponents of 3's.

			a(1) = 0, because the first Pierpont prime is 2 = 2^0 * 3^0 + 1.
a(6) = 4, because the sixth Pierpont prime is 17 = 2^4 * 3^0 + 1.
a(7) = 1, because the seventh Pierpont prime is 19 = 2^1 * 3^2 + 1.

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

Cf. A005109, A007814, A374578.

With[{lim = 10^11}, IntegerExponent[Select[Sort@ Flatten@Table[2^i*3^j + 1, {i, 0, Log2[lim]}, {j, 0, Log[3, lim/2^i]}], PrimeQ] - 1, 2]] (* Amiram Eldar, Sep 02 2024 *)

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(valuation(s[i] - 1, 2), ", ")));} \\ Amiram Eldar, Sep 02 2024

a(n) = A007814(A005109(n)-1).

More terms from Stefano Spezia, Jul 12 2024

A374578 Pierpont primes are primes of the form 2^t*3^u + 1; this sequence gives the u's in order.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 2, 2, 2, 1, 3, 4, 1, 0, 3, 5, 2, 1, 2, 4, 6, 4, 6, 3, 5, 4, 1, 7, 2, 9, 8, 0, 7, 2, 8, 4, 10, 9, 6, 1, 8, 5, 2, 6, 3, 5, 4, 9, 4, 12, 11, 3, 14, 3, 15, 5, 7, 16, 13, 3, 10, 4, 17, 10, 11, 12, 3, 4, 1, 8, 5, 8, 4, 11, 7, 15, 12, 2, 10, 1, 22, 4

Offset: 1

William C. Laursen, Jul 11 2024

nonn

This sequence gives the exponents of 3's in the Pierpont primes, A374577 gives the exponents of 2's.

			a(1) = 0, because the first Pierpont prime is 2 = 2^0 * 3^0 + 1.
a(6) = 0, because the sixth Pierpont prime is 17 = 2^4 * 3^0 + 1.
a(7) = 2, because the seventh Pierpont prime is 19 = 2^1 * 3^2 + 1.

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

Cf. A005109, A007949, A374577.

With[{lim = 10^12}, IntegerExponent[Select[Sort@ Flatten@Table[2^i*3^j + 1, {i, 0, Log2[lim]}, {j, 0, Log[3, lim/2^i]}], PrimeQ] - 1, 3]] (* Amiram Eldar, Sep 02 2024 *)

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(valuation(s[i] - 1, 3), ", ")));} \\ Amiram Eldar, Sep 02 2024

a(n) = A007949(A005109(n)-1).

More terms from Stefano Spezia, Jul 12 2024

A069358 Primes of the form 2^i*3^j + (i+j) with i, j >= 0.

Original entry on oeis.org

3, 11, 37, 53, 113, 167, 199, 439, 521, 3083, 6569, 12301, 23339, 32783, 139981, 663569, 708601, 3981331, 7558289, 20155411, 38263769, 45349651, 75497497, 483729431, 1934917657, 2717909021, 6115295261, 12397455671, 14693280793

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. A069357, A005109, A069356, A069348.

Take[ Select[ Union[ Flatten[ Table[2^i*3^j + (i + j), {i, 0, 25}, {j, 0, 18}]]], PrimeQ[ # ] &], 30]

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

A101231 a(n) = n-th prime of Erdős-Selfridge classification n-.

Original entry on oeis.org

2, 29, 67, 179, 941, 4079, 20389, 65267, 224563, 978863, 6448979, 47247763, 309550999, 2150787839, 13925010299

Offset: 1

Jonathan Vos Post, Dec 15 2004

Diagonalization of the Erdős-Selfridge classification of primes.

			a(1) = 2 because 2 is the first element of A005109.
a(2) = 29 because 29 is the 2nd element of A005110.
a(3) = 67 because 67 is the 3rd element of A005111.
a(4) = 179 because 179 is the 4th element of A005112.

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

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

More terms from R. J. Mathar, May 02 2007

A113434 Semi-Pierpont semiprimes which are also Pierpont semiprimes.

Original entry on oeis.org

4, 9, 10, 25, 49, 65, 289

Offset: 1

Jonathan Vos Post, Nov 01 2005

Semiprimes both of whose prime factors are Pierpont primes (A005109), which are primes of the form (2^K)*(3^L)+1 and where the semiprime is itself of the form (2^K)*(3^L)+1.
No more under 10^50; what is the next element of this sequence?
No more terms <= 10^100. - Robert Israel, Mar 10 2017
This sequence is complete, see Links. - Charlie Neder, Feb 04 2019

			a(1) = 4 = 2^2 = [(2^0)*(3^0)+1]*[(2^0)*(3^0)+1] = (2^0)*(3^1)+1.
a(2) = 9 = 3^2 = [(2^1)*(3^0)+1]*[(2^1)*(3^0)+1] = (2^3)*(3^0)+1.
a(3) = 10 = 2*5 = [(2^0)*(3^0)+1]*[(2^2)*(3^0)+1] = (2^0)*(3^2)+1.
a(4) = 25 = 5^2 = [(2^2)*(3^0)+1]*[(2^2)*(3^0)+1] = (2^3)*(3^1)+1.
a(5) = 49 = 7^2 = [(2^1)*(3^1)+1]*[(2^1)*(3^1)+1] = (2^4)*(3^1)+1.
a(6) = 65 = 5*13 = [(2^2)*(3^0)+1]*[(2^2)*(3^1)+1] = (2^6)*(3^0)+1.
a(7) = 289 = 17^2 = [(2^4)*(3^0)+1]*[(2^4)*(3^0)+1] = (2^5)*(3^2)+1.

Chris Caldwell, "Pierpont primes." primeform posting, Oct 25, 2005.
Chris Caldwell, "Pierpont primes." primeform posting, Oct 25, 2005. [Cached copy]
Charlie Neder, Proof of the completeness of this sequence
Eric Weisstein's World of Mathematics, Pierpont Prime
Eric Weisstein's World of Mathematics, Semiprime

Cf. A001358, A003586, A005109, A055600, A111153, A111206, A113432, A113433.

N:= 10^100: # to get all terms <= N
PP:= select(isprime, {seq(seq(1+2^i*3^j, i=0..ilog2((N-1)/3^j)),j=0..floor(log[3](N-1)))}):
SP:= select(t -> t <= N and t = 1+2^padic:-ordp(t-1,2)*3^padic:-ordp(t-1,3), [seq(seq(PP[i]*PP[j], j=1..i),i=1..nops(PP))]):
sort(convert(SP,list)); # Robert Israel, Mar 10 2017

{a(n)} = intersection of A113432 and A113433. {a(n)} = Semiprimes A001358 of the form (2^K)*(3^L)+1 both of whose factors are of the form (2^K)*(3^L)+1. {a(n)} = {integers P such that, for nonnegative integers I, J, K, L, m, n there is a solution to (2^I)*(3^J)+1 = [(2^K)*(3^L)+1]*[(2^m)*(3^n)+1] where both [(2^K)*(3^L)+1] and [(2^m)*(3^n)+1] are prime}.

A174099 Indices of primes of the form 2^t*3^u + 1 in the primes.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 12, 21, 25, 29, 38, 44, 55, 84, 93, 106, 136, 191, 211, 232, 378, 422, 483, 539, 1272, 1470, 2014, 2111, 4144, 5359, 6543, 13006, 13632, 18802, 28547, 39420, 51327, 59982, 62947, 66875, 78156, 91466, 113675, 132938, 148273, 193541

Offset: 1

Juri-Stepan Gerasimov, Mar 07 2010

nonn

Numbers k such that 2^t*3^u + 1 = prime(k).

			a(1) = 1 because 2^0 * 3^0 + 1 = 2 = prime(1).
a(2) = 2 because 2^1 * 3^0 + 1 = 3 = prime(2).

Amiram Eldar, Table of n, a(n) for n = 1..166

Cf. A000720, A005109.

With[{lim = 10^7}, PrimePi[Select[Sort@ Flatten@ Table[2^i * 3^j + 1, {i, 0, Log2[lim]}, {j, 0, Log[3, lim/2^i]}], PrimeQ]]] (* Amiram Eldar, Sep 02 2024 *)

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(primepi(s[i]), ", ")));} \\ Amiram Eldar, Sep 02 2024

a(n) = A000720(A005109(n)).

Corrected and extended by Charles R Greathouse IV, Mar 21 2010

A113420 Number of Pierpont primes less than 10^n.

Original entry on oeis.org

4, 10, 18, 25, 32, 42, 50, 58, 65, 72, 78, 83, 93, 106, 114, 125, 139, 143, 149, 157, 167, 176, 183, 194, 198, 209, 219, 221, 229, 237, 243, 250, 260, 270, 273, 278, 286, 294, 304, 314, 317, 323, 327, 337, 342, 352, 362, 371, 377, 394, 399, 405, 414, 420

Offset: 1

Eric W. Weisstein, Oct 29 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Pierpont Prime.

Cf. A005109, A113412.

A122258 Number of Pierpont primes <= n.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10

Offset: 1

Reinhard Zumkeller, Aug 29 2006

nonn

Eric Weisstein's World of Mathematics, Pierpont Prime.

Partial sums of A122257.

Cf. A000720, A005109, A071521.

smooth3Q[n_] := Times @@ ({2, 3}^IntegerExponent[n, {2, 3}]) == n; s[n_] := Boole[PrimeQ[n] && smooth3Q[n-1]]; Accumulate[Table[s[n], {n, 1, 100}]] (* Amiram Eldar, May 14 2025 *)

is3smooth(n) = my(m = n >> valuation(n, 2)); m == 3^valuation(m, 3);
f(n) = isprime(n) && is3smooth(n-1);
list(lim) = {my(s = 0); for(n = 1, lim, s += f(n); print1(s, ", "));} \\ Amiram Eldar, May 14 2025

a(n) <= A000720(n).

a(n) <= A071521(n).

A174144 Primes of the form 2^p3^q5^r*7^s + 1.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 37, 41, 43, 61, 71, 73, 97, 101, 109, 113, 127, 151, 163, 181, 193, 197, 211, 241, 251, 257, 271, 281, 337, 379, 401, 421, 433, 449, 487, 491, 541, 577, 601, 631, 641, 673, 701, 751, 757, 769, 811, 883, 1009, 1051, 1153, 1201

Offset: 1

Michel Lagneau, Mar 09 2010

nonn

Restricting to r=s=0 gives the Pierpont primes (A005109); s = 0 gives A002200.

			6301 = 2^2 * 3^2 * 5^2 * 7 + 1.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pierpont Prime

Cf. A002200, A005109.

K:=10^7;; # to get all terms <= K.
A:=Filtered([1..K],IsPrime);;    I:=[3,5,7];;
B:=List(A,i->Elements(Factors(i-1)));;
C:=List([0..Length(I)],j->List(Combinations(I,j),i->Concatenation([2],i)));
A174144:=Concatenation([2],List(Set(Flat(List([1..Length(C)],i->List([1..Length(C[i])], j->Positions(B,C[i][j]))))),i->A[i])); # Muniru A Asiru, Sep 12 2017

[p: p in PrimesUpTo(2000) | forall{d: d in PrimeDivisors(p-1) | d le 7}]; // Bruno Berselli, Sep 24 2012

with(numtheory):T:=array(0..50000000):U=array(0..50000000 ):k:=1:for a from 0 to 25 do:for b from 0 to 16 do:for c from 0 to 16 do:for d from 0 to 16 do: p:= 2^a*3^b*5^c*7^d + 1:if type(p, prime)=true then T[k]:=p:k:=k+1: else fi: od :od:od:od:mini:=T[1]:ii:=1:for p from 1 to k-1 do:for n from 1 to k-1 do: if T[n] < mini then mini:= T[n]:ii:=n: indice:=U[n]: else f i:od:print(mini):T[ii]:= 10^30: ii:=1:mini:=T[1] :od:

Take[ Select[ Sort[ Flatten[ Table[2^a*3^b*5^c*7^d + 1, {a, 0, 25}, {b, 0, 16},{c, 0, 16},{d, 0, 16}]]], PrimeQ[ # ] &], 100] (* or *) 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[3, 6300],
ClassMinusNbr[ Prime[ # ]] == 1 &]] Select[Prime /@ Range[10^5], Max @@ First /@ FactorInteger[ # - 1] < 5 &]

list(lim)={
    lim\=1;
    my(v=List([2]),s,t,p);
    for(i=0,log(lim\2+.5)\log(7),
        t=2*7^i;
        for(j=0,log(lim\t+.5)\log(5),
            s=t*5^j;
            while(s < lim,
                p=s;
                while(p < lim,
                    if(isprime(p+1),listput(v,p+1));
                    p <<= 1
                );
                s *= 3;
            )
        )
    );
    vecsort(Vec(v))
}; \\ Charles R Greathouse IV, Sep 21 2011

A174144 = list(p for p in primes(2000) if set(prime_factors(p-1)) <= set([2,3,5,7]))

Corrected and edited by D. S. McNeil, Nov 20 2010

cp's OEIS Frontend

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

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A374577 Pierpont primes are primes of the form 2^t*3^u + 1; this sequence gives the t's in order.

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A374578 Pierpont primes are primes of the form 2^t*3^u + 1; this sequence gives the u's in order.

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Crossrefs

Programs

Mathematica

PARI

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Extensions

A069358 Primes of the form 2^i*3^j + (i+j) with i, j >= 0.

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Mathematica

Extensions

A101231 a(n) = n-th prime of Erdős-Selfridge classification n-.

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Extensions

A113434 Semi-Pierpont semiprimes which are also Pierpont semiprimes.

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Maple

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A174099 Indices of primes of the form 2^t*3^u + 1 in the primes.

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A174144 Primes of the form 2^p3^q5^r*7^s + 1.