A114873 -id:A114873 - cp's OEIS frontend

A114874 Numbers representable in exactly two ways as (p-1)*p^e (where p is a prime and e >= 0) in ascending order.

2, 4, 6, 16, 18, 42, 100, 156, 162, 256, 486, 1458, 2028, 4422, 6162, 14406, 19182, 22650, 23548, 26406, 37056, 39366, 62500, 65536, 77658, 113232, 121452, 143262, 208392, 292140, 342732, 375156, 412806, 527802, 564898, 590592, 697048, 843642

Offset: 1

Franz Vrabec, Jan 03 2006

nonn

Numbers that are one less than a prime number and of the form (p-1)*p^e for some prime p and e > 0. - Jianing Song, Apr 13 2019

			6 is a member because 6 = (3-1)*3^1 = (7-1)*7^0 and 3 and 7 are primes.

Jianing Song, Table of n, a(n) for n = 1..162 (all terms below 10^8)

Cf. A114871, A114873.

s = Split@Sort@Flatten@Table[(Prime[n] - 1)Prime[n]^k, {n, 68000}, {k, 0, 16}]; Union@Flatten@Select[s, Length@# == 2 &] (* Robert G. Wilson v, Jan 05 2006 *)

isA114874(n) = if(n>1, my(v=factor(n), d=#v[, 1], p=v[d,1], e=v[d,2]); (isprime(n+1) && n==(p-1)*p^e), 0) \\ Jianing Song, Apr 13 2019

a(13)-a(38) from Robert G. Wilson v, Jan 05 2006

A134269 Number of solutions to the equation p^k - p^(k-1) = n, where k is a positive integer and p is prime.

Original entry on oeis.org

1, 2, 0, 2, 0, 2, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0

Offset: 1

Anthony C Robin, Jan 15 2008

nonn

The Euler phi function A000010 (number of integers less than n which are coprime with n) involves calculating the expression p^(k-1)*(p-1), where p is prime. For example phi(120) = phi(2^3*3*5) = (2^3-2^2)*(3-1)*(5-1) = 4*2*4 = 32.

			Notice that it is not possible to have more than 2 solutions, but say when n=4 there are two solutions, namely 5^1 - 5^0 and 2^3 - 2^2.
a(2) = 2 refers to 2^2 - 2^1 = 2 and 3^1 - 3^0 = 2.
a(6) = 2 as 6 = 3^2 - 3^1 = 7^1 - 7^0.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A014197, A114871, A114873, A114874.

A134269 := proc(n)
    local a,p,r ;
    a := 0 ;
    p :=2 ;
    while p <= n+1 do
        r := n/(p-1) ;
        if type(r,'integer') then
            if r = 1 then
                a := a+1 ;
            else
                r := ifactors(r)[2] ;
                if nops(r) = 1 then
                    if op(1,op(1,r)) = p then
                        a := a+1 ;
                    end if;
                end if;
            end if;
        end if;
        p := nextprime(p) ;
    end do:
    return a;
end proc: # R. J. Mathar, Aug 06 2013

lista(N=100) = {tab = vector(N); for (i=1, N, p = prime(i); for (j=1, N, v = p^j-p^(j-1); if (v <= #tab, tab[v]++););); for (i=1, #tab, print1(tab[i], ", "));} \\ Michel Marcus, Aug 06 2013

A134269list(up_to) = { my(v=vector(up_to)); forprime(p=2,1+up_to, for(j=1,oo,my(d = (p^j)-(p^(j-1))); if(d>up_to,break,v[d]++))); (v); };
v134269 = A134269list(up_to);
A134269(n) = v134269[n]; \\ Antti Karttunen, Nov 09 2018

a(2) corrected by Michel Marcus, Aug 06 2013

More terms from Antti Karttunen, Nov 09 2018

cp's OEIS Frontend

A114874 Numbers representable in exactly two ways as (p-1)*p^e (where p is a prime and e >= 0) in ascending order.

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A134269 Number of solutions to the equation p^k - p^(k-1) = n, where k is a positive integer and p is prime.

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Maple

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PARI

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