A114871 -id:A114871 - 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

A280681 Numbers k such that Fibonacci(k) is a totient.

Original entry on oeis.org

1, 2, 3, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 72, 84, 90, 96, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 180, 192, 198, 204, 210, 216, 222, 228, 234, 240, 252, 264, 270, 276, 288, 294, 300, 306, 312, 324, 330, 336, 342, 348, 354, 360, 372, 378, 384, 390, 396, 402, 408, 414, 420, 432, 444, 450, 456, 462, 468, 480, 492, 504, 510, 516, 522, 528, 540, 546, 552, 558, 564, 570, 576, 588, 594, 600, 612, 624, 630, 636

Offset: 1

Altug Alkan, Jan 07 2017

nonn

Respectively, corresponding Fibonacci numbers are 1, 1, 2, 8, 144, 2584, 46368, 832040, 14930352, 267914296, 4807526976, 86267571272, 1548008755920, 498454011879264, 160500643816367088, 2880067194370816120, ...
Note that sequence does not contain all the positive multiples of 6, e.g., 66 and 102. See A335976 for a related sequence.
Conjecture: Sequence is infinite. - Altug Alkan, Jul 05 2020
All terms > 2 are multiples of 3, because Fibonacci(k) is odd unless k is a multiple of 3.  Are all terms > 3 multiples of 6? If a term k is not a multiple of 6, then since Fibonacci(k) is not divisible by 4, Fibonacci(k)+1 must be in A114871. - Robert Israel, Aug 02 2020
Unless there is an odd term > 3, this sequence as a set is {1, 2, 3} U 6*(Z^+ \ A335976). - Max Alekseyev, Dec 08 2024

			12 is in the sequence because Fibonacci(12) = 144 is in A000010.

Max Alekseyev, Table of n, a(n) for n = 1..665

Cf. A000010, A000045, A114871, A280592, A335976.

select(k -> numtheory:-invphi(combinat:-fibonacci(k))<>[], [1,2,seq(i,i=3..100,3)]); # Robert Israel, Aug 02 2020

isok(k) = istotient(fibonacci(k)); \\ Altug Alkan, Jul 05 2020

a(28)-a(49) from Jinyuan Wang, Jul 08 2020

Terms a(50) onward from Max Alekseyev, Dec 08 2024

A114873 Numbers representable in exactly one way as (p-1)p^k (where p is a prime and k>=0), in ascending order.

Original entry on oeis.org

1, 8, 10, 12, 20, 22, 28, 30, 32, 36, 40, 46, 52, 54, 58, 60, 64, 66, 70, 72, 78, 82, 88, 96, 102, 106, 108, 110, 112, 126, 128, 130, 136, 138, 148, 150, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 262, 268, 270, 272, 276, 280

Offset: 1

Franz Vrabec, Jan 03 2006

nonn

			(2-1)*2^3 is the only representation of 8 in the required form.

Cf. A114871, A114874.

s = Split@ Sort@ Flatten@ Table[(Prime[n] - 1)Prime[n]^k, {n, 60}, {k, 0, 6}]; Take[Union@ Flatten@ Select[s, Length@# == 1 &], 80] (* Robert G. Wilson v *)

More terms 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

A328413 Numbers k such that (Z/mZ)* = C_2 X C_(2k) has solutions m, where (Z/mZ)* is the multiplicative group of integers modulo m.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 14, 15, 16, 18, 20, 21, 22, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36, 39, 40, 41, 42, 44, 45, 46, 48, 50, 51, 53, 54, 55, 56, 58, 60, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 78, 81, 82, 83, 86, 87, 88, 89, 90, 95, 96, 98, 99, 102, 105, 106, 110, 111

Offset: 1

Jianing Song, Oct 14 2019

nonn

For n > 1, it is easy to see A114871(n)/2 is a term of this sequence. The smallest term here not of the form A114871(k)/2 is 24: 48 is not of the form (p-1)*p^k for any prime p, but (Z/mZ)* = C_2 X C_48 has solutions m = 119, 153, 238, 306.

			(Z/mZ)* = C_2 X C_2 has solutions m = 8, 12; (Z/mZ)* = C_2 X C_4 has solutions m = 15, 16, 20, 30; (Z/mZ)* = C_2 X C_6 has solutions m = 21, 28, 36, 42; (Z/mZ)* = C_2 X C_8 has solutions m = 32; (Z/mZ)* = C_2 X C_10 has solutions m = 33, 44, 66; (Z/mZ)* = C_2 X C_12 has solutions m = 35, 39, 45, 52, 70, 78, 90. So 1, 2, 3, 4, 5, 6 are all terms.

Cf. A328412. Complement of A328414.

Cf. also A114871.

isA328413(n) = my(r=4*n, N=floor(exp(Euler)*r*log(log(r^2))+2.5*r/log(log(r^2)))); for(k=r+1, N+1, if(eulerphi(k)==r && lcm(znstar(k)[2])==r/2, return(1)); if(k==N+1, return(0)))
for(n=1, 100, if(isA328413(n), print1(n, ", ")))

A114872 Even numbers not representable as (p-1)p^k (where p is a prime and k>=0) in ascending order.

Original entry on oeis.org

14, 24, 26, 34, 38, 44, 48, 50, 56, 62, 68, 74, 76, 80, 84, 86, 90, 92, 94, 98, 104, 114, 116, 118, 120, 122, 124, 132, 134, 140, 142, 144, 146, 152, 154, 158, 160, 164, 168, 170, 174, 176, 182, 184, 186, 188, 194, 200, 202, 204, 206, 208, 212, 214, 216, 218

Offset: 1

Franz Vrabec, Jan 03 2006

nonn

			It is easy to check there is no prime p with 14=(p-1)*p^k and k>=0.

Cf. A114871.

s = Split@ Sort@ Flatten@ Table[(Prime[n] - 1)Prime[n]^k, {n, 60}, {k, 0, 7}]; Complement[ 2Range@116, Take[Union@ Flatten@ s, {2, 58}]] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Jan 05 2006

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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A280681 Numbers k such that Fibonacci(k) is a totient.

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A114873 Numbers representable in exactly one way as (p-1)p^k (where p is a prime and k>=0), in ascending order.

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Mathematica

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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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A328413 Numbers k such that (Z/mZ)* = C_2 X C_(2k) has solutions m, where (Z/mZ)* is the multiplicative group of integers modulo m.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A114872 Even numbers not representable as (p-1)p^k (where p is a prime and k>=0) in ascending order.

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Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions