Search: a114871 -id:a114871
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A114874
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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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+10
6
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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6 is a member because 6 = (3-1)*3^1 = (7-1)*7^0 and 3 and 7 are primes.
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MATHEMATICA
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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 *)
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PROG
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(PARI) 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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A114873
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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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+10
2
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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
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OFFSET
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1,2
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LINKS
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EXAMPLE
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(2-1)*2^3 is the only representation of 8 in the required form.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A134269
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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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+10
2
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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;
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PROG
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(PARI) 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
(PARI)
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);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A280681
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Numbers k such that Fibonacci(k) is a totient.
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+10
2
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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EXAMPLE
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12 is in the sequence because Fibonacci(12) = 144 is in A000010.
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MAPLE
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select(k -> numtheory:-invphi(combinat:-fibonacci(k))<>[], [1, 2, seq(i, i=3..100, 3)]); # Robert Israel, Aug 02 2020
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PROG
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(PARI) isok(k) = istotient(fibonacci(k)); \\ Altug Alkan, Jul 05 2020
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A328413
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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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+10
2
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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(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.
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PROG
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(PARI) 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, ", ")))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A114872
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Even numbers not representable as (p-1)p^k (where p is a prime and k>=0) in ascending order.
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+10
0
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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
(list;
graph;
refs;
listen;
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text;
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OFFSET
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1,1
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LINKS
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EXAMPLE
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It is easy to check there is no prime p with 14=(p-1)*p^k and k>=0.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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