The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A336297 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A336297

Prime numbers p such that equation x = p*sopf(x) (where sopf(x) is the sum of distinct prime factors of x) has exactly 1 solution in positive integers.
(history; published version)

#11 by Michel Marcus at Mon Jul 20 11:51:25 EDT 2020

STATUS

editing

proposed

Discussion

Mon Jul 20

11:52

Michel Marcus: and add one to A336296 ?

12:36

Vladimir Letsko: I'm not  sure that it is necessary at the description of the  sequence.
Because it works for large primes only.
Here is this program:

with(NumberTheory):
sopf := proc (n) options operator, arrow; add(i, i in PrimeFactors(n)) end proc:

fnsol := proc (p) local x, S, q; S := {}; for x from p^2 by p to 1.175*p^2 do if x = sopf(x)*p then S := {op(S), x} end if end do; q := (1/4)*p+5/4; if type(q, posint) and isprime(q) then S := {5*q*p, op(S)} end if; q := (1/3)*p+2/3; if type(q, posint) and isprime(q) then S := {4*q*p, op(S)} end if; q := (1/2)*p+3/2; if type(q, posint) and isprime(q) then S := {3*q*p, op(S)} end if; q := p+2; if isprime(q) then S := {2*q*p, op(S)} end if; S end proc

#10 by Michel Marcus at Mon Jul 20 11:51:18 EDT 2020

PROG

lista(nn) = {my(lim = pp(ip(nn))); my(v = vector(lim, k, k++; k/sopf(k))); my(w = vector(nn-1, k, #select(x->(x==k+1), v))); select(x->isprime(x), apply(x->x+1, Vec(select(x->(x==1), w, 1)))); } \\ Michel Marcus, Jul 19 2020~

STATUS

proposed

editing

#9 by Michel Marcus at Sun Jul 19 07:55:48 EDT 2020

STATUS

editing

proposed

Discussion

Sun Jul 19

10:57

Vladimir Letsko: I have quick program. But it works only for primes more than 1000.

Mon Jul 20

00:51

Michel Marcus: what language ?

01:13

Vladimir Letsko: maple

11:50

Michel Marcus: so you could add it here ?

#8 by Michel Marcus at Sun Jul 19 07:51:50 EDT 2020

PROG

(PARI) sopf(n) = vecsum(factor(n)[, 1]); \\ A008472

pp(n) = prod(k=1, n, prime(k)); \\ A002110

sp(n) = sum(k=1, n, prime(k)); \\ A007504

ip(n) = {my(k=1); while (pp(k)/sp(k) <= n, k++); k+1; }

STATUS

proposed

editing

Discussion

Sun Jul 19

07:55

Michel Marcus: ther emust be a better way, I can go up to lista(700) to get terms up to 683

#7 by Michel Marcus at Sun Jul 19 06:34:48 EDT 2020

STATUS

editing

proposed

#6 by Michel Marcus at Sun Jul 19 06:34:45 EDT 2020

EXAMPLE

4 is the unique integer x such that x = 2*sopf(x), a prime, so 2 is a term.

STATUS

proposed

editing

#5 by Michel Marcus at Thu Jul 16 12:31:47 EDT 2020

STATUS

editing

proposed

#4 by Michel Marcus at Thu Jul 16 12:31:43 EDT 2020

NAME

Prime numbres numbers p such that equation x = p*sopf(x) (where sopf(x) is the sum of distinct prime factors of x) has exactly 1 solution in positive integers.

LINKS

Vladimir Letsko, <a href="https://dxdy.ru/post1257616.html#p1257616">Mathematical Marathon, Problem 227</a> (in Russian).

CROSSREFS

Cf. A008472, A336098, A336099, A336296.

STATUS

proposed

editing

#3 by Vladimir Letsko at Thu Jul 16 12:26:16 EDT 2020

STATUS

editing

proposed

#2 by Vladimir Letsko at Thu Jul 16 12:24:01 EDT 2020

NAME

allocated for Vladimir LetskoPrime numbres p such that equation x = p*sopf(x) (where sopf(x) is the sum of distinct prime factors of x) has exactly 1 solution in positive integers.

DATA

2, 61, 97, 113, 151, 173, 241, 277, 317, 353, 389, 449, 457, 593, 601, 607, 653, 673, 683, 727, 733, 797, 907, 929, 941, 947, 953, 977, 997, 1021, 1051, 1087, 1153, 1181, 1193, 1217, 1249, 1307, 1321, 1361, 1373, 1409, 1433, 1489, 1493, 1523, 1553, 1579, 1597, 1609, 1627

OFFSET

1,1

LINKS

Vladimir Letsko, <a href="https://dxdy.ru/post1257616.html#p1257616">Mathematical Marathon, Problem 227</a> (in Russian)

CROSSREFS

A008472, A336098, A336099, A336296.

KEYWORD

allocated

nonn

AUTHOR

Vladimir Letsko, Jul 16 2020

STATUS

approved

editing