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A259562 Numbers n such that the sum of the distinct prime factors of prime(n)-1 and prime(n+1)-1 are the same. 1
2, 414, 556, 3962, 4972, 6151, 6521, 8440, 8665, 13769, 13909, 15576, 16696, 17176, 19926, 20630, 21541, 27090, 30822, 62118, 65349, 74014, 94203, 98600, 101231, 103058, 108333, 112332, 136036, 142714, 145588, 147150, 160730, 162366, 169137, 194681, 200837 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Although there are more terms than A105403 so far, these numbers are still fairly uncommon.
Is this sequence infinite?
It would follow from the generalized Bunyakovsky conjecture that, e.g., there are infinitely many primes p such that p+2, p+12, p+14, 6*p^2+84*p+1 and 6*p^2+84*p+145 are all prime, and there are no primes between 6*p^2+84*p+1 and 6*p^2+84*p+145. If so, then the sequence is infinite, because it contains n where prime(n) = 6*p^2+84*p+1, with prime(n)-1 having distinct prime factors 2,3,p,p+14 and prime(n+1) having distinct prime factors 2,3,p+2,p+12. - Robert Israel, Jun 30 2015
LINKS
EXAMPLE
The prime factors of prime(414)-1 are 2,3,5,5,19 and the prime factors of prime(415)-1 are 2,2,2,3,7,17. The sum of the distinct entries in each of these lists is 29.
MAPLE
Primes:= select(isprime, [2, seq(2*i+1, i=1..10^6)]):
spf:= map(p -> convert(numtheory:-factorset(p-1), `+`), Primes):
select(t -> spf[t+1]=spf[t], [$1..nops(Primes)-1]);
MATHEMATICA
Select[Range@ 250000, Total[First /@ FactorInteger[Prime@ # - 1]] == Total[First /@ FactorInteger[Prime[# + 1] - 1]] &] (* Michael De Vlieger, Jul 01 2015 *)
PROG
(PARI) spf(n) = {my(f=factor(n)); sum(k=1, #f~, f[k, 1]); }
lista(nn) = {forprime(p=2, nn, if (spf(p-1)==spf(nextprime(p+1)-1), print1(primepi(p), ", ")); ); } \\ Michel Marcus, Jun 30 2015
CROSSREFS
Sequence in context: A352848 A349764 A153911 * A177321 A365520 A080392
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Alois P. Heinz, Jun 30 2015
STATUS
approved

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Last modified August 8 15:25 EDT 2024. Contains 375022 sequences. (Running on oeis4.)