[seqfan] A083371 = A124583

Don Reble djr at nk.ca
Sat Jan 22 19:41:15 CET 2022


> %N A083371 Primes satisfying f(2*p)=p when f(1)=7 (see comment).
> %C ...start from any initial value f(1) >= 2 and define f(n) as the
>    largest prime factor of f(1)+f(2)+...+f(n-1)...
> %C Is this the same sequence as A124583?

> %N A124583 Primes p such that q-p >= 8, where q is the next prime
>            after p.

(Here, X' is the first prime after X, 'X is the first prime before X.)

The F sequence starting at "7" has 11 "7"s, then 6 "11"s, 6 "13"s,
6 "17"s, 6 "19"s, 10 "23"s, ...
One easily sees that the F sequence starting at prime S has S' instances
of S; then for each prime P after S, it has (P'-'P) instances of P.
(A076973 is the F sequence starting at "2".)

The primes from S to P occupy the first
[S' + (S''-S) + (S'''-S') + ... + (P' - 'P)] terms of F.
That sum telescopes to P'+P-S, and so
    F(P'+P-S) = P;  F(P'+P-S+1) = P';
    F(P+'P-S) = 'P; F(P+'P-S+1) = P.

If F(X) =P, then P+'P-S < X   <= P'+P-S.
If F(2P)=P, then P+'P-S < 2P  <= P'+P-S
                     'P < P+S <= P'
                            S <= P'-P

So A083371 has the primes P for which P'-P >= 7;
and since P'-P is even (both primes are odd), P'-P >= 8.
That's A124583.

Don Reble  djr at nk.ca

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