A080612 - OEIS

A080612

Numbers m such that (1/p(2*m+1)) * Sum_{k=1..m} p(2*k+1)-p(2*k) >= (1/p(2*m))* Sum_{k=1..m} p(2*k)-p(2*k-1) where p(k) denotes the k-th prime.

1, 2, 3, 4, 5, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

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OFFSET

1,2

COMMENTS

Conjectured to be finite with last term = 314.

This conjecture is false, 1494 is the first term after 314. - Sean A. Irvine, Sep 24 2025

Other conjecture log(n)^2 * (1/p(2*n+1) * Sum_{k=1..n} (p(2*k+1)-p(2*k)) - (1/p(2*n)) * Sum_{k=1..n} (p(2*k)-p(2*k-1))) -> constant. Weaker : previous formula is bounded.

LINKS

Sean A. Irvine, Table of n, a(n) for n = 1..1000

MATHEMATICA

Select[Range[120], (1/Prime[2*# + 1])*Sum[ Prime[2*k + 1] - Prime[2*k], {k, #}] >= (1/Prime[2*#])*Sum[Prime[2*k] - Prime[2*k - 1], {k, #} ] &] (* Michael De Vlieger, Sep 24 2025 *)

CROSSREFS

Sequence in context: A115928 A247665 A117331 * A039261 A039201 A039151

Adjacent sequences: A080609 A080610 A080611 * A080613 A080614 A080615

KEYWORD

nonn

AUTHOR

Benoit Cloitre, Feb 25 2003

STATUS

approved