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A247246 Differences of consecutive Achilles numbers. 1
36, 92, 88, 104, 40, 68, 148, 27, 125, 64, 104, 4, 153, 27, 171, 29, 20, 196, 232, 144, 56, 312, 280, 108, 188, 199, 113, 67, 189, 72, 344, 16, 112, 232, 268, 63, 45, 392, 292, 32, 76, 8, 80, 587, 50, 147, 456, 184, 288, 488, 115, 772, 137, 36, 40, 212, 248 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

29 is the first prime in this sequence, and it equals 1352 - 1323. Clearly, if the difference is prime, then these two Achilles numbers must be relatively prime, so primes appear in this sequence rarely. However, are there infinitely many n such that a(n) is prime?

The number 1 can also appear in this sequence, because it equals 5425069448 - 5425069447 = (2^3 * 26041^2) - (7^3 * 41^2 * 97^2). Does every natural number appear in this sequence? If so, do they appear infinitely often?

LINKS

Table of n, a(n) for n=1..57.

Carlos Rivera, Problem 53

Eric Weisstein's World of Mathematics, Achilles number

Wikipedia, Achilles number

FORMULA

a(n) = A052486(n+1) - A052486(n).

MAPLE

f:= proc(n) local E; E:= map(t->t[2], ifactors(n)[2]); min(E)>1 and igcd(op(E))=1 end proc:

Achilles:= select(f, [$1..10^5]):

seq(Achilles[i+1]-Achilles[i], i=1..nops(Achilles)-1); # Robert Israel, Dec 13 2014

PROG

(PARI) isA052486(n) = { n>1 & vecmin(factor(n)[, 2])>1 & !ispower(n); }

lista(nn) = {v = select(n->isA052486(n), vector(nn, i, i)); vector(#v-1, n, v[n+1] - v[n]); } \\ Michel Marcus, Nov 29 2014

CROSSREFS

Cf. A052486, A076446, A053289.

Sequence in context: A305068 A182467 A060936 * A242238 A200866 A324297

Adjacent sequences:  A247243 A247244 A247245 * A247247 A247248 A247249

KEYWORD

nonn,easy

AUTHOR

Eric Chen, Nov 28 2014

STATUS

approved

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Last modified June 2 18:17 EDT 2020. Contains 334787 sequences. (Running on oeis4.)