A002540 Increasing gaps between prime-powers. (Formerly M1431 N0565)

1, 5, 13, 19, 32, 53, 89, 139, 199, 293, 887, 1129, 1331, 5591, 8467, 9551, 15683, 19609, 31397, 155921, 360653, 370261, 492113, 1349533, 1357201, 2010733, 4652353, 17051707, 20831323, 47326693, 122164747, 189695659, 191912783

Offset

1,2

Comments

List of prime-powers where A057820 increases.

The entry K=a(k) is the start of the smallest chain of m=A121492(k) consecutive numbers such that lcm(1,2,...,K) = lcm(1,2,...,K,K+1) = lcm(1,2,...,K,K+1,K+2) = ... = lcm(1,2,...,K,...,K+m-1). See A121493. - Lekraj Beedassy, Aug 03 2006

References

J. P. Gram, Undersoegelser angaaende maengden af primtal under en given graense, Det Kongelige Danskevidenskabernes Selskabs Skrifter, series 6, vol. 2 (1884), 183-288; see p. 255.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Donovan Johnson, Table of n, a(n) for n = 1..79

J. P. Gram, Undersoegelser angaaende maengden af primtal under en given graense (1884) [Scanned copy of page 255 with annotations by Victor Meally and N. J. A. Sloane]

Des MacHale and Joseph Manning, Maximal runs of strictly composite integers, The Mathematical Gazette, Vol. 99, No. 545 (2015), pp. 213-219.

Victor Meally, Letter to N. J. A. Sloane, Mar 17, 1980.

Index entries for primes, gaps between.

Mathematica

s = {}; gap = 0; p1 = 1; Do[If[PrimePowerQ[p2], If[(d = p2 - p1) > gap, gap = d; AppendTo[s, p1]]; p1 = p2], {p2, 2, 10^6}]; s (* Amiram Eldar, Dec 12 2022 *)

Prog

(PARI) /* calculates smaller terms - see Donovan Johnson link for larger terms */
isA000961(n) = (omega(n) == 1 || n == 1)
d_max=0; n_prev=1; for(n=2, 1e6, if(isA000961(n), d=n-n_prev; if(d>d_max, print(n_prev); d_max=d); n_prev=n)) \\ Michael B. Porter, Oct 31 2009

Crossrefs

Cf. A000961 (prime-powers), A057820 (gaps), A002386 (prime equivalent), A094158, A121493.

Sequence in context: A166575 A265808 A265790 * A290515 A082093 A045455

Adjacent sequences: A002537 A002538 A002539 * A002541 A002542 A002543

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

Second term corrected by Donovan Johnson, Nov 13 2008 (cf. A094158)

a(28)-a(79) and b-file from Donovan Johnson, Nov 14 2008

Status

approved