A002540 Increasing gaps between prime-powers.

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, 387096133, 436273009, 1294268491

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

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

• Jan Kristian Haugland, Table of n, a(n) for n = 1..87 (terms 1..79 from Donovan Johnson). The extra terms are copied from A002386 as the associated prime gaps do not contain any prime powers.
• 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.

Crossrefs

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

Programs

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 *)
Join[{1},Rest[Module[{nn=5*10^6,pps},pps=Select[Range[nn],PrimePowerQ]; DeleteDuplicates[ Thread[{Most[ pps],Differences[ pps]}],GreaterEqual[ #1[[2]],#2[[2]]]&]][[;;,1]]]] (* The program generates the first 27 terms of the sequence. *) (* Harvey P. Dale, Aug 20 2024 *)

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

Extensions

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

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