A121492 -id:A121492 - cp's OEIS frontend

A167236 Larger prime power associated with gaps in A121492.

2, 7, 16, 23, 37, 59, 97, 149, 211, 307, 907, 1151, 1361, 5623, 8501, 9587, 15727, 19661, 31469, 156007, 360749, 370373, 492227, 1349651, 1357333, 2010881, 4652507, 17051887, 20831533, 47326913, 122164969, 189695893, 191913031, 387096383, 436273291, 1294268779

Offset: 1

Michael B. Porter, Nov 01 2009, Nov 03 2009

nonn

			59 is in the sequence since 53 and 59 are consecutive prime powers with a difference of 6 and no smaller pair of consecutive prime powers differ by 6 or more.

Jan Kristian Haugland, Table of n, a(n) for n = 1..87. The extra terms after a(33) are copied from A000101 as the associated prime gaps do not contain any prime powers.

Size of gap: A121492
Smaller prime power (start of gap): A002540
Gaps between prime powers: A057820
List of prime powers: A000961

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);d_max=d);n_prev=n))

a(34) onwards from Jan Kristian Haugland, Oct 18 2024

A121493 Start of the smallest m-string of equal entries in A003418(n)=lcm(1,2,...,n), where m=A121492.

Original entry on oeis.org

2, 60, 360360, 232792560, 144403552893600, 164249358725037825439200, 71876675494548945530447225706507529440, 33312720618553145840562713089120360606823375590405920630576000

Offset: 1

Lekraj Beedassy, Aug 03 2006

nonn

Cf. A002540.

A002540 Increasing gaps between prime-powers.

Original entry on oeis.org

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

N. J. A. Sloane

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

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).

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.

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

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 *)

/* 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

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

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

A167186 Record gaps between nonprime prime powers.

Original entry on oeis.org

3, 4, 7, 9, 17, 40, 41, 74, 151, 307, 312, 408, 424, 912, 1032, 1217, 1872, 2518, 3713, 4920, 5208, 8400, 8520, 8892, 9297, 12840, 16008, 21840, 24360, 35880, 38808, 80760, 102168, 129480, 167160, 183960, 201072, 258720, 290760, 301242, 358848, 375468, 415920

Offset: 1

Michael B. Porter, Oct 29 2009, Oct 31 2009, Nov 03 2009

nonn

			17 is in the sequence since A025475(9) - A025475(8) = 49 - 32 = 17, and no previous gap is larger.
A025475(10) - A025475(9) = 64 - 49 = 15, but the previous gap is larger, so 15 is not in the sequence.

Donovan Johnson, Table of n, a(n) for n = 1..200

List of nonprime prime powers: A025475.
Gaps between nonprime prime powers: A053707.
Record gaps between prime powers including primes: A121492.

Join[{3},DeleteDuplicates[Differences[Select[Range[10^6],PrimePowerQ[#] && !PrimeQ[ #]&]], GreaterEqual]] (* Harvey P. Dale, Feb 28 2023 *)

isA025475(n) = (omega(n) == 1 & !isprime(n)) || (n == 1)
d_max=0;n_prev=1;for(n=2,32e6,if(isA025475(n),d=n-n_prev;n_prev=n;if(d>d_max,print(d);d_max=d)))

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A167236 Larger prime power associated with gaps in A121492.

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A121493 Start of the smallest m-string of equal entries in A003418(n)=lcm(1,2,...,n), where m=A121492.

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A002540 Increasing gaps between prime-powers.

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A167186 Record gaps between nonprime prime powers.

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