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URL: https://oeis.org/A133812

⇱ A133812 - OEIS

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A133812
Numbers that are primally tight and have weakly descending powers.
6
1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 23, 24, 25, 27, 29, 30, 31, 32, 35, 36, 37, 41, 43, 45, 47, 48, 49, 53, 59, 60, 61, 64, 67, 71, 72, 73, 77, 79, 81, 83, 89, 96, 97, 101, 103, 105, 107, 109, 113, 120, 121, 125, 127, 128, 131, 135, 137, 139, 143, 144
OFFSET
1,2
COMMENTS
All numbers of the form p_1^k1*p_2^k2*...*p_n^k_n, where k1 >= k2 >= ... >= k_n and the p_i are n successive primes.
Differs from A073491 starting n=16.
LINKS
EXAMPLE
105 is in the sequence as 105 = 3^1*5^1*7^1 and so it has 3 prime factors. Those three prime factors are consecutive primes and their exponents (1, 1, 1) are weakly decreasing. - David A. Corneth, Jan 30 2026
MATHEMATICA
aQ[n_] := Module[{f=FactorInteger[n]}, p=f[[;; , 1]]; e=f[[;; , 2]]; PrimePi[p[[-1]]]-PrimePi[p[[1]]] == Length[p]-1 && AllTrue[Differences[e], #<=0 &]]; Join[{1}, Select[Range[2, 144], aQ]] (* Amiram Eldar, Jun 20 2019 *)
PROG
(PARI) is(n) = {my(f = factor(n)); if(#f~ <= 1, return(1)); my(t = 1); forprime(q = f[1, 1]+1, oo, t++; if(f[t, 1] != q, return(0)); if(f[t, 2] > f[t-1, 2], return(0)); if(t >= #f~, return(1)))} \\ David A. Corneth, Jan 29 2026
KEYWORD
nonn
AUTHOR
Olivier Gérard, Sep 23 2007
EXTENSIONS
Cross-references from Charles R Greathouse IV, Dec 04 2009
STATUS
approved