login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A339794 a(n) is the least integer k satisfying rad(k)^2 < sigma(k) and whose prime factors set is the same as the prime factors set of A005117(n+1). 2
4, 9, 25, 18, 49, 80, 121, 169, 112, 135, 289, 361, 441, 352, 529, 416, 841, 360, 961, 891, 1088, 875, 1369, 1216, 1053, 1681, 672, 1849, 1472, 2209, 2601, 2809, 3025, 3249, 1856, 3481, 3721, 1984, 4225, 1584, 4489, 4761, 1960, 5041, 5329, 4736, 5929, 2496, 6241 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Equivalently, subsequence of terms of A339744 excluding terms whose prime factor set has already been encountered.
a(n) = A005117(n + 1)^2 when A005117(n + 1) is prime. Proof: if A005117(n + 1) is a prime p then rad(A005117(n + 1))^2 = rad(p)^2 = p^2 and so integers whose prime factors set is the same as the prime factors set of A005117(n + 1) = p are p^m where m >= 1. p^2 > sigma(p^1) = p + 1 but p^2 < sigma(p^2) = p^2 + p + 1. Q.E.D. - David A. Corneth, Dec 19 2020
From Bernard Schott, Jan 19 2021: (Start)
Indeed, a(n) satisfies the double inequality A005117(n+1) < a(n) <= A005117(n+1)^2.
It is also possible that a(n) = A005117(n+1)^2, even when A005117(n+1) is not prime; the smallest such example is for a(13) = 441 = 21^2 = A005117(14)^2. (End)
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Michel Marcus)
FORMULA
a(n) <= A005117(n+1)^2. - David A. Corneth, Dec 19 2020
EXAMPLE
n a(n) prime factor set
1 4 [2] A000079
2 9 [3] A000244
3 25 [5] A000351
4 18 [2, 3] A033845
5 49 [7] A000420
6 80 [2, 5] A033846
7 121 [11] A001020
8 169 [13] A001022
9 112 [2, 7] A033847
10 135 [3, 5] A033849
11 289 [17] A001026
12 361 [19] A001029
13 441 [3, 7] A033850
14 352 [2, 11] A033848
15 529 [23] A009967
16 416 [2, 13] A288162
17 841 [29] A009973
18 360 [2, 3, 5] A143207
PROG
(PARI) u(n) = {my(fn=factor(n)[, 1]); for (k = n, n^2, my(fk = factor(k)); if (fk[, 1] == fn, if (factorback(fk[, 1])^2 < sigma(fk), return (k)); ); ); }
lista(nn) = {for (n=2, nn, if (issquarefree(n), print1(u(n), ", "); ); ); }
CROSSREFS
Cf. A000203 (sigma), A007947 (rad).
Cf. A005117 (squarefree numbers), A027748, A265668, A339744.
Subsequence: A001248 (squares of primes).
Sequence in context: A131826 A366786 A051961 * A251544 A175119 A093867
KEYWORD
nonn
AUTHOR
Michel Marcus, Dec 17 2020
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 22 08:12 EDT 2024. Contains 371893 sequences. (Running on oeis4.)