A255334 Numbers n for which there exists k > n such that A000203(k) = A000203(n) and A007947(k) = A007947(n), where A000203 gives the sum of divisors, and A007947 gives the squarefree kernel of n.

1512, 7560, 16632, 19656, 25704, 28728, 34776, 37800, 43848, 44928, 46872, 55944, 61992, 65016, 71064, 80136, 83160, 89208, 92232, 98280, 101304, 107352, 110376, 119448, 125496, 128520, 134568, 143640, 146664, 152712, 155736, 161784, 164808, 170856, 173880, 182952, 189000, 192024, 198072, 207144, 210168, 216216

Offset: 1

Author

Antti Karttunen, Mar 23 2015

Keywords

• nonn

Comments

None of the terms are squarefree, because if there were such x, then we would have rad(x) = x, and for any value k > x such that rad(k) = x we would have k = y*x, for some strictly positive integer y, and in that case sigma(k) > sigma(x). Thus all terms are members of sequence A013929.

None of the terms in range a(1) .. a(6589) occur in A255335. Are the sequences disjoint forever?

Links

• Antti Karttunen, Table of n, a(n) for n = 1..6589

Crossrefs

Subsequence of A013929.

Cf. A000203, A007947.

Cf. also A255423 (gives the corresponding k), A255335 (same sequence sorted into ascending order, with duplicates removed), A255412 [gives sigma(a(n))], A255424 [gives rad(a(n))], A255425, A254035, A254791.

Programs

PARI

A007947(n) = factorback(factorint(n)[, 1]); \\ Andrew Lelechenko, May 09 2014
isA255334(n) = { my(r=A007947(n), s=sigma(n), k=n+r); while(kA007947(k) == r), return(1), k = k+r)); return(0); };
i=0; for(n=1, 2^25, if(isA255334(n), i++; write("b255334.txt", i, " ", n)))

Scheme

;; With Antti Karttunen's IntSeq-library. Quite naive and slow implementation.
(define A255334 (MATCHING-POS 1 1 isA255334?))
(define (isA255334? n) (let ((sig_n (A000203 n)) (rad_n (A007947 n))) (let loop ((try (+ n rad_n))) (cond ((>= try sig_n) #f) ((and (= sig_n (A000203 try)) (= rad_n (A007947 try))) #t) (else (loop (+ try rad_n)))))))

Formula

a(n) = A255424(n) * A255425(n).