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

1,1

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

Formula

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

Prog

(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(k<s, if((sigma(k) == s)&&(A007947(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)))))))

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.

Sequence in context: A060676 A230402 A248718 * A255778 A236091 A239175

Adjacent sequences: A255331 A255332 A255333 * A255335 A255336 A255337

Keyword

nonn

Author

Antti Karttunen, Mar 23 2015

Status

approved