A255335 -id:A255335 - cp's OEIS frontend

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

Antti Karttunen, Mar 23 2015

nonn

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?

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

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.

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)))

;; 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)))))))

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

A255423 The least number k > A255334(n) for which A000203(k) = A000203(A255334(n)) and A007947(k) = A007947(A255334(n)), where A000203 gives the sum of divisors, and A007947 gives the squarefree kernel of n.

Original entry on oeis.org

2058, 10290, 22638, 26754, 34986, 39102, 47334, 51450, 59682, 52728, 63798, 76146, 84378, 88494, 96726, 109074, 113190, 121422, 125538, 133770, 137886, 146118, 150234, 162582, 170814, 174930, 183162, 195510, 199626, 207858, 211974, 220206, 224322, 232554, 236670, 249018, 257250, 261366, 269598, 281946, 286062, 294294

Offset: 1

Antti Karttunen, Apr 06 2015

nonn

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

Cf. A000203, A007947, A255334.

Cf. also A255335 (same sequence sorted into ascending order), A255424 (squarefree kernel of a(n)), A255426 (same terms with but with their squarefree kernel divided out of them).

A007947(n) = factorback(factorint(n)[, 1]);
nextone(n) = { if(!n,return(0)); my(r=A007947(n), s=sigma(n), k=n+r); while(kA007947(k) == r), return(k), k = k+r)); return(0); };
i=0; for(n=1, 2^25, k = nextone(n); if(k, i++; write("b255423.txt", i, " ", k))); \\ Andrew Lelechenko, May 09 2014

a(n) = A255424(n) * A255426(n).

A254035 Sequence A255412 sorted into ascending order, with duplicates removed.

Original entry on oeis.org

4800, 28800, 57600, 67200, 86400, 96000, 115200, 142800, 144000, 148800, 153600, 182400, 201600, 211200, 230400, 259200, 288000, 297600, 326400, 345600, 355200, 384000, 403200, 432000, 470400, 489600, 499200, 518400, 528000, 547200, 576000, 614400, 633600, 638400, 662400, 672000, 691200, 720000, 729600

Offset: 1

Naohiro Nomoto, Jan 23 2015

nonn

Numbers n such that n = A000203(j) = A000203(k) and A007947(j) = A007947(k), where j != k.
In other words, numbers n such that sigma(x) = n has at least two distinct solutions, with each x having the same squarefree kernel, where sigma(x) is the sum of divisor function (A000203).
Equally, sequence A000203(A255335(n)) sorted into ascending order, with duplicates removed.

			4800 is the sum of divisors of 1512 and 2058, and rad(1512) = rad(2058) = 42, hence 4800 is in the sequence with j=1512 and k=2058.

Subsequence of A159886.
Cf. A000203 (sum of divisors of n), A007947 (squarefree kernel of n).
Cf. A254791 (a subsequence).
Cf. A252997, A255334, A255335, A255412.

a(n) = A000203(A255334(n)) = A000203(A255335(n)) for n = 1 .. 7. - Antti Karttunen, Apr 05 2015

More terms from Antti Karttunen, Apr 13 2015

cp's OEIS Frontend

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.

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A255423 The least number k > A255334(n) for which A000203(k) = A000203(A255334(n)) and A007947(k) = A007947(A255334(n)), where A000203 gives the sum of divisors, and A007947 gives the squarefree kernel of n.

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A254035 Sequence A255412 sorted into ascending order, with duplicates removed.

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