A255423 -id:A255423 - cp's OEIS frontend

A255426 a(n) = A003557(A255423(n)) = A255423(n) / A255424(n).

49, 49, 49, 49, 49, 49, 49, 245, 49, 676, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 539, 1225, 49, 49, 49, 49, 49, 49, 676, 49, 49, 49, 49, 49, 49, 637, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 676, 49, 49

Offset: 1

Antti Karttunen, Apr 06 2015

nonn

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

Cf. A003557, A254791, A255423, A255424, A255425.

a(n) = A003557(A255423(n)) = A255423(n) / A255424(n).

For all n, a(n) > A255425(n) > 1.

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.

Original entry on oeis.org

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).

A255424 Squarefree kernel of A255334: a(n) = A007947(A255334(n)).

Original entry on oeis.org

42, 210, 462, 546, 714, 798, 966, 210, 1218, 78, 1302, 1554, 1722, 1806, 1974, 2226, 2310, 2478, 2562, 2730, 2814, 2982, 3066, 3318, 3486, 3570, 3738, 3990, 4074, 4242, 4326, 4494, 4578, 4746, 4830, 462, 210, 5334, 5502, 5754, 5838, 6006, 6090, 390, 6258, 6342, 6510, 6594, 6846, 7014, 546, 7266, 7518, 7602, 7770, 7854, 8022, 8106, 8274, 8358, 8610

Offset: 1

Antti Karttunen, Apr 06 2015

nonn

Sequence gives value of A007947(n) for 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. The sequence is ordered according to the magnitude of n, and contains duplicates, because there are cases of multiple such pairs having same squarefree kernel.

The first duplicate occurs as a(2) = a(8) = 210.

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

Cf. A007947, A255334, A255412, A255423, A255425, A255426.

(define (A255424 n) (A007947 (A255334 n)))

a(n) = A007947(A255334(n)).

a(n) = A007947(A255423(n)). [Equally, squarefree kernel of A255423(n).]

A255335 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.

Original entry on oeis.org

2058, 10290, 22638, 26754, 34986, 39102, 47334, 51450, 52728, 59682, 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, 263640, 269598, 281946, 286062, 294294

Offset: 1

Antti Karttunen, Mar 23 2015, suggested by Michel Marcus, Feb 23 2015

nonn

Sequence A255423 sorted into ascending order.
Note that both for u = a(17) = 113190 and v = a(22) = 146118, A000203(u) = A000203(v) = 345600.
Also, both for w = a(20) = 133770 and x = a(25) = 170814, A000203(w) = A000203(x) = 403200.
Question: Does this have any common terms with A255334 ?

Antti Karttunen, Table of n, a(n) for n = 1..2434

Subsequence of A013929.
Cf. A000203, A007947.
Cf. also A255334, A255423, A254035.

allocatemem(234567890);
A007947(n) = factorback(factorint(n)[, 1]); \\ Andrew Lelechenko, May 09 2014
upto = (2^24)-4;
bigvec = vector(upto);
i=0; for(n=1, upto, bigvec[n] = Set([]); my(r=A007947(n), s=sigma(n)); if(setsearch(bigvec[r],s), i++; write("b255335.txt", i, " ", n), bigvec[r] = setunion(Set([s]),bigvec[r])));

;; With Antti Karttunen's IntSeq-library. Quite naive implementation.
(define A255335 (MATCHING-POS 1 1 isA255335?))
(define (isA255335? n) (let ((sig_n (A000203 n)) (rad_n (A007947 n))) (let loop ((try (- n rad_n))) (cond ((< try rad_n) #f) ((and (= sig_n (A000203 try)) (= rad_n (A007947 try))) #t) (else (loop (- try rad_n)))))))

A255412 a(n) = A000203(A255334(n)).

Original entry on oeis.org

4800, 28800, 57600, 67200, 86400, 96000, 115200, 148800, 144000, 142800, 153600, 182400, 201600, 211200, 230400, 259200, 345600, 288000, 297600, 403200, 326400, 345600, 355200, 384000, 403200, 518400, 432000, 576000, 470400, 489600, 499200, 518400, 528000, 547200, 691200, 638400, 748800, 614400, 633600, 662400, 672000, 806400, 864000, 856800, 720000

Offset: 1

Antti Karttunen, Apr 05 2015

nonn

Sequence gives value of sigma(n) for 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. The sequence is ordered by the magnitude of n, and contains duplicates, because there are cases of multiple such pairs having the same value of sigma.

The first such duplicate values occur as a(17) = a(22) = 345600 and a(20) = a(25) = 403200.

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

Cf. A254035 (same sequence sorted into ascending order, with duplicates removed).

Cf. A000203, A007947, A255334, A255423, A255424.

(define (A255412 n) (A000203 (A255334 n)))

a(n) = A000203(A255334(n)).

a(n) = A000203(A255423(n)).

cp's OEIS Frontend

A255426 a(n) = A003557(A255423(n)) = A255423(n) / A255424(n).

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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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A255424 Squarefree kernel of A255334: a(n) = A007947(A255334(n)).

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A255335 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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Programs

PARI

Scheme

A255412 a(n) = A000203(A255334(n)).

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