cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Antti Karttunen, Mar 23 2015

Keywords

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

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).

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

Views

Author

Antti Karttunen, Apr 06 2015

Keywords

Comments

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.

Links

Crossrefs

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

Programs

Formula

a(n) = A007947(A255334(n)).
a(n) = A007947(A255423(n)). [Equally, squarefree kernel of A255423(n).]

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

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 06 2015

Keywords

Links

Crossrefs

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

Formula

a(n) = A003557(A255423(n)) = A255423(n) / A255424(n).
For all n, a(n) > A255425(n) > 1.

A254791 Nontrivial solutions to n = sigma(a) = sigma(b) (A000203) and rad(a) = rad(b) (A007947) with a != b.

Original entry on oeis.org

4800, 142800, 1909440, 32948784, 210313800, 993938400, 1069286400, 1264808160, 1309463064, 2281635216, 3055104000, 3250790400
Offset: 1

Views

Author

Fred Schneider, Feb 07 2015

Keywords

Comments

On the term "nontrivial":
If a !=b, sigma(a) = sigma(b) and rad(a) = rad(b) then sigma(a*x) = sigma(b*x) and rad(n*x) = rad(m*x) when gcd(a, b) = gcd(a,x) = gcd(b,x) = 1. So each general solution to the stated problem could generate an infinitude of constructed, "trivial" solutions. So we will limit ourselves to the more interesting "nontrivial" solutions. Precisely, if rad(a) = rad(b) = Product(p(i)), we can write a = Product(p(i)^a(i)), b = Product(p(i)^b(i)) and in this context, a(i) != b(i) for each i in order to have a nontrivial solution.
There is another type of trivial solution, if n can be expressed as the product of two or more smaller solutions, it would be considered a composite solution but still trivial.
The smallest composite solution is below:
210313800: 131576362 = 2 * 17 * 157^3 and 98731648 = 2^7 * 17^3 * 1573250790400: 2196937295 = 5 * 7^3 * 31^3 * 43 and 2156627375 = 5^3 * 7 * 31 * 43^3. Note: the common rads for the two pairs have no factors in common so we have these "trivial" composite solutions below.
sigma(131576362 * 2196937295) = sigma(98731648 * 2156627375) = sigma(131576362 * 2156627375) = sigma(98731648 * 2196937295) = 683686082027520000.

Examples

			Sigma => Pair of distinct integers 4800 => 2058 = 2 * 3 * 7^3 and 1512 = 2^3 * 3^3 * 7142800 => 52728 = 2^3 * 3 * 13^3 and 44928 = 2^7 * 3^3 * 131909440 => 1038230 = 2 * 5 * 47^3 and 752000 = 2^7 * 5^3 * 4732948784 => 10825650 = 2 * 3^9 * 5^2 * 11 and 8624880 = 2^4 * 3^4 * 5 * 11^3210313800 => 131576362 = 2 * 17 * 157^3 and 98731648 = 2^7 * 17^3 * 157993938400 => 336110688 = 2^5 * 3^3 * 73^3 and 326965248 = 2^11 * 3^7 * 73.
The pairs that contribute to the solution each have the same rad or squarefree kernel and they are "nontrivial" because within a pair for the same prime, none of the exponents match.

Crossrefs

Cf. A000203, A007947.
Subsequence of A254035. Cf. also A255334, A255425, A255426.
Showing 1-4 of 4 results.

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