A255334 -id:A255334 - cp's OEIS frontend

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.

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).

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).]

A255425 a(n) = A003557(A255334(n)) = A255334(n) / A255424(n).

Original entry on oeis.org

36, 36, 36, 36, 36, 36, 36, 180, 36, 576, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 396, 900, 36, 36, 36, 36, 36, 36, 576, 36, 36, 36, 36, 36, 36, 468, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 576, 36, 36

Offset: 1

Antti Karttunen, Apr 06 2015

nonn

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

Cf. A003557, A254791, A255334, A255424, A255426.

a(n) = A003557(A255334(n)) = A255334(n) / A255424(n).

For all n, a(n) > 1 and a(n) < A255426(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)).

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

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

Fred Schneider, Feb 07 2015

nonn

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.

			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.

Cf. A000203, A007947.

Subsequence of A254035. Cf. also A255334, A255425, A255426.

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)))))))

cp's OEIS Frontend

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

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A255425 a(n) = A003557(A255334(n)) = A255334(n) / A255424(n).

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

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

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

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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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Scheme