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 1-5 of 5 results.

A325635 a(n) = A091255(2n, sigma(n)).

Original entry on oeis.org

1, 1, 2, 1, 6, 12, 2, 1, 1, 6, 2, 4, 2, 4, 6, 1, 6, 3, 2, 2, 2, 4, 6, 12, 1, 2, 10, 56, 6, 12, 2, 1, 6, 10, 2, 1, 2, 4, 2, 10, 2, 4, 2, 4, 10, 12, 2, 4, 1, 1, 6, 2, 6, 20, 2, 8, 10, 6, 2, 8, 2, 4, 2, 1, 42, 12, 2, 6, 2, 28, 6, 9, 2, 2, 2, 4, 6, 4, 2, 2, 1, 2, 14, 56, 10, 12, 2, 4, 6, 2, 2, 8, 2, 4, 10, 12, 2, 1, 10, 1, 6, 20, 2, 2, 6
Offset: 1

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Author

Antti Karttunen, May 21 2019

Keywords

Links

Crossrefs

Cf. A000203, A091255, A318467, A325634, A325636, A325638 (n such that a(n) = 2n).

Programs

  • PARI
    A091255sq(a,b) = fromdigits(Vec(lift(gcd(Pol(binary(a))*Mod(1, 2),Pol(binary(b))*Mod(1, 2)))),2);
A325635(n) = A091255sq(n+n, sigma(n));

Formula

a(n) = A091255(2n, A000203(n)).
a(n) = A091255(2n, A318467(n)) = A091255(A000203(n), A318467(n)).

A325637 Numbers k for which gcd(2k, sigma(k)) = 2k.

Original entry on oeis.org

6, 28, 496, 8128, 30240, 32760, 2178540, 23569920, 33550336, 45532800, 142990848, 1379454720, 8589869056, 43861478400, 66433720320, 137438691328, 153003540480, 403031236608, 704575228896, 181742883469056, 6088728021160320, 14942123276641920, 20158185857531904, 275502900594021408, 622286506811515392, 2305843008139952128
Offset: 1

Views

Author

Antti Karttunen, May 21 2019

Keywords

Comments

Multiply-perfect numbers (A007691) k with an even abundancy index sigma(k)/k. - Amiram Eldar, Jun 26 2024

Links

Crossrefs

Cf. A000203, A033879, A224832, A325636, A325638, A325654.
Subsequences: A000396, A336702 (after its initial 1).
Subsequence of A007691.

Programs

  • PARI
    isA325637(n) = ((n+n)==gcd(n+n,sigma(n)));

Formula

a(n) = A224832(n)/2. - Amiram Eldar, Jun 26 2024

A325639 Numbers n for which A091255(n, sigma(n)) = n.

Original entry on oeis.org

1, 6, 28, 120, 312, 428, 456, 496, 504, 672, 760, 6552, 8128, 30240, 31452, 32760, 429240, 523776, 2178540, 5009850, 7505976, 23569920, 33550336, 45532800, 142990848, 186076800, 379975680
Offset: 1

Views

Author

Antti Karttunen, May 21 2019

Keywords

Comments

Numbers n for which A000203(n) = A048720(n, k) for some k. The value of k for the initial terms is: 1, 2, 2, 7, 3, 3, 6, 2, 5, 3, 3, 6, 2, 4, 6, 4, 6, 7, 4, 3, 6, 4, 2, 4, 4, 7, 7, ...
Conjecture: all terms after the initial one are even. If this is true, then there are no odd perfect numbers.
A007691(11) = 2178540 is the first term of A007691 which is not present in this sequence.

Links

Crossrefs

Fixed points of A325632 and A325634.
Cf. A000203, A007691, A048720, A091255.
Cf. A000396, A325638 (subsequences).

Programs

A379114 Numbers k for which A379113(k) > 1, i.e., k that have a proper unitary divisor d > 1 such that A048720(A065621(sigma(d)),sigma(k/d)) is equal to sigma(k).

Original entry on oeis.org

6, 10, 12, 14, 15, 20, 21, 22, 24, 26, 28, 30, 33, 35, 39, 40, 42, 44, 46, 48, 50, 51, 52, 55, 56, 57, 58, 60, 62, 63, 65, 66, 69, 70, 72, 75, 76, 77, 78, 80, 84, 86, 87, 88, 91, 92, 93, 94, 96, 100, 102, 104, 105, 108, 111, 112, 114, 115, 116, 118, 119, 120, 122, 123, 124, 126, 129, 132, 133, 136, 138, 140, 141, 143, 144
Offset: 1

Views

Author

Antti Karttunen, Dec 17 2024

Keywords

Links

Crossrefs

Positions of terms > 1 in A379113.
Cf. A000396 (subsequence, at least the even terms are), A379118 (characteristic function).
Cf. also A325638, A325639 (not subsequences).

Programs

A325808 Numbers n such that sigma(n) can be obtained as the base-3 carryless product of 2n and some k.

Original entry on oeis.org

1, 6, 28, 120, 259, 496, 8128, 18990, 667296, 1858939, 2097414, 2383279, 4843717, 33550336, 150588313, 186695863, 188908297
Offset: 1

Views

Author

Antti Karttunen, May 22 2019

Keywords

Comments

Numbers n that satisfy A000203(n) = A325820(2n, k) for some k.
Numbers n such that polynomial p divides polynomial q over GF(3), where p and q are obtained from the base-3 representations of 2n and sigma(n). (See the examples).
Conjecture: If we select only those n of these for which sigma(n) >= 2n, then we get a subsequence which contains only even terms: 6, 28, 120, 496, 8128, 18990, 667296, 2097414, 33550336, etc. If this is true, then there are no odd perfect numbers. See also conjectures in A325638 and A325639.

Examples

			2*120 has ternary representation (A007089) 22220_3, thus it encodes polynomial 2*x^4 + 2*x^3 + 2*x^2 + 2*x, while sigma(120) = 360 = 111100_3, encodes polynomial x^5 + x^4 + x^3 + x^2 which is a multiple of the former as it is equal to 2x(x^4 + x^3 + x^2 + x) when polynomial multiplication is done over GF(3). Thus 120 is included in this sequence.
2*259 = 201012_3 encodes polynomial 2*x^5 + x^3 + x + 2, while sigma(259) = 304 = 102021_3 encodes polynomial x^5 + 2*x^3 + 2*x + 1 = 2(2*x^5 + x^3 + x + 2), thus 259 is included.
2*18990 = 1221002200_3 encodes polynomial x^9 + 2*x^8 + 2*x^7 + x^6 + 2*x^3 + 2*x^2, while sigma(18990) = 49608 = 2112001100_3 encodes polynomial 2*x^9 + x^8 + x^7 + 2*x^6 + x^3 + x^2 = 2(x^9 + 2*x^8 + 2*x^7 + x^6 + 2*x^3), thus 18990 is included.
2*667296 = 2111210201100_3 encodes polynomial 2*x^12 + x^11 + x^10 + x^9 + 2*x^8 + x^7 + 2*x^5 + x^3 + x^2, while sigma(667296) = 2175264 = 11002111220100_3 encodes polynomial x^13 + x^12 + 2*x^9 + x^8 + x^7 + x^6 + 2*x^5 + 2*x^4 + x^2 = (2*x + 1)(2*x^12 + x^11 + x^10 + x^9 + 2*x^8 + x^7 + 2*x^5 + x^3 + x^2) [when polynomial multiplication is done over GF(3)], thus 667296 is included.

Links

Crossrefs

Cf. A000203, A325638, A325820, A325825, A325827.
Cf. A000396 (a subsequence).

Programs

  • PARI
    isA325808(n) = { my(p=Pol(digits(n+n,3))*Mod(1, 3), q=Pol(digits(sigma(n),3))*Mod(1, 3)); !(q%p); };
Showing 1-5 of 5 results.

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