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.

A084292 a(n) = 6n + A054904(n).

Original entry on oeis.org

110, 77, 38, 104, 74, 161, 87, 111, 94, 159, 122, 142, 374, 209, 178, 206, 206, 253, 326, 302, 206, 302, 471, 249, 519, 341, 346, 303, 354, 481, 542, 377, 2057, 533, 386, 411, 5138, 662, 846, 527, 386, 437, 1034, 519, 794, 689, 626, 493, 566, 629, 873, 527, 638
Offset: 1

Views

Author

Labos Elemer, May 26 2003

Keywords

Comments

Composite solutions y to sigma(y-6n) = sigma(y) - 6n. For terms x of A054904, where sigma(x+6n) = sigma(x) + 6n, replacing x+6n = y, x = y-6n, we get sigma(y) - 6n = sigma(y-6n).

Links

Crossrefs

Cf. A000203 (sigma), A054904, A084293.
For several analogous sequences such corresponding "mirror-solutions" can be easily constructed. See, e.g., A015913-A015918, A050507, A054799, A054903-A054906, A054982-A054987, A059118, A055009, A055458, A063500, etc.

A054982 a(n) = least composite number such that sigma(a(n)+n!) = sigma(a(n))+n! where sigma() = A000203.

Original entry on oeis.org

434, 104, 80, 182, 427, 1727, 4147, 7163, 42031, 165841, 569257, 2683909, 10040081, 39094849, 155533969, 717519401, 3041377519, 16076525809, 71749935913
Offset: 2

Views

Author

Labos Elemer, May 29 2000

Keywords

Comments

a(21) <= 328823468719, a(22) <= 1542201899569, a(23) <= 9325753929619. - Donovan Johnson, Sep 22 2013

Examples

			a(7) = 1727 = 11*157, 4 divisors, sigma(1727)+5040 = 1896+5040 = 6936, sigma(1727+5040) = sigma(6767) = 1+67+101+6767 = 6936.
a(2) = A054799(24) = 434, a(3) = A015914(19) = 104, the first composites in that series.

Crossrefs

Cf. A054904, A054905, A054799, A015914, A033560.

Programs

  • Mathematica
    L = {}; Do[i = 1; While[ ! ((Plus @@ Divisors[i + j! ] == j! + Plus @@ Divisors[i]) && ! PrimeQ[i]), i++ ]; L = Append[L, i], {j, 2, 13}]; L (from Vit Planocka)

Extensions

More terms from Vit Planocka (planocka(AT)mistral.cz), Sep 22 2003
a(14)-a(19) from Donovan Johnson, Nov 30 2008
a(20) from Donovan Johnson, Sep 19 2013

A063500 a(n) is the least composite solution of phi(k + 6n) = phi(k) + 6n.

Original entry on oeis.org

21, 45, 63, 95, 224, 135, 301, 95, 189, 161, 260, 115, 400, 165, 476, 195, 620, 145, 644, 203, 640, 285, 343, 155, 728, 185, 567, 155, 560, 301, 860, 185, 1000, 329, 1892, 235
Offset: 1

Views

Author

Labos Elemer, Jul 30 2001

Keywords

Comments

Next term (a(37)) exceeds 10^5. - Matthew Conroy, Sep 13 2001
Next term (a(37)), if it exists, exceeds 10^7. - David Wasserman, Jul 03 2002
If a(37) exists, it is larger than 7.6 * 10^11. - Jud McCranie, Aug 29 2013
a(37) > 10^13, if it exists. - Giovanni Resta, Feb 25 2020

Examples

			n=100, d=600=6n, a(100)=671=11*61, phi(671)=600, phi(671+600)=phi(1271)=(31-1)*(41-1)=600+600=phi(671)+d.

Crossrefs

Cf. A000010, A054904, A054905.

Programs

  • PARI
    for (n=1, 36, a=1; while (isprime(a) || eulerphi(a + 6*n) != eulerphi(a) + 6*n, a++); write("b063500.txt", n, " ", a) ) \\ Harry J. Smith, Aug 24 2009

Formula

Smallest values satisfying A000010(a(n)+6n)=A000010(a(n))+6n relation.

A055036 Min[x] composite zero site for sigma(x+6^n) - sigma(x) - 6^n.

Original entry on oeis.org

104, 125, 195, 415, 2743, 2935, 3535, 19735, 22645, 108703, 977353, 1921033, 2523433, 2425175, 4227575, 85969345, 32606935, 224917033, 1362833713, 716210677, 1557843865, 6226853857, 20369543065
Offset: 1

Views

Author

Labos Elemer, Jun 01 2000

Keywords

Examples

			n = 6: d = 6^6 = 46656, a(n) = a(6) = 2935 because sigma(2935) + 46656 = 1 + 5 + 587 + 2935 + 46656 = sigma(2935 + 46656) = sigma(49591) = 1 + 101 + 491 + 49591 = 50184.

Crossrefs

Subsequence of A054904.
Cf. A054902-A054905, A054799, A054982-A054985, A054987, A055009, A015913-A015518.

Programs

  • Mathematica
    L = {}; Do[i = 1; While[ ! ((Plus @@ Divisors[i + 6^j] == 6^j + Plus @@ Divisors[i]) && ! PrimeQ[i]), i++ ]; L = Append[L, i], {j, 1, 11}]; L (from Vit Planocka)

Formula

a(n) = Min(x) solution for A000203(x+A000400(n)) = A000203(x) + A000400(n) Diophantine equation.

Extensions

One more term from Vit Planocka (planocka(AT)mistral.cz), Sep 23 2003
a(12)-a(23) from Donovan Johnson, Nov 30 2008

A063519 Least composite k such that phi(k+12n) = phi(k)+12n and sigma(k+12n) = sigma(k) + 12n where phi is the Euler totient function and sigma is the sum of divisors function.

Original entry on oeis.org

65, 95, 341, 95, 161, 115, 629, 203, 145, 203, 365, 155, 185, 155, 301, 185, 329, 235, 1541, 287, 185, 287, 413, 205, 329, 215, 469, 215, 905, 371, 365, 305, 553, 371, 1037, 235, 1145, 623, 445, 371, 35249, 295, 1133, 371, 497, 515, 749, 413, 305, 671, 565
Offset: 1

Views

Author

Labos Elemer, Aug 01 2001

Keywords

Comments

No such simultaneous solutions were found if d=12n+6.

Examples

			a(97)=10217 because 10217 is composite, phi(10217)+1164 = 9600+1164 = 10764 = phi(11381) and sigma(10217)+1164 = 10836+1164 = 12000 = sigma(11381) with 1164 = 12*97 and there is no smaller composite with these properties.

Crossrefs

A000010, A000203, A054904, A054905, A055458.

Formula

a(n) = Min{k: phi(k+12n) = phi(k)+12n and sigma(k+12n) = sigma(k)+12n and k is composite} with phi(k) = A000010(k) and sigma(k) = A000203(k).

Extensions

Name corrected by Sean A. Irvine, Apr 30 2023
Showing 1-5 of 5 results.

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