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 1-3 of 3 results.

A046399 Smallest squarefree palindrome with exactly n distinct prime factors.

Original entry on oeis.org

1, 2, 6, 66, 858, 6006, 222222, 22444422, 244868442, 6434774346, 438024420834, 50146955964105, 2415957997595142, 495677121121776594, 22181673755737618122, 5521159517777159511255, 477552751050050157255774
Offset: 0

Views

Author

Patrick De Geest, Jun 15 1998

Keywords

Comments

Initial terms of sequences A046392-A046398.

Examples

			a(4) = 858 = 2*3*11*13.

References

  • J.-P. Delahaye, Merveilleux nombres premiers ("Amazing primes"), p. 315, Pour la Science, Paris 2000.

Crossrefs

Cf. A046385, A076886.
Cf. A046392, A046393, A046394, A046395, A046396, A046397, A046398.

Programs

  • Mathematica
    r[n_] := FromDigits[Reverse[IntegerDigits[n]]]; Do[k = 1; While[r[k] != k || !SquareFreeQ[k] || Length[Select[Divisors[k], PrimeQ]] != n, k++ ]; Print[k], {n, 0, 30}] (* Ryan Propper, Sep 16 2005 *)

Extensions

Edited by N. J. A. Sloane, Dec 06 2008 at the suggestion of R. J. Mathar
a(10)-a(13) from Donovan Johnson, Oct 03 2011
a(14)-a(15) from David A. Corneth, Oct 03 2020
a(15) corrected by Daniel Suteu, Feb 05 2023
a(16) from Michael S. Branicky, Feb 08 2023

A373465 Palindromes with exactly 5 distinct prime divisors.

Original entry on oeis.org

6006, 8778, 20202, 28182, 40404, 41514, 43134, 50505, 60606, 63336, 66066, 68586, 80808, 83538, 86268, 87978, 111111, 141141, 168861, 171171, 202202, 204402, 209902, 210012, 212212, 219912, 225522, 231132, 232232, 239932, 246642, 249942, 252252, 258852, 262262, 266662, 272272
Offset: 1

Views

Author

M. F. Hasler, Jun 06 2024

Keywords

Examples

			a(1) = 6006 = 2 * 3 * 7 * 11 * 13 is a palindrome (A002113) with 5 prime divisors.
a(5) = 40404 = 2^2 * 3 * 7 * 13 * 37 also is a palindrome with 5 prime divisors, although the divisor 2 occurs twice as a factor in the factorization.

Crossrefs

Cf. A002113 (palindromes), A051270 (omega(.) = 5).
Cf. A046331 (same but counting prime factors with multiplicity), A046395 (same but squarefree), A373466 (same with omega = 6), A373467 (with omega = 7).

Programs

  • Mathematica
    Select[Range[300000],PalindromeQ[#]&&Length[FactorInteger[#]]==5&] (* James C. McMahon, Jun 08 2024 *)
Select[Range[300000],PalindromeQ[#]&&PrimeNu[#]==5&] (* Harvey P. Dale, Sep 01 2024 *)
  • PARI
    A373465_upto(N, start=1, num_fact=5)={ my(L=List()); while(N >= start = nxt_A002113(start), omega(start)==num_fact && listput(L, start)); L}

Formula

Intersection of A002113 and A051270.

A253382 Triangle read by rows: T(n,k) appears in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} T(n,k)*(x-2k)^k.

Original entry on oeis.org

1, 5, 2, 5, 26, 3, 5, 170, 75, 4, 5, 810, 1035, 164, 5, 5, 3210, 10635, 3764, 305, 6, 5, 11274, 91275, 64244, 10385, 510, 7, 5, 36362, 693387, 910964, 261265, 24030, 791, 8, 5, 110090, 4822155, 11361908, 5422225, 830430, 49175, 1160, 9, 5, 317450, 31364235, 128935028, 98319505, 23510430, 2226455, 91880, 1629, 10
Offset: 0

Views

Author

Derek Orr, Dec 30 2014

Keywords

Comments

Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = T(n,0)*(x-0)^0 + T(n,1)*(x-2)^1 + T(n,2)*(x-4)^2 + ... + T(n,n)*(x-2n)^n, for n >= 0.

Examples

			From _Wolfdieter Lang_, Jan 14 2015: (Start)
The triangle T(n,k) starts:
n\k 0      1        2         3        4        5       6     7    8  9 ...
0:  1
1:  5
2:  5     26        3
3:  5    170       75         4
4:  5    810     1035       164        5
5:  5   3210    10635      3764      305        6
6:  5  11274    91275     64244    10385      510       7
7:  5  36362   693387    910964   261265    24030     791     8
8:  5 110090  4822155  11361908  5422225   830430   49175  1160    9
9:  5 317450 31364235 128935028 98319505 23510430 2226455 91880 1629 10
... Reformatted.
----------------------------------------------------------------------------
n = 3: 1 + 2*x + 3*x^2 + 4*x^3 = 5*(x-0)^0 +  170*(x-2)^1 + 75*(x-4)^2 + 4*(x-6)^3. (End)

Crossrefs

Cf. A253381, A247236, A247237.

Programs

  • PARI
    T(n, k)=(k+1)-sum(i=k+1, n, (-2*i)^(i-k)*binomial(i, k)*T(n, i))
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")))

Formula

T(n,n) = n+1, n >= 0.
T(n,n-1) = n + 2*n^2 + 2*n^3 = A046395(n), for n >= 1.
T(n,n-2) = (n-1)*(2*n^4-2*n^3-2*n^2-2*n+1), for n >= 2.
T(n,n-3) = (n-2)*(4*n^6-24*n^5+38*n^4-6*n^3+12*n^2-36*n+15)/3, for n >= 3.

Extensions

Edited. - Wolfdieter Lang, Jan 14 2015
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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