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 1-7 of 7 results.

A048268 Smallest palindrome greater than n in bases n and n+1.

Original entry on oeis.org

6643, 10, 46, 67, 92, 121, 154, 191, 232, 277, 326, 379, 436, 497, 562, 631, 704, 781, 862, 947, 1036, 1129, 1226, 1327, 1432, 1541, 1654, 1771, 1892, 2017, 2146, 2279, 2416, 2557, 2702, 2851, 3004, 3161, 3322, 3487, 3656, 3829, 4006, 4187, 4372, 4561
Offset: 2

Views

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Keywords

Comments

From A.H.M. Smeets, Jun 19 2019: (Start)
In the following, dig(expr) stands for the digit that represents the value of expression expr, and . stands for concatenation.
As for the naming of this sequence, the trivial 1 digit palindromes 0..dig(n-1) are excluded.
If a number m is palindromic in bases n and n+1, then m has an odd number of digits when represented in base n.
All three digit numbers in base n, that are palindromic in bases n and n+1 are given by:
101_3 22_4 for n = 3,
232_n 1.dig(n).1_(n+1)
343_n 2.dig(n-1).2_(n+1)
up to and including
dig(n-2).dig(n-1).dig(n-2)n dig(n-3).4.dig(n-3)(n+1) for n > 3, and
dig(n-1).0.dig(n-1)n dig(n-3).5.dig(n-3)(n+1) for n > 4.
Let d_L(n) be the number of integers with L digits in base n (L being odd), being palindromic in bases n and n+1, then:
d_1(n) = n for n >= 2 (see above),
d_3(n) = n-2 for n >= 5 (see above),
d_5(n) = n-1 for n >= 7 and n == 1 (mod 3),
d_5(n) = n-4 for n >= 7 and n in {0, 2} (mod 3), and
it seems that d_7(n) is of order O(n^2*log(n)) for n large enough. (End)

Examples

			a(14) = 2*14^2 + 3*14 + 2 = 436, which is 232_14 and 1e1_15.

Links

Crossrefs

Cf. A029965, A029966, A048269, A060792, A097928, A097929, A097930, A097931, A099145, A099146, A099147, A099153.

Programs

  • Mathematica
    Do[ k = n + 2; While[ RealDigits[ k, n + 1 ][ [ 1 ] ] != Reverse[ RealDigits[ k, n + 1 ][ [ 1 ] ] ] || RealDigits[ k, n ][ [ 1 ] ] != Reverse[ RealDigits[ k, n ][ [ 1 ] ] ], k++ ]; Print[ k ], {n, 2, 75} ]
palQ[n_Integer, base_Integer] := Block[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]]; f[n_] := Block[{k = n + 2}, While[ !palQ[k, n] || !palQ[k, n + 1], k++ ]; k]; Table[ f[n], {n, 2, 48}] (* Robert G. Wilson v, Sep 29 2004 *)
  • PARI
    isok(j, n) = my(da=digits(j,n), db=digits(j,n+1)); (Vecrev(da)==da) && (Vecrev(db)==db);
a(n) = {my(j = n); while(! isok(j, n), j++); j;} \\ Michel Marcus, Nov 16 2017
  • PARI
    Vec(x^2*(6643 - 19919*x + 19945*x^2 - 6684*x^3 + 19*x^4) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Jun 30 2019

Formula

a(n) = 2n^2 + 3n + 2 for n >= 4 (which is 232_n and 1n1_(n+1)).
a(n) = A130883(n+1) for n > 3. - Robert G. Wilson v, Oct 08 2014
From Colin Barker, Jun 30 2019: (Start)
G.f.: x^2*(6643 - 19919*x + 19945*x^2 - 6684*x^3 + 19*x^4) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>6.
(End)

Extensions

More terms from Robert G. Wilson v, Aug 14 2000

A099145 Numbers in base 10 that are palindromic in bases 7 and 8.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 121, 178, 235, 292, 300, 2997, 6953, 7801, 10658, 13459, 16708, 428585, 431721, 444713, 447849, 450985, 502457, 626778, 786435, 10453500, 27924649
Offset: 1

Views

Author

Robert G. Wilson v, Sep 30 2004

Keywords

Comments

Intersection of A029954 and A029803. - Michel Marcus, Oct 09 2014

Examples

			178 is in the sequence because 178_10 = 343_7 = 262_8.

Links

Crossrefs

Cf. A048268, A060792, A097928, A097929, A097930, A097931, A099146, A029965, A029966, A099147, A099153.

Programs

  • Mathematica
    palQ[n_Integer, base_Integer] := Block[{idn = IntegerDigits[n, base]}, idn == Reverse[ idn]]; Select[ Range[ 150000000], palQ[ #, 7] && palQ[ #, 8] &]

A099146 Numbers in base 10 that are palindromic in bases 8 and 9.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 154, 227, 300, 373, 446, 455, 11314, 12547, 17876, 27310, 889435, 894619, 899803, 926371, 1257716, 1262900, 1268084, 1273268, 1294652, 1368461, 1373645, 1405397, 2067519, 63367795, 71877268, 98383349
Offset: 1

Views

Author

Robert G. Wilson v, Sep 30 2004

Keywords

Comments

Intersection of A029803 and A029955. - Michel Marcus, Oct 09 2014

Examples

			227 is in the sequence because 227_10 = 343_8 = 272_9.

Links

Crossrefs

Cf. A048268, A060792, A097928, A097929, A097930, A097931, A099145, A029965, A029966, A099147, A099153.

Programs

  • Mathematica
    palQ[n_Integer, base_Integer] := Block[{idn = IntegerDigits[n, base]}, idn == Reverse[ idn]]; Select[ Range[ 250000000], palQ[ #, 8] && palQ[ #, 9] &]

Extensions

Term 0 prepended by Robert G. Wilson v, Oct 08 2014

A114517 Numbers k such that the k-th heptagonal number is semiprime.

Original entry on oeis.org

4, 5, 10, 13, 14, 17, 22, 26, 29, 34, 41, 46, 53, 61, 62, 73, 74, 94, 97, 101, 109, 113, 118, 122, 146, 158, 166, 173, 178, 194, 197, 218, 229, 241, 257, 262, 274, 277, 281, 298, 314, 326, 334, 353, 358, 382, 389, 397, 398, 409, 421, 454, 458, 461, 521, 538
Offset: 1

Views

Author

Jonathan Vos Post, Feb 15 2006

Keywords

Comments

Hep(2) = 7 is the only prime heptagonal number.

Examples

			a(1) = 4 because Hep(4) = 4*(5*4-3)/2 = 34 = 2 * 17 is semiprime.
a(2) = 5 because Hep(5) = 5*(5*5-3)/2 = 55 = 5 * 11 is semiprime.
a(10) = 34 because Hep(34) = 2839 = 17 * 167 is semiprime and this is also the first iterated heptagonal semiprime Hep(34) = Hep(Hep(4)).
a(20) = 101 because Hep(101) = 25351 = 101 * 251 is semiprime [and brilliant].

Links

Crossrefs

Cf. A000040, A000566, A001222, A001358, A099153.

Programs

  • Mathematica
    Select[Range[700],PrimeOmega[(#(5#-3))/2]==2&] (* Harvey P. Dale, Jul 24 2011 *)

Formula

Numbers k such that Hep(k) = k*(5*k-3)/2 is semiprime.
Numbers k such that A000566(k) is a term of A001358.
Numbers k such that A001222(A000566(k)) = 2.
Numbers k such that A001222(k*(5*k-3)/2) = 2.
Numbers k such that [k/2 is prime and 5*k-3 is prime] or [k is prime and (5*k-3)/2 is prime].

Extensions

More terms from Harvey P. Dale, Jul 24 2011

A114548 Numbers k such that k-th heptagonal number is 3-almost prime.

Original entry on oeis.org

3, 8, 11, 19, 20, 25, 28, 37, 38, 43, 52, 58, 59, 67, 68, 70, 77, 82, 83, 85, 86, 89, 92, 98, 106, 110, 116, 124, 130, 131, 133, 134, 137, 139, 142, 149, 157, 161, 169, 172, 179, 181, 182, 185, 188, 190, 193, 202, 206, 209, 211, 214, 217, 227, 233, 238, 244
Offset: 1

Views

Author

Jonathan Vos Post, Feb 15 2006

Keywords

Examples

			a(1) = 3 because Hep(3) = 3*(5*3-3)/2 = 18 = 2 * 3^2 is 3-almost prime.
a(2) = 8 because Hep(8) = 8*(5*8-3)/2 = 148 = 2^2 * 37 is 3-almost prime.
a(3) = 11 because Hep(11) = 11*(5*11-3)/2 = 286 = 2 * 11 * 13 is 3-almost prime.
a(17) = 82 because Hep(82) = 82*(5*82-3)/2 = 16687 = 11 * 37 * 41 is 3-almost prime (and 3-brilliant).

Links

Crossrefs

Cf. A000040, A000566, A001222, A001358, A014612, A099153.

Programs

  • Mathematica
    Select[Range[400], PrimeOmega[# (5 # - 3)/2] == 3 &] (* Giovanni Resta, Jun 14 2016 *)
Select[Range[250],PrimeOmega[PolygonalNumber[7,#]]==3&] (* Harvey P. Dale, Sep 04 2020 *)

Formula

Numbers k such that Hep(k) = k*(5*k-3)/2 is 3-almost prime.
Numbers k such that A000566(k) is a term of A014612.
Numbers k such that A001222(A000566(k)) = 3.
Numbers k such that A001222(k*(5*k-3)/2) = 3.

Extensions

Corrected and extended by Giovanni Resta, Jun 14 2016

A114554 Numbers k such that the k-th heptagonal number is 4-almost prime.

Original entry on oeis.org

6, 9, 12, 18, 21, 31, 35, 40, 44, 47, 49, 50, 56, 57, 65, 66, 76, 91, 107, 121, 125, 127, 129, 136, 138, 145, 148, 152, 154, 155, 163, 164, 187, 196, 201, 205, 212, 220, 221, 223, 226, 230, 235, 236, 237, 239, 242, 246, 248, 260, 268, 284, 289, 292, 299, 309
Offset: 1

Views

Author

Jonathan Vos Post, Feb 15 2006

Keywords

Examples

			a(1) = 6 because Hep(6) = 6*(5*6-3)/2 = 81 = 3^4 is 4-almost prime.
a(2) = 9 because Hep(9) = 9*(5*9-3)/2 = 189 = 3^3 * 7 is 4-almost prime.
a(3) = 12 because Hep(12) = 12*(5*12-3)/2 = 342 = 2 * 3^2 * 19 is 4-almost prime.
a(4) = 18 because Hep(18) = 18*(5*18-3)/2 = 783 = 3^3 * 29 is 4-almost prime.
[also 783 = Hep(18) = Hep(Hep(3)) is the smallest 4-almost prime iterated heptagonal number].
a(11) = 49 because Hep(49) = 49*(5*49-3)/2 = 5929 = 7^2 * 11^2 is 4-almost prime (and the smallest such square heptagonal number A046196).
a(27) = 148 because Hep(148) = 148*(5*148-3)/2 = 54538 = 2 * 11 * 37 * 67 is 4-almost prime [also 54538 = Hep(148) = Hep(Hep(8)) is the second smallest 4-almost prime iterated heptagonal number].

Links

Crossrefs

Cf. A000040, A000566, A001222, A001358, A014613, A099153.

Programs

  • Mathematica
    Select[Range[500],PrimeOmega[(#(5#-3))/2]==4&] (* Harvey P. Dale, Aug 04 2016 *)

Formula

Numbers k such that Hep(k) = k*(5*k-3)/2 is 4-almost prime.
Numbers k such that A000566(k) is a term of A014613.
Numbers k such that A001222(A000566(k)) = 4.
Numbers k such that A001222(k*(5*k-3)/2) = 4.

Extensions

More terms from Harvey P. Dale, Aug 04 2016

A296374 a(0) = 3; a(n) = a(n-1)*(a(n-1)^2 - 3*a(n-1) + 4)/2.

Original entry on oeis.org

3, 6, 66, 137346, 1295413937737986, 1086915296274625337063297033180803022465442306
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 11 2017

Keywords

Comments

The next term is too large to include.

Examples

			a(0) = 3;
a(1) = 6 and 6 is the 3rd triangular number;
a(2) = 66 and 66 is the 6th hexagonal number;
a(3) = 137346 and 137346 is the 66th 66-gonal number, etc.

Links

Crossrefs

Cf. A001146, A007501, A057145, A060354, A097419, A099137, A099147, A099153.

Programs

  • Mathematica
    RecurrenceTable[{a[0] == 3, a[n] == a[n - 1] (a[n - 1]^2 - 3 a[n - 1] + 4)/2}, a[n], {n, 5}]

Formula

a(0) = 3; a(n) = [x^a(n-1)] x*(1 - 2*x + 4*x^2)/(1 - x)^4.
a(0) = 3; a(n) = a(n-1)! * [x^a(n-1)] exp(x)*x*(1 + x^2/2).
Showing 1-7 of 7 results.

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