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 1-10 of 11 results. Next

A109031 Numbers that have exactly eleven prime factors counted with multiplicity (A069272) whose digit reversal is different and also has 11 prime factors (with multiplicity).

Original entry on oeis.org

295245, 426816, 542592, 618624, 2112480, 2116224, 2150064, 2154816, 2196000, 2302560, 2327616, 2342277, 2388672, 2555280, 2576896, 2599200, 2768832, 2952288, 2952576, 4017216, 4074240, 4074840, 4076160, 4076568, 4078848
Offset: 1

Views

Author

Jonathan Vos Post, Jun 16 2005

Keywords

Comments

This sequence is the k = 11 instance of the series which begins with k = 1 (emirps), k = 2, k = 3 (A109023), k = 4 (A109024), k = 5 (A109025), k = 6 (A109026), k = 7 (A109027), k = 8 (A109028), k = 9 (A109029), k = 10 (A109030).

Examples

			a(1) = 295245 is in this sequence because 295245 = 3^10 * 5 has exactly 11 prime factors counted with multiplicity and reverse(295245) = 542592 = 2^7 * 3^3 * 157 also has 11 prime factors counted with multiplicity.

Links

Crossrefs

Cf. A069272, A006567, A097393, A109018, A109023, A109024, A109025, A109026, A109027, A109028, A109029, A109030.

Programs

  • PARI
    is(n) = {
	my(r = fromdigits(Vecrev(digits(n))));
	n!=r && bigomega(n) == 11 && bigomega(r) == 11
} \\ David A. Corneth, Mar 07 2024

Extensions

a(5)-a(25) from Donovan Johnson, Apr 09 2010

A109023 3-almost primes (A014612) whose digit reversal is different and also has 3 prime factors (with multiplicity).

Original entry on oeis.org

117, 147, 165, 244, 246, 285, 286, 290, 338, 366, 369, 406, 418, 425, 435, 438, 442, 475, 498, 506, 507, 508, 524, 534, 539, 548, 561, 574, 582, 604, 605, 609, 628, 642, 663, 670, 682, 705, 711, 741, 759, 805, 814, 826, 833, 834, 845, 890, 894, 906, 935
Offset: 1

Views

Author

Jonathan Vos Post, Jun 16 2005

Keywords

Comments

This sequence is the k = 3 instance of the series which begins with k = 1, k = 2.

Examples

			1066 is in this sequence because 1066 = 2 * 13 * 41, making it a 3-almost prime and reverse(1066) = 6601 = 7 * 23 * 41, also a 3-almost prime.
2001 is in this sequence because 2001 = 3 * 23 * 29 and reverse(2001) = 1002 = 2 * 3 * 167.

References

  • W. W. R. Ball and H. S. M. Coxeter, Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 14-15, 1987.
  • J. Edalj, Problem 1622. L'Intermédiaire des Mathématiciens, 16, 34, 1909.

Links

Crossrefs

Cf. A006567, A097393, A109018, A109024, A109025, A109026, A109027, A109028, A109029, A109030, A109031.

Programs

  • Mathematica
    Select[Range[1000],PrimeOmega[#]==3&&PrimeOmega[FromDigits[Reverse[IntegerDigits[#]]]]==3&&!PalindromeQ[#]&] (* James C. McMahon, Mar 06 2024 *)
  • PARI
    is(n) = {
	my(r = fromdigits(Vecrev(digits(n))));
	n!=r && bigomega(n) == 3 && bigomega(r) == 3
} \\ David A. Corneth, Mar 07 2024

Extensions

1002 replaced by 935 - R. J. Mathar, Dec 14 2009

A109024 Numbers that have exactly four prime factors counted with multiplicity (A014613) whose digit reversal is different and also has 4 prime factors (with multiplicity).

Original entry on oeis.org

126, 225, 294, 315, 459, 488, 492, 513, 522, 558, 621, 650, 738, 837, 855, 884, 954, 1035, 1062, 1098, 1107, 1197, 1206, 1236, 1287, 1305, 1422, 1518, 1617, 1665, 1917, 1926, 1956, 1962, 1989, 2004, 2034, 2046, 2068, 2104, 2148, 2170, 2180, 2223, 2226
Offset: 1

Views

Author

Jonathan Vos Post, Jun 16 2005

Keywords

Comments

This sequence is the k = 4 instance of the series which begins with k = 1, k = 2, k = 3 (A109023).

Examples

			a(1) = 126 is in this sequence because 126 = 2 * 3^2 * 7 is a 4-almost prime and reverse(126) = 621 = 3^3 * 23 is also a 4-almost prime.
a(2) = 225 is in this sequence because 225 = 3^2 * 5^2 is a 4-almost prime and reverse(225) = 522 = 2 * 3^2 * 29 is also a 4-almost prime. (That 225 and 522 are concatenated from entirely prime digits is a coincidence, as with 2223).

Links

Crossrefs

Cf. A006567, A097393, A109018, A109023, A109025, A109026, A109027, A109028, A109029, A109030, A109031.

Programs

  • Mathematica
    Select[Range[2226],PrimeOmega[#]==4 && PrimeOmega[FromDigits[Reverse[IntegerDigits[#]]]]==4 &&!PalindromeQ[#]&] (* James C. McMahon, Mar 07 2024 *)
  • PARI
    is(n) = {
	my(r = fromdigits(Vecrev(digits(n))));
	n!=r && bigomega(n) == 4 && bigomega(r) == 4
} \\ David A. Corneth, Mar 07 2024

A109025 Numbers that have exactly five prime factors counted with multiplicity (A014614) whose digit reversal is different and also has 5 prime factors (with multiplicity).

Original entry on oeis.org

270, 1386, 1575, 2070, 2136, 2142, 2295, 2300, 2394, 2412, 2475, 2508, 2550, 2565, 2568, 2610, 2844, 2964, 3087, 3267, 3465, 3654, 3708, 3924, 4008, 4016, 4068, 4185, 4208, 4290, 4293, 4347, 4446, 4482, 4563, 4692, 4779, 4875, 4932, 5049, 5238, 5355
Offset: 1

Views

Author

Jonathan Vos Post, Jun 16 2005

Keywords

Comments

This sequence is the k = 5 instance of the series which begins with k = 1, k = 2, k = 3 (A109023), k = 4 (A109024).

Examples

			a(2) = 1386 is in this sequence because 1386 = 2 * 3^2 * 7 * 11 has exactly 5 prime factors counted with multiplicity and reverse(1386) = 6831 = 3^3 * 11 * 23 is also has exactly 5 prime factors counted with multiplicity.
5355 is in this sequence because 5355 = 3^2 * 5 * 7 * 17 and reverse(5355) = 5535 = 3^3 * 5 * 41.

Links

Crossrefs

Cf. A006567, A014614, A097393, A109018, A109023, A109024, A109026, A109027, A109028, A109029, A109030, A109031.

Programs

  • Mathematica
    Select[Range[6000],!PalindromeQ[#]&&Total[FactorInteger[#][[All,2]]]==Total[ FactorInteger[ IntegerReverse[#]][[All,2]]]==5&] (* Harvey P. Dale, Nov 20 2022 *)
  • PARI
    is(n) = {
	my(r = fromdigits(Vecrev(digits(n))));
	n!=r && bigomega(n) == 5 && bigomega(r) == 5
} \\ David A. Corneth, Mar 07 2024

Extensions

Typo in definition corrected by Harvey P. Dale, Nov 20 2022

A109026 Numbers that have exactly six prime factors counted with multiplicity (A046306) whose digit reversal is different and also has 6 prime factors (with multiplicity).

Original entry on oeis.org

2576, 2970, 4284, 4356, 4410, 4600, 4698, 4824, 5265, 5625, 6534, 6752, 6900, 8250, 8964, 10710, 10890, 13140, 13986, 16236, 16335, 17577, 18504, 19494, 20286, 20574, 21114, 21150, 21160, 21336, 21492, 21576, 21609, 21900, 21996, 22392, 22770
Offset: 1

Views

Author

Jonathan Vos Post, Jun 16 2005

Keywords

Comments

This sequence is the k = 6 instance of the series which begins with k = 1, k = 2, k = 3 (A109023), k = 4 (A109024), k = 5 (A109025).

Examples

			a(1) = 2576 is in this sequence because 2576 = 2^4 * 7 * 23 has exactly 6 prime factors counted with multiplicity reverse(2576) = 6752 = 2^5 * 211 is also has exactly 6 prime factors counted with multiplicity.

Links

Crossrefs

Cf. A046306, A006567, A097393, A109018, A109023, A109024, A109025, A109027, A109028, A109029, A109030, A109031.

Programs

  • Mathematica
    Select[Range[23000],!PalindromeQ[#]&&Total[FactorInteger[#][[All,2]]] == Total[FactorInteger[IntegerReverse[#]][[All,2]]]==6&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 10 2017 *)
Select[Range[25000],PrimeOmega[#]==PrimeOmega[IntegerReverse[#]]==6&&!PalindromeQ[#]&] (* Harvey P. Dale, Mar 11 2025 *)
  • PARI
    is(n) = {
	my(r = fromdigits(Vecrev(digits(n))));
	n!=r && bigomega(n) == 6 && bigomega(r) == 6
} \\ David A. Corneth, Mar 07 2024

A109027 Numbers that have exactly seven prime factors counted with multiplicity (A046308) whose digit reversal is different and also has 7 prime factors (with multiplicity).

Original entry on oeis.org

8820, 21240, 21708, 21780, 21920, 23280, 23472, 23625, 23800, 25560, 25584, 25758, 26280, 27432, 27504, 27888, 27900, 28836, 29250, 29403, 29736, 29970, 30492, 34884, 36828, 40338, 40572, 40950, 41976, 42228, 42984, 43659, 43956, 44128
Offset: 1

Views

Author

Jonathan Vos Post, Jun 16 2005

Keywords

Comments

This sequence is the k = 7 instance of the series which begins with k = 1 (emirps), k = 2, k = 3, k = 4, k = 5 (A109025), k = 6 (A109026).

Examples

			a(20) = 29403 is in this sequence because 29403 = 3^5 * 11^2 has exactly 7 prime factors counted with multiplicity and reverse(29403) = 30492 = 2^2 * 3^2 * 7 * 11^2 also has exactly 7 prime factors counted with multiplicity.

Links

Crossrefs

Cf. A046308, A006567, A097393, A109018, A109023, A109024, A109025, A109026, A109028, A109029, A109030, A109031.

Programs

  • Mathematica
    Select[Range[45000],!PalindromeQ[#]&&PrimeOmega[#]==PrimeOmega[ IntegerReverse[ #]] ==7&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 02 2019 *)
  • PARI
    is(n) = {
	my(r = fromdigits(Vecrev(digits(n))));
	n!=r && bigomega(n) == 7 && bigomega(r) == 7
} \\ David A. Corneth, Mar 07 2024

A109028 Numbers that have exactly eight prime factors counted with multiplicity (A046310) whose digit reversal is different and also has 8 prime factors (with multiplicity).

Original entry on oeis.org

16560, 25515, 27864, 42480, 46872, 51552, 57348, 61488, 65448, 67797, 69408, 69840, 79776, 80496, 84375, 84456, 88416, 105336, 119448, 125928, 160416, 167076, 202032, 204984, 206928, 210960, 211104, 211464, 213750, 213792, 213920, 213984
Offset: 1

Views

Author

Jonathan Vos Post, Jun 16 2005

Keywords

Comments

This sequence is the k = 8 instance of the series which begins with k = 1 (emirps), k = 2, k = 3 (A109023), k = 4 (A109024), k = 5 (A109025), k = 6 (A109026), k = 7 (A109027).

Examples

			a(2) = 25515 is in this sequence because 25515 = 3^6 * 5 * 7 has exactly 8 prime factors counted with multiplicity and reverse(25515) = 51552 = 2^5 * 3^2 * 179 also has exactly 8 prime factors counted with multiplicity.

Links

Crossrefs

Cf. A046310, A006567, A097393, A109018, A109023, A109024, A109025, A109026, A109027, A109029, A109030, A109031.

Programs

  • PARI
    is(n) = {
	my(r = fromdigits(Vecrev(digits(n))));
	n!=r && bigomega(n) == 8 && bigomega(r) == 8
} \\ David A. Corneth, Mar 07 2024

A109029 Numbers that have exactly nine prime factors counted with multiplicity (A046312) whose digit reversal is different and also has 9 prime factors (with multiplicity).

Original entry on oeis.org

21168, 23424, 23616, 27456, 41184, 42432, 48114, 61632, 65472, 86112, 211410, 212256, 213192, 215232, 217440, 219072, 230208, 232512, 236925, 236928, 238656, 238680, 251100, 251505, 251748, 253824, 255024, 255960, 257856, 259968, 270912
Offset: 1

Views

Author

Jonathan Vos Post, Jun 16 2005

Keywords

Comments

This sequence is the k = 8 instance of the series which begins with k = 1 (emirps), k = 2, k = 3 (A109023), k = 4 (A109024), k = 5 (A109025), k = 6 (A109026), k = 7 (A109027), k = 8 (A109028).

Examples

			a(1) = 21168 is in this sequence because 21168 = 2^4 * 3^3 * 7^2 has exactly 9 prime factors counted with multiplicity and reverse(21168) = 86112 = 2^5 * 3^2 * 13 * 23 also has exactly 9 prime factors counted with multiplicity.

Links

Crossrefs

Cf. A046312, A006567, A097393, A109018, A109023, A109024, A109025, A109026, A109027, A109028, A109030, A109031.

Programs

  • Mathematica
    okQ[n_]:=Module[{idn=IntegerDigits[n],ridn},ridn=Reverse[idn];idn!=ridn && PrimeOmega[n]==9&&PrimeOmega[FromDigits[ridn]]==9]; Select[Range[ 271000],okQ] (* Harvey P. Dale, Sep 24 2011 *)
  • PARI
    is(n) = {
	my(r = fromdigits(Vecrev(digits(n))));
	n!=r && bigomega(n) == 9 && bigomega(r) == 9
} \\ David A. Corneth, Mar 07 2024

A109030 Numbers that have exactly ten prime factors counted with multiplicity (A046314) whose digit reversal is different and also has 10 prime factors (with multiplicity).

Original entry on oeis.org

46848, 84864, 217152, 219456, 232848, 251712, 257664, 259776, 274104, 276048, 401472, 415584, 422820, 428160, 428736, 447360, 466752, 485514, 637824, 650160, 654912, 677952, 808320, 840672, 846369, 848232, 963648
Offset: 1

Views

Author

Jonathan Vos Post, Jun 16 2005

Keywords

Comments

This sequence is the k = 10 instance of the series which begins with k = 1 (emirps), k = 2, k = 3 (A109023), k = 4 (A109024), k = 5 (A109025), k = 6 (A109026), k = 7 (A109027), k = 8 (A109028), k = 9 (A109029).

Examples

			a(1) = 46848 is in this sequence because 46848 = 2^8 * 3 * 61 has exactly 10 prime factors counted with multiplicity and reverse(46848) = 84864 = 2^7 * 3 * 13 * 17 also has exactly 10 prime factors counted with multiplicity.

Links

Crossrefs

Cf. A046314, A006567, A097393, A109018, A109023, A109024, A109025, A109026, A109027, A109028, A109029, A109031.

Programs

  • Mathematica
    taQ[n_]:=Module[{idn=IntegerDigits[n],rev},rev=Reverse[idn];rev!=idn&&PrimeOmega[n] == 10 == PrimeOmega[FromDigits[rev]]]; Select[Range[ 1000000], taQ] (* Harvey P. Dale, May 03 2013 *)
  • PARI
    is(n) = {
	my(r = fromdigits(Vecrev(digits(n))));
	n!=r && bigomega(n) == 10 && bigomega(r) == 10
} \\ David A. Corneth, Mar 07 2024

A050699 Nonprime numbers n such that n and n-reversed (<> n and no leading zeros) have the same number of prime factors (counted with multiplicity).

Original entry on oeis.org

15, 26, 39, 49, 51, 58, 62, 85, 93, 94, 115, 117, 122, 123, 126, 129, 143, 147, 155, 158, 159, 165, 169, 177, 178, 183, 185, 187, 203, 205, 221, 225, 226, 244, 246, 265, 285, 286, 289, 294, 302, 314, 315, 319, 321, 326, 327, 329, 335, 338, 339, 341, 355, 366
Offset: 1

Views

Author

Patrick De Geest, Aug 15 1999

Keywords

Examples

			E.g., 321 = 3*107 and 123 = 3*41 -> both 321 and 123 have two prime factors.

Links

Crossrefs

Cf. A050700, A050701, A050702, A006567, A097393, A109018-A109031.

Programs

  • Maple
    with(numtheory): read(transforms): for n from 12 to 366 do r:=digrev(n): if(not isprime(n) and not n=r and not n mod 10 = 0 and bigomega(n)=bigomega(r))then printf("%d, ", n); fi: od: # Nathaniel Johnston, Jun 23 2011
  • Mathematica
    nrnQ[n_]:=Module[{idn=IntegerDigits[n],rev},rev=Reverse[idn];!PrimeQ[n] &&First[rev]!=0&&idn!=rev&&PrimeOmega[n]==PrimeOmega[FromDigits[rev]]]; Select[Range[400],nrnQ] (* Harvey P. Dale, Jun 23 2011 *)

Extensions

Definition clarified by Harvey P. Dale, Jun 23 2011
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