A109026 -id:A109026 - cp's OEIS frontend

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).

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

Jonathan Vos Post, Jun 16 2005

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).

			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.

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Emirp.
Eric Weisstein and Jonathan Vos Post, Emirpimes.

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

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

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

A109018 Least number with exactly n prime factors counted with multiplicity which gives a different number with exactly n prime factors counted with multiplicity when digits are reversed.

Original entry on oeis.org

13, 15, 117, 126, 270, 2576, 8820, 16560, 21168, 46848, 295245, 441600, 846720, 4078080, 80663040, 40590720, 2173236480, 4011724800, 21122906112, 40915058688, 274148425728, 63769149440, 2707602702336, 6167442456576, 21586195906560, 29798871072768, 420127895977984, 631722992467968

Offset: 1

Jonathan Vos Post, Jun 16 2005

An emirp ("prime" spelled backwards) is a prime whose (base 10) reversal is also prime, but which is not a palindromic prime. The first few are 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, ... (A006567).
An emirpimes ("semiprime" spelled backwards) is a semiprime whose (base 10) reversal is a different semiprime. A list of the first emirpimeses (or "semirpimes") are 15, 26, 39, 49, 51, 58, 62, 85, 93, 94, 115, 122, 123, ... (A097393).
An "emirp tsomla-3" ("3-almost prime" spelled backwards) is the k=3 sequence of the series for which k=1 are emirps and k=2 are emirpimes, a list of these being A109023. The union of these for k=1 through k = 13 is A109019.
The primes correspond to the "1-almost prime" numbers 2, 3, 5, 7, 11, ... (A000040). The 2-almost prime numbers correspond to semiprimes 4, 6, 9, 10, 14, 15, 21, 22, ... (A001358).
The first few 3-almost primes are 8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, ... (A014612). The first few 4-almost primes are 16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, ... (A014613).
The first few 5-almost primes are 32, 48, 72, 80, ... (A014614).
The Mathematica code for this was written by Ray Chandler, who has coauthorship credit for this sequence.

			a(1) = 13 because 13 is the smallest "emirp" (prime which, digits reversed, becomes a different prime) since reverse(13) = 31 is prime.
a(2) = 15 because 15 is the smallest emirpimes ("semiprime" spelled backwards) as a semiprime whose (base 10) reversal is a different semiprime. The first such number is 15, since 15 reversed is 51 and both 15 and 51 are semiprimes (i.e. 15 = 3 * 5 and 51 = 3 * 17).
a(3) = 117 because 117 is the smallest "emirp tsomla-3" ("3-almost prime" spelled backwards) since 117 reversed is 711 and 117 = 3^2 * 13 and 711 = 3^2 * 79.

Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Emirp.
Eric Weisstein and Jonathan Vos Post, Emirpimes.

Cf. A006567, A097393, A109023, A109023, A109024, A109025, A109026, A109027, A109028, A109029, A109030, A109031.

kAlmost[n_] := Plus @@ Last /@ FactorInteger@n; fQ[n_] := Block[{id = IntegerDigits@n, k = kAlmost@n}, If[id != Reverse@id && k == kAlmost@FromDigits@Reverse@id, k, -1]]; t = Table[0, {20}]; Do[ a = fQ@n; If[a < 20 && t[[a]] == 0, t[[a]] = n; Print[{a, n}]], {n, 10, 150000000}] (* Robert G. Wilson v, Jan 06 2008 *)
Table[Select[Range[41*10^5],!PalindromeQ[#]&&PrimeOmega[#]==PrimeOmega[ IntegerReverse[ #]] ==n&][[1]],{n,14}] (* The program generates the first 14 terms of the sequence. *) (* Harvey P. Dale, Oct 15 2023 *)

a(14)-a(16) from Robert G. Wilson v, Jan 06 2008
a(17)-a(24) from Donovan Johnson, Nov 17 2008
a(25)-a(28) from Michael S. Branicky, Jun 04 2024

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

Jonathan Vos Post, Jun 16 2005

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

			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.

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.

David A. Corneth, Table of n, a(n) for n = 1..10000
J. Jonesco, Query 1622, L'Intermédiaire des Mathématiciens, 200, Tome VI, 1899.
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Emirp.
Eric Weisstein and Jonathan Vos Post, Emirpimes.

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

Select[Range[1000],PrimeOmega[#]==3&&PrimeOmega[FromDigits[Reverse[IntegerDigits[#]]]]==3&&!PalindromeQ[#]&] (* James C. McMahon, Mar 06 2024 *)

is(n) = {
	my(r = fromdigits(Vecrev(digits(n))));
	n!=r && bigomega(n) == 3 && bigomega(r) == 3
} \\ David A. Corneth, Mar 07 2024

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

Jonathan Vos Post, Jun 16 2005

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

			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).

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Emirp.
Eric Weisstein's World of Mathematics, Emirpimes.

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

Select[Range[2226],PrimeOmega[#]==4 && PrimeOmega[FromDigits[Reverse[IntegerDigits[#]]]]==4 &&!PalindromeQ[#]&] (* James C. McMahon, Mar 07 2024 *)

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

Jonathan Vos Post, Jun 16 2005

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

			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.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Emirp.
Eric Weisstein and Jonathan Vos Post, Emirpimes.

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

Select[Range[6000],!PalindromeQ[#]&&Total[FactorInteger[#][[All,2]]]==Total[ FactorInteger[ IntegerReverse[#]][[All,2]]]==5&] (* Harvey P. Dale, Nov 20 2022 *)

is(n) = {
	my(r = fromdigits(Vecrev(digits(n))));
	n!=r && bigomega(n) == 5 && bigomega(r) == 5
} \\ David A. Corneth, Mar 07 2024

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

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

Jonathan Vos Post, Jun 16 2005

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).

			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.

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Emirp.
Eric Weisstein and Jonathan Vos Post, Emirpimes.

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

Select[Range[45000],!PalindromeQ[#]&&PrimeOmega[#]==PrimeOmega[ IntegerReverse[ #]] ==7&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 02 2019 *)

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

Jonathan Vos Post, Jun 16 2005

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).

			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.

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Emirp.
Eric Weisstein and Jonathan Vos Post, Emirpimes.

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

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

Jonathan Vos Post, Jun 16 2005

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).

			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.

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Emirp.
Eric Weisstein and Jonathan Vos Post, Emirpimes.

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

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 *)

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

Jonathan Vos Post, Jun 16 2005

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).

			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.

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Emirp.
Eric Weisstein and Jonathan Vos Post, Emirpimes.

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

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 *)

is(n) = {
	my(r = fromdigits(Vecrev(digits(n))));
	n!=r && bigomega(n) == 10 && bigomega(r) == 10
} \\ David A. Corneth, Mar 07 2024

cp's OEIS Frontend

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).

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A109018 Least number with exactly n prime factors counted with multiplicity which gives a different number with exactly n prime factors counted with multiplicity when digits are reversed.

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

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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).

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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).

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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).

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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).

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