A036302 -id:A036302 - cp's OEIS frontend

A036325 Composite numbers whose prime factors have no digits other than 8 and 9.

7921, 704969, 800911, 8001011, 8009021, 8802011, 8810911, 8899021, 62742241, 71281079, 79120021, 80001121, 80982001, 88109911, 88910021, 712089979, 712802869, 783378979, 784171079, 791120021, 791200121, 792012869, 800020021, 800109911, 800901121, 800991011, 809001101, 809811011, 880111121

Offset: 1

Patrick De Geest, Dec 15 1998

All terms are a product of at least two terms of A020472. - David A. Corneth, Apr 30 2018

			7921 is in the sequence because it's composite and its only prime factor is 89, only having digits 8 or 9. - _David A. Corneth_, Apr 30 2018

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 5375 terms from Robert Israel)
Index entries for sequences related to prime factors.

Cf. A020472, A036302-A036324.

N:= 9: # to get all terms of <= N digits
R:= 10^N: G:= {9}: S:= {1}:
for n from 1 to N-1 do
  G:= map(t -> (t+8*10^n,t+9*10^n), G);
  newprimes:= select(isprime, G);
  for p in newprimes do
    S:= map(s -> seq(s*p^i,i=0..floor(log[p](R/s))), S)
  od
od:
sort(convert(remove(isprime, S minus {1}),list)); # Robert Israel, Apr 30 2018

Sum_{n>=1} 1/a(n) = Product_{p in A020472} (p/(p - 1)) - Sum_{p in A020472} 1/p - 1 = 0.0001296249159... . - Amiram Eldar, May 22 2022

More terms from Robert Israel, Apr 29 2018

A036303 Composite numbers whose prime factors contain no digits other than 1 and 3.

Original entry on oeis.org

9, 27, 33, 39, 81, 93, 99, 117, 121, 143, 169, 243, 279, 297, 339, 341, 351, 363, 393, 403, 429, 507, 729, 837, 891, 933, 939, 961, 993, 1017, 1023, 1053, 1089, 1179, 1209, 1243, 1287, 1331, 1441, 1469, 1521, 1573, 1703, 1859, 2187, 2197, 2511, 2673, 2799

Offset: 1

Patrick De Geest, Dec 15 1998

All terms are a product of at least two terms of A020451. - David A. Corneth, Oct 09 2020

			The composite 117 = 3^2 * 13 is in the sequence as the digits of the prime factors are either 1 or 3. - _David A. Corneth_, Oct 17 2020

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Alois P. Heinz)
Index entries for sequences related to prime factors.

Cf. A003597, A020451, A036302-A036325.

Select[Range[3000],CompositeQ[#]&&SubsetQ[{1,3},Union[Flatten[IntegerDigits/@FactorInteger[#][[;;,1]]]]]&] (* Harvey P. Dale, Jan 08 2025 *)

from sympy import factorint
def ok(n):
    f = factorint(n)
    return sum(f.values()) > 1 and all(set(str(p)) <= set("13") for p in f)
print(list(filter(ok, range(2800)))) # Michael S. Branicky, Sep 27 2021

Sum_{n>=1} 1/a(n) = Product_{p in A020451} (p/(p - 1)) - Sum_{p in A020451} 1/p - 1 = 0.3374936085... . - Amiram Eldar, May 18 2022

A036319 Composite numbers whose prime factors have no digits other than 4's and 9's.

Original entry on oeis.org

201601, 224051, 249001, 2244551, 2494501, 4467101, 4964551, 19957601, 22180051, 22225051, 22449551, 24700001, 24949501, 24990001, 42632101, 42654551, 47379551, 47404501, 49735051, 90518849, 98982601, 100598899, 111801449, 124251499, 199557601, 221780051, 222200551, 247445501

Offset: 1

Patrick De Geest, Dec 15 1998

Closed under multiplication. - David A. Corneth, Sep 21 2020
From M. F. Hasler, Sep 22 2020: (Start)
Also closed under LCM, but not under GCD.
All terms are congruent to 1 or 9 (mod 10), depending on the parity of their number of prime factors counted with multiplicity, A001222. (End)

			The smallest prime made up of 4's and 9's is 449 (see A020466), so the smallest term here is 449^2 = 201601. - _N. J. A. Sloane_, Sep 21 2020

David A. Corneth, Table of n, a(n) for n = 1..10000
Index entries for sequences related to prime factors.

Cf. A001222, A020466, A036302-A036325.

cn49Q[n_]:=Module[{fi=FactorInteger[n][[All,1]]},CompositeQ[n]&&Union[ Flatten[ IntegerDigits/@fi]]=={4,9}&&AllTrue[fi,PrimeQ]]; Select[Range[ 1,1006*10^5,2],cn49Q] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 21 2020 *)

is(N)={!isprime(N)&& !#setminus(Set(concat(apply (digits, factor(N)[,1]))), [4,9])} \\ M. F. Hasler, Sep 22 2020

Sum_{n>=1} 1/a(n) = Product_{p in A020466} (p/(p - 1)) - Sum_{p in A020466} 1/p - 1 = 0.00001523788893... . - Amiram Eldar, May 22 2022

More terms from David A. Corneth, Sep 21 2020

A036324 Composite numbers whose prime factors have no digits other than 7's and 9's.

Original entry on oeis.org

49, 343, 553, 679, 2401, 3871, 4753, 5579, 6241, 6839, 6979, 7663, 9409, 16807, 27097, 33271, 39053, 43687, 47873, 48853, 53641, 62963, 65863, 77183, 77309, 78763, 94769, 96709, 117649, 189679, 232897, 273371, 305809, 335111, 341971

Offset: 1

Patrick De Geest, Dec 15 1998

All terms are a product of at least two terms of A020471. - David A. Corneth, Oct 09 2020

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 100 terms from Harvey P. Dale)
Index entries for sequences related to prime factors.

Cf. A020471, A036302-A036325.

Select[Range[342000],CompositeQ[#]&&SubsetQ[{7,9},Union[ Flatten[ IntegerDigits/@ FactorInteger[#][[All,1]]]]]&] (* Harvey P. Dale, Aug 01 2019 *)

Sum_{n>=1} 1/a(n) = Product_{p in A020471} (p/(p - 1)) - Sum_{p in A020471} 1/p - 1 = 0.0287747452... . - Amiram Eldar, May 22 2022

Definition clarified by Harvey P. Dale, Aug 01 2019

A036304 Composite numbers whose prime factors contain no digits other than 1 and 4.

Original entry on oeis.org

121, 451, 1331, 1681, 4961, 14641, 18491, 45221, 48851, 54571, 68921, 125521, 158521, 161051, 168551, 182081, 203401, 452551, 455521, 467851, 485221, 497431, 537361, 590851, 600281, 758131, 1380731, 1686781, 1697851, 1743731, 1771561

Offset: 1

Patrick De Geest, Dec 15 1998

All terms are a product of at least two terms of A020452. - David A. Corneth, Oct 09 2020

			The composite 4961 = 11^2 * 41 is in the sequence as the digits of its prime factors are either 1 or 4. - _David A. Corneth_, Oct 17 2020

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Alois P. Heinz)
Index entries for sequences related to prime factors.

Cf. A020452, A036302-A036325.

Sum_{n>=1} 1/a(n) = Product_{p in A020452} (p/(p - 1)) - Sum_{p in A020452} 1/p - 1 = 0.0122909749... . - Amiram Eldar, May 18 2022

A036305 Composite numbers whose prime factors contain no digits other than 1 and 5.

Original entry on oeis.org

25, 55, 121, 125, 275, 605, 625, 755, 1331, 1375, 1661, 3025, 3125, 3775, 5755, 6655, 6875, 7555, 8305, 12661, 14641, 15125, 15625, 16621, 18271, 18875, 22801, 28775, 33275, 34375, 37775, 41525, 57755, 63305, 73205, 75625, 77555, 77755, 78125

Offset: 1

Patrick De Geest, Dec 15 1998

All terms are a product of at least two terms of A020453. - David A. Corneth, Oct 09 2020

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Alois P. Heinz)
Index entries for sequences related to prime factors.

Cf. A003598, A020453, A036302-A036325.

Sum_{n>=1} 1/a(n) = Product_{p in A020453} (p/(p - 1)) - Sum_{p in A020453} 1/p - 1 = 0.0873463128... . - Amiram Eldar, May 18 2022

A036306 Composite numbers whose prime factors contain no digits other than 1 and 6.

Original entry on oeis.org

121, 671, 1331, 3721, 7271, 7381, 14641, 40321, 40931, 73271, 79981, 81191, 122771, 161051, 177221, 183271, 226981, 406321, 436921, 443531, 450241, 680821, 727771, 805981, 879791, 893101, 982771, 1016321, 1227721, 1350481, 1771561

Offset: 1

Patrick De Geest, Dec 15 1998

All terms are a product of at least two terms of A020454. - David A. Corneth, Oct 09 2020

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Alois P. Heinz)
Index entries for sequences related to prime factors.

Cf. A020454, A036302-A036325.

pf16Q[n_]:=Module[{pfs=Union[Flatten[IntegerDigits/@Transpose[ FactorInteger[ n]][[1]]]]},pfs=={1}||pfs=={1,6}]; Select[Range[ 2*10^6],CompositeQ[#]&&pf16Q[#]&] (* Harvey P. Dale, Jul 12 2014 *)

Sum_{n>=1} 1/a(n) = Product_{p in A020454} (p/(p - 1)) - Sum_{p in A020454} 1/p - 1 = 0.0112607346... . - Amiram Eldar, May 18 2022

A036307 Composite numbers whose prime factors contain no digits other than 1 and 7.

Original entry on oeis.org

49, 77, 119, 121, 187, 289, 343, 497, 539, 781, 833, 847, 1207, 1309, 1331, 2023, 2057, 2401, 3179, 3479, 3773, 4913, 5041, 5467, 5831, 5929, 7819, 8197, 8449, 8591, 9163, 9317, 12287, 12439, 12881, 13277, 14161, 14399, 14641, 16807, 18989, 19547

Offset: 1

Patrick De Geest, Dec 15 1998

All terms are a product of at least two terms of A020455. - David A. Corneth, Oct 09 2020

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Alois P. Heinz)
Index entries for sequences related to prime factors.

Cf. A003599, A020455, A036302-A036325.

Sum_{n>=1} 1/a(n) = Product_{p in A020455} (p/(p - 1)) - Sum_{p in A020455} 1/p - 1 = 0.0775663737... . - Amiram Eldar, May 18 2022

A036308 Composite numbers whose prime factors contain no digits other than 1 and 8.

Original entry on oeis.org

121, 1331, 1991, 8921, 9691, 12991, 14641, 19921, 21901, 32761, 89221, 98131, 106601, 142901, 146791, 159461, 161051, 199991, 213761, 219131, 240911, 327791, 360371, 657721, 714491, 776161, 892991, 957791, 976921, 981431, 1040461, 1079441, 1172611, 1394761, 1468091

Offset: 1

Patrick De Geest, Dec 15 1998

All terms are a product of at least two terms of A020456. - David A. Corneth, Oct 09 2020

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 58 terms from Vincenzo Librandi)
Index entries for sequences related to prime factors.

Cf. A020456, A036302-A036325.

[n: n in [4..1500000] | not IsPrime(n) and forall{f: f in PrimeDivisors(n) | Intseq(f) subset [1,8]}]; // Bruno Berselli, Aug 26 2013

dpfQ[n_]:=Module[{d=Union[Flatten[IntegerDigits/@Transpose[FactorInteger[n]][[1]]]]}, !PrimeQ[n]&&(d == {1}||d == {8}||d == {1, 8})]; Select[Range[2, 1500000], dpfQ] (* Vincenzo Librandi, Aug 25 2013 *)

Sum_{n>=1} 1/a(n) = Product_{p in A020456} (p/(p - 1)) - Sum_{p in A020456} 1/p - 1 = 0.0101097220... . - Amiram Eldar, May 18 2022

More terms from Vincenzo Librandi, Aug 25 2013

A036309 Composite numbers whose prime factors contain no digits other than 1 and 9.

Original entry on oeis.org

121, 209, 361, 1331, 2101, 2189, 2299, 3629, 3781, 3971, 6859, 10021, 10109, 10901, 14641, 17309, 17461, 18829, 21989, 23111, 24079, 25289, 36481, 37981, 38009, 39601, 39919, 41591, 43681, 68951, 71839, 75449, 101189, 110231, 111199, 119911

Offset: 1

Patrick De Geest, Dec 15 1998

All terms are a product of at least two terms of A020457. - David A. Corneth, Oct 09 2020

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 360 terms from Harvey P. Dale)
Index entries for sequences related to prime factors.

Cf. A020457, A036302-A036325.

Select[Range[120000],CompositeQ[#]&&SubsetQ[{1,9},Union[Flatten[ IntegerDigits /@ FactorInteger[ #][[All,1]]]]]&] (* Harvey P. Dale, Mar 30 2019 *)

Sum_{n>=1} 1/a(n) = Product_{p in A020457} (p/(p - 1)) - Sum_{p in A020457} 1/p - 1 = 0.0200389643... . - Amiram Eldar, May 18 2022

cp's OEIS Frontend

A036325 Composite numbers whose prime factors have no digits other than 8 and 9.

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A036303 Composite numbers whose prime factors contain no digits other than 1 and 3.

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A036319 Composite numbers whose prime factors have no digits other than 4's and 9's.

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A036324 Composite numbers whose prime factors have no digits other than 7's and 9's.

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A036304 Composite numbers whose prime factors contain no digits other than 1 and 4.

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A036305 Composite numbers whose prime factors contain no digits other than 1 and 5.

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A036306 Composite numbers whose prime factors contain no digits other than 1 and 6.

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A036307 Composite numbers whose prime factors contain no digits other than 1 and 7.

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