A195945 -id:A195945 - cp's OEIS frontend

A195943 Zeroless prime powers: Intersection of A000961 and A052382.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, 256, 257, 263, 269, 271, 277, 281, 283, 289, 293, 311

Offset: 1

M. F. Hasler, Sep 25 2011

In contrast to A195942, we also allow for primes (p^n with n=1) in this sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
C. Rivera, Puzzle 607. A zeroless Prime power, on primepuzzles.net, Sept. 24, 2011.

Cf. A195942, A195944, A195945, A195946, A195908, A195948, A007377, A008839, A030700, A030701, A030702, A030703, A030704, A030705, A030706.

Cf. A038618 (subsequence).

a195943 n = a195943_list !! (n-1)
a195943_list = filter ((== 1) . a010055) a052382_list
-- Reinhard Zumkeller, Sep 27 2011

for( n=1,9999, is_A000961(n) && is_A052382(n) && print1(n","))

A195943 = A000961 intersect A052382 = A000961 \ A011540.

A010055(a(n)) * A168046(a(n)) = 1. - Reinhard Zumkeller, Sep 27 2011

A195942 Zeroless prime powers (excluding primes): Intersection of A025475 and A052382.

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1331, 1369, 1681, 1849, 2187, 2197, 3125, 3481, 3721, 4489, 4913, 5329, 6241, 6561, 6859, 6889, 7921, 8192

Offset: 1

M. F. Hasler, Sep 25 2011

Reinhard Zumkeller and T. D. Noe, Table of n, a(n) for n = 1..4094 (terms < 10^10)
C. Rivera, Puzzle 607. A zeroless Prime power, on primepuzzles.net, Sept. 24, 2011.

Cf. A195985, A195943, A195944, A195945, A195946, A195908, A195948, A007377, A008839, A030700, A030701, A030702, A030703, A030704, A030705, A030706.

a195942 n = a195942_list !! (n-1)
a195942_list = filter (\x -> a010051 x == 0 && a010055 x == 1) a052382_list
-- Reinhard Zumkeller, Sep 27 2011

mx = 10^10; t = {1}; p = 2; While[pw = 2; While[n = p^pw; n <= mx, If[Union[IntegerDigits[n]][[1]] > 0, AppendTo[t, n]]; pw++]; pw > 2, p = NextPrime[p]]; t = Sort[t] (* T. D. Noe, Sep 27 2011 *)

for( n=1,9999, is_A025475(n) && is_A052382(n) && print1(n","))

A195942 = A025475 intersect A052382.

A010055(a(n)) * (1 - A010051(a(n))) * A168046(a(n)) = 1. - Reinhard Zumkeller, Sep 27 2011

A195944 Numbers k such that 13^k has no zero in its decimal expansion.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 10, 14

Offset: 1

M. F. Hasler, Sep 25 2011

Probably finite. Is 14 the largest term?

Carlos Rivera, Puzzle 607. A zeroless Prime power

Cf. A195942, A195943, A195945, A195946, A195908, A195948, A052382, A007377, A008839, A030700, A030701, A030702, A030704, A030705, A030706.

[n: n in [0..1000] | not 0 in Intseq(13^n) ]; // Vincenzo Librandi, May 06 2015

Select[Range[0,20],DigitCount[13^#,10,0]==0&] (* Harvey P. Dale, May 24 2023 *)

for( n=0,9999, is_A052382(13^n) && print1(n","))

Equals { n | A001022(n) is in A052382 }.

Keyword:fini removed by Jianing Song, Jan 28 2023 as finiteness is only conjectured.

A195948 Powers of 5 which have no zero in their decimal expansion.

Original entry on oeis.org

1, 5, 25, 125, 625, 3125, 15625, 78125, 1953125, 9765625, 48828125, 762939453125, 3814697265625, 931322574615478515625, 116415321826934814453125, 34694469519536141888238489627838134765625

Offset: 1

M. F. Hasler, Sep 25 2011

Probably finite. Is 34694469519536141888238489627838134765625 the largest term?

C. Rivera, Puzzle 607. A zeroless Prime power, on primepuzzles.net, Sept. 24, 2011.

Cf. A195943, A195944, A195945, A195946, A195908, A007377, A008839, A030700, A030701, A030702, A030703, A030704, A030705, A030706.

Select[5^Range[0,60],DigitCount[#,10,0]==0&] (* Harvey P. Dale, Aug 30 2016 *)

for( n=0,9999, is_A052382(5^n) && print1(5^n,","))

a(n) = 5^A008839(n).

A000351 intersect A052382.

Keyword:fini removed by Jianing Song, Jan 28 2023 as finiteness is only conjectured.

A238938 Powers of 2 without the digit '0' in their decimal expansion.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 8192, 16384, 32768, 65536, 262144, 524288, 16777216, 33554432, 134217728, 268435456, 2147483648, 4294967296, 8589934592, 17179869184, 34359738368, 68719476736, 137438953472, 549755813888, 562949953421312, 2251799813685248, 147573952589676412928

Offset: 1

M. F. Hasler, Mar 07 2014

Conjectured to be finite and complete. See the OEIS wiki page for further information, references and links.

			256 = 2^8 is in the sequence because 256 has a 2, a 5 and a 6 but no 0's.
512 = 2^9 is also in because it has a 1, a 2 and a 5 but no 0's.
1024 = 2^10 is not in the sequence because it has a 0.

Daniel Mondot, Table of n, a(n) for n = 1..36
M. F. Hasler, Zeroless powers, OEIS wiki, Mar 07 2014

Cf. A007377, A030700, A030701, A008839, A030702, A030703, A030704, A030705, A030706, A195944.
Cf. A195908, A195946, A195945.
Cf. A195942, A195943, A027870, A102483.

Select[2^Range[0, 127], DigitCount[#, 10, 0] == 0 &] (* Alonso del Arte, Mar 07 2014 *)

for(n=0,99,vecmin(digits(2^n))&& print1(2^n","))

a(n) = 2^A007377(n).

'fini' keyword removed as finiteness is only conjectured by Max Alekseyev, Apr 10 2019

A238939 Powers of 3 without the digit '0' in their decimal expansion.

Original entry on oeis.org

1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 177147, 531441, 1594323, 4782969, 1162261467, 94143178827, 282429536481, 2541865828329, 7625597484987, 22876792454961, 617673396283947, 16677181699666569, 278128389443693511257285776231761

Offset: 1

M. F. Hasler, Mar 07 2014

Conjectured to be finite and complete. See the OEIS wiki page for further information, references and links.

M. F. Hasler, Zeroless powers, OEIS wiki, Mar 07 2014.

For the zeroless numbers (powers x^n), see A238938, A238939, A238940, A195948, A238936, A195908, A195946, A195945, A195942, A195943, A103662.
For the corresponding exponents, see A007377, A008839, A030700, A030701, A008839, A030702, A030703, A030704, A030705, A030706, A195944.
For other related sequences, see A052382, A027870, A102483, A103663.

Select[3^Range[0,100],DigitCount[#,10,0]==0&] (* Paolo Xausa, Oct 07 2023 *)

for(n=0,99,vecmin(digits(3^n))&& print1(3^n","))

a(n) = 3^A030700(n).

Keyword:fini removed by Jianing Song, Jan 28 2023 as finiteness is only conjectured.

A238936 Powers of 6 without the digit '0' in their decimal expansion.

Original entry on oeis.org

1, 6, 36, 216, 1296, 7776, 46656, 279936, 1679616, 2176782336, 16926659444736, 4738381338321616896, 36845653286788892983296, 17324272922341479351919144385642496

Offset: 1

M. F. Hasler, Mar 07 2014

Conjectured to be finite and complete. See the OEIS wiki page for further information, references and links.

M. F. Hasler, Zeroless powers, OEIS wiki, Mar 07 2014

Cf. A007377, A030700, A030701, A008839, A030702, A030703, A030705, A030706, A195944.
Cf. A195908, A195946, A195945.
Cf. A195942, A195943, A027870, A102483.

Select[6^Range[0,50],DigitCount[#,10,0]==0&] (* Harvey P. Dale, Dec 03 2020 *)

for(n=0,99,vecmin(digits(6^n))&& print1(6^n","))

a(n)=6^A030702(n).

Keyword:fini and keyword:full removed by Jianing Song, Jan 28 2023 as finiteness is only conjectured.

A238940 Powers of 4 without the digit '0' in their decimal expansion.

Original entry on oeis.org

1, 4, 16, 64, 256, 16384, 65536, 262144, 16777216, 268435456, 4294967296, 17179869184, 68719476736, 4722366482869645213696, 75557863725914323419136, 77371252455336267181195264

Offset: 1

M. F. Hasler, Mar 07 2014

Conjectured to be finite and complete. See the OEIS wiki page for further information, references and links.

M. F. Hasler, Zeroless powers, OEIS Wiki, Mar 07 2014

For the zeroless numbers (powers x^n), see A238938, A238939, A238940, A195948, A238936, A195908, A195946, A195945, A195942, A195943.
For the corresponding exponents, see A007377, A008839, A030700, A030701, A030702, A030703, A030704, A030705, A030706, A195944.
For other related sequences, see A052382, A027870, A102483.

Select[4^Range[0,50],DigitCount[#,10,0]==0&] (* Harvey P. Dale, Aug 31 2021 *)

for(n=0,99,vecmin(digits(4^n))&& print1(4^n","))

a(n)=4^A030701(n).

Keyword:fini removed by Jianing Song, Jan 28 2023 as finiteness is only conjectured.

A305939 Number of powers of 9 having exactly n digits '0' (in base 10), conjectured.

Original entry on oeis.org

12, 7, 18, 3, 9, 13, 11, 11, 6, 9, 17, 15, 12, 9, 11, 6, 9, 9, 9, 13, 16, 9, 10, 7, 7, 9, 9, 13, 14, 15, 14, 15, 9, 9, 8, 8, 15, 11, 11, 12, 5, 12, 14, 5, 7, 14, 10, 8, 5, 16, 12

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) = 12 is the number of terms in A030705 and in A195945, which includes the power 7^0 = 1.

These are the row lengths of A305929. It remains an open problem to provide a proof that these rows are complete (as for all terms of A020665), but the search has been pushed to many orders of magnitude beyond the largest known term, and the probability of finding an additional term is vanishing, cf. Khovanova link.

M. F. Hasler, Zeroless powers, OEIS Wiki, March 2014, updated 2018.
T. Khovanova, The 86-conjecture, Tanya Khovanova's Math Blog, Feb. 2011.
W. Schneider, No Zeros, 2000, updated 2003. (On web.archive.org--see A007496 for a cached copy.)

Cf. A030705 = row 0 of A305929: k such that 9^k has no 0's; A195945: these powers 9^k.
Cf. A020665: largest k such that n^k has no '0's.
Cf. A063626 = column 1 of A305929: least k such that 9^k has n digits 0 in base 10.
Cf. A305942 (analog for 2^k), ..., A305947, A305938 (analog for 8^k).

A305939(n,M=99*n+199,x=9)=sum(k=0,M,#select(d->!d,digits(x^k))==n)

A305939_vec(nMax,M=99*nMax+199,x=9,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(x^k)),nMax)]++);a[^-1]}

A245853 Powers of 12 without the digit '0' in their decimal expansion.

Original entry on oeis.org

1, 12, 144, 1728, 248832, 2985984, 429981696, 61917364224, 1283918464548864, 3833759992447475122176, 11447545997288281555215581184

Offset: 1

Vincenzo Librandi, Aug 04 2014

Conjectured to be finite.

Cf. Powers of k without the digit '0' in their decimal expansion: A238938 (k=2), A238939 (k=3), A238940 (k=4), A195948 (k=5), A238936 (k=6), A195908 (k=7), A245852 (k=8), A240945 (k=9), A195946 (k=11), this sequence (k=12), A195945 (k=13).

[12^n: n in [0..3*10^4] | not 0 in Intseq(12^n)];

Select[12^Range[0, 2*10^5], DigitCount[#, 10, 0]==0 &]

cp's OEIS Frontend

A195943 Zeroless prime powers: Intersection of A000961 and A052382.

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A195942 Zeroless prime powers (excluding primes): Intersection of A025475 and A052382.

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A195944 Numbers k such that 13^k has no zero in its decimal expansion.

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A195948 Powers of 5 which have no zero in their decimal expansion.

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A238938 Powers of 2 without the digit '0' in their decimal expansion.

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A238939 Powers of 3 without the digit '0' in their decimal expansion.

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A238936 Powers of 6 without the digit '0' in their decimal expansion.

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