A007496 -id:A007496 - cp's OEIS frontend

A305947 Number of powers of 7 having exactly n digits '0' (in base 10), conjectured.

10, 11, 12, 13, 9, 10, 9, 7, 10, 14, 21, 10, 18, 7, 11, 11, 12, 15, 17, 10, 11, 6, 10, 16, 13, 9, 7, 9, 11, 12, 10, 16, 7, 16, 9, 14, 13, 13, 9, 17, 14, 12, 11, 9, 13, 9, 12, 12, 9, 12, 14

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) = 10 is the number of terms in A030703 and in A195908, which includes the power 7^0 = 1.

These are the row lengths of A305927. 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. A030703 (= row 0 of A305927): k such that 7^k has no 0's; A195908: these powers 7^k.
Cf. A020665: largest k such that n^k has no '0's.
Cf. A063606 (= column 1 of A305927): least k such that 7^k has n digits '0' in base 10.
Cf. A305942 (analog for 2^k), ..., A305946, A305938, A305939 (analog for 9^k).

A305947(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(7^k))==n)

A305947_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(7^k)),nMax)]++);a[^-1]}

A306112 Largest k such that 2^k has exactly n digits 0 (in base 10), conjectured.

Original entry on oeis.org

86, 229, 231, 359, 283, 357, 475, 476, 649, 733, 648, 696, 824, 634, 732, 890, 895, 848, 823, 929, 1092, 1091, 1239, 1201, 1224, 1210, 1141, 1339, 1240, 1282, 1395, 1449, 1416, 1408, 1616, 1524, 1727, 1725, 1553, 1942, 1907, 1945, 1870, 1724, 1972, 1965, 2075, 1983, 2114, 2257, 2256

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) is the largest term in A007377: exponents of powers of 2 without digit 0.

There is no proof for any of the terms, just as for any term of A020665 and many similar / related sequences. However, the search has been pushed to many magnitudes beyond the largest known term, and the probability of any of the terms being wrong is extremely small, cf., e.g., the 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. A031146: least k such that 2^k has n digits 0 in base 10.
Cf. A305942: number of k's such that 2^k has n digits 0.
Cf. A305932: row n lists exponents of 2^k with n digits 0.
Cf. A007377: { k | 2^k has no digit 0 } : row 0 of the above.
Cf. A238938: { 2^k having no digit 0 }.
Cf. A027870: number of 0's in 2^n (and A065712, A065710, A065714, A065715, A065716, A065717, A065718, A065719, A065744 for digits 1 .. 9).
Cf. A102483: 2^n contains no 0 in base 3.

A306112_vec(nMax,M=99*nMax+199,x=2,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(x^k)),nMax)]=k);a[^-1]}

A305938 Number of powers of 8 having exactly n digits '0' (in base 10), conjectured.

Original entry on oeis.org

14, 11, 15, 11, 6, 12, 10, 7, 14, 21, 9, 9, 15, 8, 6, 10, 8, 13, 11, 13, 7, 10, 12, 8, 16, 10, 10, 10, 9, 14, 18, 11, 15, 12, 9, 9, 10, 17, 8, 12, 8, 12, 9, 8, 8, 12, 10, 17, 12, 6, 16

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) = 14 is the number of terms in A030704 and in A195946, which includes the power 7^0 = 1.

These are the row lengths of A305928. It remains an open problem to provide a proof that these rows are complete (as are 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. A030704 (= row 0 of A305928): k such that 8^k has no 0's; A195946: these powers 8^k.
Cf. A020665: largest k such that n^k has no '0's.
Cf. A063616 (= column 1 of A305928): least k such that 8^k has n digits '0' in base 10.
Cf. A305942 (analog for 2^k), ..., A305947, A305939 (analog for 9^k).

A305947(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(8^k))==n)

A305947_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(8^k)),nMax)]++);a[^-1]}

A306119 Largest k such that 9^k has exactly n digits 0 (in base 10), conjectured.

Original entry on oeis.org

34, 36, 68, 56, 65, 106, 144, 134, 119, 138, 154, 186, 194, 191, 219, 208, 247, 267, 199, 314, 292, 263, 319, 303, 307, 345, 431, 401, 375, 388, 413, 498, 488, 504, 465, 513, 565, 464, 481, 541, 568, 532, 588, 542, 600, 677, 649, 633, 613, 734, 627

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) is the largest term in A030705: exponents of powers of 9 without digit 0 in base 10.

There is no proof for any of the terms, just as for any term of A020665 and many similar / related sequences. However, the search has been pushed to many magnitudes beyond the largest known term, and the probability of any of the terms being wrong is extremely small, cf., e.g., the 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. A063626: least k such that 9^k has n digits 0 in base 10.
Cf. A305939: number of k's such that 9^k has n digits 0.
Cf. A305929: row n lists exponents of 9^k with n digits 0.
Cf. A030705: { k | 9^k has no digit 0 } : row 0 of the above.
Cf. A020665: largest k such that n^k has no digit 0 in base 10.
Cf. A071531: least k such that n^k contains a digit 0 in base 10.
Cf. A103663: least x such that x^n has no digit 0 in base 10.
Cf. A306112, ..., A306118: analog for 2^k, ..., 8^k.

A306119_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)]=k);a[^-1]}

Data corrected thanks to a remark by R. J. Mathar, by M. F. Hasler, Feb 11 2023

A305946 Number of powers of 6 having exactly n digits '0' (in base 10), conjectured.

Original entry on oeis.org

14, 10, 17, 16, 11, 14, 10, 8, 12, 19, 9, 16, 13, 11, 10, 10, 11, 10, 10, 17, 7, 15, 14, 16, 13, 22, 12, 17, 15, 17, 7, 6, 14, 22, 13, 19, 14, 12, 15, 7, 11, 14, 6, 12, 9, 12, 9, 14, 13, 15, 21

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) = 14 is the number of terms in A030702 and in A195948, which includes the power 6^0 = 1.

These are the row lengths of A305926. 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. A030702 = row 0 of A305926: k such that 6^k has no 0's; A238936: these powers 6^k.
Cf. A020665: largest k such that n^k has no '0's.
Cf. A063596 = column 1 of A305926: least k such that 6^k has n digits '0' in base 10.
Cf. A305942 (analog for 2^k), ..., A305947, A305938, A305939 (analog for 9^k).

A305946(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(6^k))==n)

A305946_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(6^k)),nMax)]++);a[^-1]}

A305943 Number of powers of 3 having exactly n digits '0' (in base 10), conjectured.

Original entry on oeis.org

23, 15, 31, 13, 18, 23, 23, 25, 16, 17, 28, 25, 22, 20, 18, 21, 19, 19, 18, 24, 33, 17, 17, 18, 17, 14, 21, 26, 25, 23, 24, 29, 17, 22, 18, 21, 27, 26, 20, 21, 13, 27, 24, 12, 18, 24, 16, 17, 15, 30, 24, 32, 24, 12, 16, 16, 23, 23, 20, 23, 19, 23, 10, 21, 20, 21, 23, 20, 19, 23, 23, 22, 16, 18, 20, 20, 13, 15, 25, 24, 28, 24, 21, 16, 14, 23, 21, 19, 23, 19, 27, 26, 22, 18, 27, 16, 31, 21, 18, 25, 24

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) = 23 is the number of terms in A030700 and in A238939, which include the power 3^0 = 1.

These are the row lengths of A305933. 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 vanishingly small, 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. A030700 = row 0 of A305933: k s.th. 3^k has no '0'; A238939: these powers 3^k.
Cf. A305931, A305934: powers of 3 with at least / exactly one '0'.
Cf. A020665: largest k such that n^k has no '0's.
Cf. A063555 = column 1 of A305933: least k such that 3^k has n digits '0' in base 10.
Cf. A305942 (analog for 2^k), ..., A305947, A305938, A305939 (analog for 9^k).

A305943(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(3^k))==n)

A305943_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(3^k)),nMax)]++);a[^-1]}

A305944 Number of powers of 4 having exactly n digits '0' (in base 10), conjectured.

Original entry on oeis.org

16, 22, 17, 14, 11, 19, 15, 15, 21, 20, 17, 22, 12, 13, 17, 24, 16, 19, 8, 17, 11, 15, 17, 15, 20, 17, 18, 20, 17, 21, 16, 19, 16, 14, 15, 19, 20, 24, 7, 16, 13, 14, 13, 14, 22, 22, 15, 18, 16, 16, 25

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) = 16 is the number of terms in A030701 and in A238940, which includes the power 4^0 = 1.

These are the row lengths of A305924. It remains an open problem to provide a proof that these rows are complete (as are 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. A030701 = row 0 of A305924: k such that 4^k has no 0's; A238940: these powers 4^k.
Cf. A020665: largest k such that n^k has no '0's.
Cf. A063575 = column 1 of A305924: least k such that 4^k has n digits '0' in base 10.
Cf. A305942 (analog for 2^k), ..., A305947, A305938, A305939 (analog for 9^k).

A305944(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(4^k))==n)

A305944_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(4^k)),nMax)]++);a[^-1]}

A305945 Number of powers of 5 having exactly n digits '0' (in base 10), conjectured.

Original entry on oeis.org

16, 16, 12, 11, 21, 12, 17, 14, 16, 17, 14, 13, 16, 18, 13, 14, 10, 10, 21, 7, 19, 13, 15, 13, 10, 15, 12, 15, 11, 11, 15, 10, 9, 15, 17, 16, 13, 12, 12, 11, 14, 9, 14, 15, 16, 14, 13, 14, 15, 24, 14

Offset: 0

M. F. Hasler, Jun 22 2018

a(0) = 16 is the number of terms in A008839 and in A195948, which includes the power 5^0 = 1.

These are the row lengths of A305925. It remains an open problem to provide a proof that these rows are complete (as are 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. A030701 (= row 0 of A305925): k such that 5^k has no 0's; A195948: these powers 4^k.
Cf. A020665: largest k such that n^k has no '0's.
Cf. A063585 (= column 1 of A305925): least k such that 5^k has n digits '0' in base 10.
Cf. A305942 (analog for 2^k), ..., A305947, A305938, A305939 (analog for 9^k).

A305945(n,M=99*n+199)=sum(k=0,M,#select(d->!d,digits(5^k))==n)

A305945_vec(nMax,M=99*nMax+199,a=vector(nMax+=2))={for(k=0,M,a[min(1+#select(d->!d,digits(5^k)),nMax)]++);a[^-1]}

A195985 Least prime such that p^2 is a zeroless n-digit number.

Original entry on oeis.org

2, 5, 11, 37, 107, 337, 1061, 3343, 10559, 33343, 105517, 333337, 1054133, 3333373, 10540931, 33333359, 105409309, 333333361, 1054092869, 3333333413, 10540925639, 33333333343, 105409255363, 333333333367, 1054092553583, 3333333333383, 10540925534207

Offset: 1

M. F. Hasler, Sep 26 2011

			a(1)^2=4, a(2)^2=25, a(3)^2=121, a(4)^2=1369 are the least squares of primes with 1, 2, 3 resp. 4 digits, and these digits are all nonzero.
a(5)=107 since 101^2=10201 and 103^2=10609 both contain a zero digit, but 107^2=11449 does not.
a(1000)=[10^500/3]+10210 (500 digits), since primes below sqrt(10^999) = 10^499*sqrt(10) ~ 3.162e499 have squares of less than 1000 digits, between sqrt(10^999) and 10^500/3 = sqrt(10^1000/9) ~ 3.333...e499 they have at least one zero digit. Finally, the 7 primes between 10^500/3 and a(1000) also happen to have a "0" digit in their square, but not so
  a(1000)^2 = 11111...11111791755555...55555659792849
  = [10^500/9]*(10^500+5) + 6806*10^500+104237294.

M. F. Hasler, Table of n, a(n) for n = 1..158
C. Rivera, Puzzle 607. A zeroless Prime power, on primepuzzles.net, Sept. 24, 2011.
W. Schneider, NoZeros: Powers n^k without Digit Zero (local copy of www.wschnei.de/digit-related-numbers/nozeros.html), as of Jan 30 2003.

Cf. A195942, A195943, A195944, A195945, A195946, A195908, A195948, A007377, A008839, A030700, A030701, A030702, A030703, A030704, A030705, A030706.

a(n)={ my(p=sqrtint(10^n\9)-1); until( is_A052382(p^2), p=nextprime(p+2));p}

A152493 Numbers k such that the decimal expansion of 2^k+5^k contains no 0's (probably 58 is last term).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 16, 17, 18, 30, 37, 58

Offset: 0

Zak Seidov, Oct 25 2009

			{n,2^n+5^n}: {0,2}, {1,7}, {2,29}, {3,133}, {4,641}, {5,3157}, {6,15689}, {7,78253}, {9,1953637}, {10,9766649}, {12,244144721}, {16,152587956161}, {17,762939584197}, {18,3814697527769}, {30,931322574616552257449}, {37,72759576141834396472156597}, {58,34694469519536141888238777858214286477369}.

Cf. A074600, A007496.

Do[p=2^n+5^n;If[FreeQ[IntegerDigits[p],0],Print[{n,p}]],{n,0,2000}]
Select[Range[0,10000],DigitCount[2^#+5^#,10,0]==0&] (* Harvey P. Dale, Oct 14 2011 *)

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A305947 Number of powers of 7 having exactly n digits '0' (in base 10), conjectured.

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A306112 Largest k such that 2^k has exactly n digits 0 (in base 10), conjectured.

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A305938 Number of powers of 8 having exactly n digits '0' (in base 10), conjectured.

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A306119 Largest k such that 9^k has exactly n digits 0 (in base 10), conjectured.

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A305946 Number of powers of 6 having exactly n digits '0' (in base 10), conjectured.

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A305943 Number of powers of 3 having exactly n digits '0' (in base 10), conjectured.

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A305944 Number of powers of 4 having exactly n digits '0' (in base 10), conjectured.

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A305945 Number of powers of 5 having exactly n digits '0' (in base 10), conjectured.

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