A328095 -id:A328095 - cp's OEIS frontend

A328561 Numbers in A328095 whose product of digits is not a power of 10.

0, 5, 6, 77, 87, 375, 376, 736, 792, 2174, 8772, 9375, 11628, 9859155, 23255814, 62227496, 398472522, 3691262781, 6886826188, 517322161894, 774773248793, 2675959368829, 51964667728417, 52446797239186

Offset: 1

Chai Wah Wu, Alois P. Heinz and David A. Corneth, Oct 19 2019

A subsequence of A328095. All other terms in A328095 have a product of digits that is a power of 10.

a(25) > 10^14. - Giovanni Resta, Oct 27 2019

Cf. A328095, A328560.

a(20)-a(22) from Giovanni Resta, Oct 23 2019

a(23)-a(24) from Giovanni Resta, Oct 24 2019

A339144 a(n) is the smallest positive integer such that n*a(n) contains n+a(n) as a substring. If no such number exists then a(n) = -1.

Original entry on oeis.org

-1, 2, -1, 68, -1, -1, -1, 44, -1, 890, 110, 60, 44, 35, 30, 27, 25, 23, 22, 20, 929, 19, 18, 88, 17, -1, 16, 16, 68, 15, 15, 60, 58, 56, 14, 14, 14, 371, 48, 360, 336, 562, 9104, 8, 13, 13, 283, 39, 269, 450, 37, 452, 245, 18, 679, 34, 225, 33, 2053, 12, 12, 12, 12, 12, 30, 369, 889, 4, 16961

Offset: 1

Scott R. Shannon, Nov 25 2020

For n = 1, 3, 5, 6, 7, 9, 26 no value has been found for which n*a(n) contains n + a(n) as a substring (obviously true for n = 1) for a(n) up to 5x10^10. It is likely, although unproven, that this is the complete list of values for which a(n) = -1.

The sequence values display erratic behavior. Most of the term values appear random but there are ranges of n values with the same value. The largest such range for the first one million terms is a(501000) to a(501499), 500 terms, all of which have a(n) = 1002. Patterns also appear for n value corresponding to multiples of powers-of-ten. For example if n=10^k then a(n) = 89*10^k. The largest value in the first one million terms is a(554635) = 879948670.

			a(2) = 2 as 2*2 = 4 which contains 2 + 2 = 4 as a substring.
a(4) = 68 as 4*68 = 272 which contains 4+68 = 72 as a substring.
a(69) = 16961 as 69*16961 = 1170309 which contains 69+16961 = 17030 as a substring.
a(501000) = 1002 as 501000*1002 = 502002000 which contains 501000+1002 = 502002 as a substring. This is the first of 500 consecutive terms with a(n) = 1002.
a(554635) = 879948670 as 554635*879948670 = 488050330585450 which contain 554635+879948670 = 880503305 as a substring. This is the largest value of a(n) for the first one million terms.

Scott R. Shannon, Table of n, a(n) for n = 1..20000.

Cf. A339107, A063427, A063428, A328095, A333923, A332580.

isok(n, k) = #strsplit(Str(n*k), Str(n+k)) > 1;
a(n) = {if (vecsearch([1, 3, 5, 6, 7, 9, 26], n), return (-1)); my(k=1); while (! isok(k, n), k++); k;} \\ Michel Marcus, Dec 02 2020 and Jan 23 2021

A328560 Numbers whose product of digits is a power of 10.

Original entry on oeis.org

1, 11, 25, 52, 111, 125, 152, 215, 251, 455, 512, 521, 545, 554, 1111, 1125, 1152, 1215, 1251, 1455, 1512, 1521, 1545, 1554, 2115, 2151, 2255, 2511, 2525, 2552, 4155, 4515, 4551, 5112, 5121, 5145, 5154, 5211, 5225, 5252, 5415, 5451, 5514, 5522, 5541, 5558, 5585

Offset: 1

Chai Wah Wu, Alois P. Heinz and David A. Corneth, Oct 19 2019

All terms must have only 1, 2, 4, 5, 8 as digits.

A subsequence of A328095.

Alois P. Heinz, Table of n, a(n) for n = 1..27446 (all terms with up to 9 digits)

Cf. A028846, A276037, A284375, A316315, A326833, A328095.

q:= n-> (m-> m>0 and m=10^ilog[10](m))(mul(i, i=convert(n, base, 10))):
select(q, [$1..6000])[];

A326806 Numbers k such that k multiplied by the sum of all its digits contains k as a substring.

Original entry on oeis.org

0, 1, 5, 6, 10, 19, 28, 37, 46, 50, 55, 60, 64, 73, 82, 91, 100, 109, 118, 127, 136, 145, 154, 163, 172, 181, 190, 208, 217, 226, 235, 244, 253, 262, 271, 280, 307, 316, 325, 334, 343, 352, 361, 370, 406, 415, 424, 433, 442, 451, 460, 500, 505, 514, 523, 532

Offset: 1

Alois P. Heinz, Oct 19 2019

Inspired by A328095.
Contains all numbers whose digit sum is a power of 10.
Contains all numbers of the form 5*10^k and 6*10^k. Terms which are not in the sets above are: 0, 6667, 58824, 8823529412, ... - Chai Wah Wu, Oct 19 2019

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A007953, A011557, A326811, A328095.

a:= proc(n) option remember; local k; if n=1 then 0 else
      for k from 1+a(n-1) while searchtext(cat(k), cat(k*
      add(i, i=convert(k, base, 10))))=0 do od: k fi
    end:
seq(a(n), n=1..75);

n, A326806_list = 0, []
while len(A326806_list) < 10000:
    sn = str(n)
    if sn in str(n*sum(int(d) for d in sn)):
        A326806_list.append(n)
    n += 1 # Chai Wah Wu, Oct 19 2019

A339403 a(n) is the smallest positive integer such that n*a(n) contains the string n+a(n) in reverse as a substring. If no such number exists then a(n) = -1.

Original entry on oeis.org

0, -1, 2, 24, 37, 26, 34, 35, 57, 9, -1, 12, 11, 45, 193, 228, 28, 51, 23, 44, 841, 11, 27, 18, 3, 626, 5, 22, 16, 46716, 56, 41, 33, 32, 6, 7, 21, 4, 3, 24, 592, 31, 7, 619, 19, 13, 38, 2, 117, 5, 463, 17, 34, 308, 33, 36, 30, 8, 31, 4, 23, 21, 648, 124, 921, 903, 386, 395, 4, 334, 755, 31, 563

Offset: 0

Scott R. Shannon, Dec 03 2020

This is a variation of A339144 where, instead of the n*a(n) containing n+a(n) as a substring, it contains the reverse of the string n+a(n), including any leading zeros.
Based on a search limit of 5x10^9 up to n = 100000 the values of n for which no a(n) is found are n = 10^k, with k>=0, and 17500. A test of 175000 and 1750000 also found no a(n) indicating that all values of the form 17500*10^k may have no term for a(n).
It is found that when n = 200*10^k, with k>=0, the corresponding value for a(n) is significantly larger than neighboring terms. As an example a(20000) = 666843331, which is the largest term up to n = 100000.
Unlike A339144, which contains multiple consecutive terms with the same value of a(n), in this sequence the largest consecutive run of the same a(n) in the first 100000 terms is only two. The first term of these pairs occurs at n = 110, 121, 2717, 4368, 7916, 10100, 11211, 13231, 17271, 44573, 63529.

			a(3) = 24 as 3*24 = 72 which contains reverse(3+24) = reverse(27) = 72 as a substring.
a(6) = 34 as 6*34 = 204 which contains reverse(6+34) = reverse(40) = 04 as a substring. Note the leading zero is included.
a(29) = 46716 as 29*46716 = 1354764 which contains reverse(29+4671) = reverse(46745) = 54764 as a substring.
a(110) = 11 as 110*11 = 1210 which contains reverse(110+11) = reverse(121) = 121 as a substring. This is the first of two consecutive terms with a(n) = 11.
a(20000) = 666843331 as 20000*666843331 = 13336866620000 which contains reverse(20000+666843331) = reverse(666863331) = 133368666 as a substring. This is the largest value in the first 100000 terms.

Scott R. Shannon, Table of n, a(n) for n = 0..10000

Cf. A339144, A339107, A063427, A063428, A328095, A333923, A332580.

isok(n, k) = #strsplit(Str(n*k), concat(Vecrev(Str(n+k)))) > 1;
ispt(n) = my(t); ispower(n,,&t) && (t==10);
a(n) = {if ((n==1) || (n==10) || ispt(n), return (-1)); my(k=0); while (! isok(n, k), k++); k;} \\ Michel Marcus, Jan 22 2021

A341035 a(n) is the smallest positive integer such that n+a(n) contains the string n-a(n), in both forward and reverse directions, as a substring. If no such number exists then a(n) = -1.

Original entry on oeis.org

-1, -1, -1, -1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 10, 10, 10, 10, 10, 15, 15, 15, 15, 15, 20, 20, 20, 20, 20, 25, 25, 25, 25, 25, 29, 30, 30, 30, 30, 33, 34, 35, 35, 35, 37, 38, 39, 40, 40, 41, 42, 43, 44, 45, 50, 50, 50, 50, 50, 55, 50, 51, 52, 53, 54, 60, 60, 60, 60, 65, 50, 50, 65, 65, 70, 70, 70

Offset: 1

Scott R. Shannon, Feb 03 2021

Based on a search limit of 5*10^9 up to n = 300000 the values of n for which no a(n) is found are n = 1,2,3,4. This is likely the complete list of values for which no a(n) exists.

The longest run of consecutive terms with the same value in the first 300000 terms is the run of 5's at the beginning of the sequence, ten in all. This is likely the longest run for all numbers.

			a(5) = 5 as 5+5 = 10 which contains both 5-5 = 0 and reverse(0) = 0 as a substring.
a(15) = 10 as 15+10 = 25 which contains both 15-10 = 5 and reverse(5) = 5 as a substring.
a(61) = 50 as 61+50 = 111 which contains both 51-50 = 11 and reverse(11) = 11 as a substring.
a(71) = 50 as 71+50 = 121 which contains both 71-50 = 21 and reverse(21) = 12 as a substring.
a(1902) = 1829 as 1902+1829 = 3731 which contains both 1902-1829 = 73 and reverse(73) = 37 as a substring.

Scott R. Shannon, Image of the terms for n=5..100000.

Cf. A341034 (forward), A341028 (reverse), A339403, A339144, A328095, A333410, A332703.

A328544 Numbers k such that k multiplied by any of its digits contains k as a substring.

Original entry on oeis.org

0, 1, 5, 6, 11, 111, 1111, 11111, 111111

Offset: 1

N. J. A. Sloane, Oct 19 2019

Conjecture: consists of 0,5,6, and the numbers 11...1.

Suggested by A328095 (which is a much more interesting sequence).

isA328544 := proc(n)
    local dgs ,d,ndgs,nddgs;
    ndgs := convert(n,base,10) ;
    dgs := convert(ndgs,set) ;
    for d in dgs do
        nddgs := convert(n*d,base,10) ;
        if not verify(ndgs,nddgs,'sublist') then
            return false;
        end if;
    end do:
    true ;
end proc:
for n from 0 do
    if isA328544(n) then
        print(n) ;
    end if;
end do: # R. J. Mathar, Oct 21 2019

A341028 a(n) is the smallest positive integer such that n+a(n) contains the string n-a(n) in reverse as a substring. If no such number exists then a(n) = -1.

Original entry on oeis.org

-1, -1, -1, -1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 10, 10, 10, 10, 10, 15, 15, 9, 15, 15, 20, 20, 20, 20, 20, 25, 25, 25, 9, 25, 29, 30, 30, 30, 30, 33, 34, 35, 35, 9, 37, 38, 39, 40, 40, 41, 42, 43, 44, 45, 9, 50, 50, 50, 50, 55, 41, 51, 52, 53, 54, 9, 60, 60, 60, 65, 50, 32, 52, 53, 54, 70, 9

Offset: 1

Scott R. Shannon, Feb 02 2021

Based on a search limit of 5*10^9 up to n = 300000 the values of n for which no a(n) is found are n = 1,2,3,4. This is likely the complete list of values for which a(n) = -1.
The longest run of consecutive terms with the same value in the first 300000 terms is the run of 5's at the beginning of the sequence, ten in all. This is likely the longest run for all numbers.
Numerous patterns exist in the values of a(n), e.g., when a(n) consists of all 9's and n is not a power of 10 then n is palindromic.

			a(5) = 5 as 5+5 = 10 which contains reverse(5-5) = reverse(0) = 0 as a substring.
a(6) = 5 as 6+5 = 11 which contains reverse(6-5) = reverse(1) = 1 as a substring.
a(15) = 10 as 15+10 = 25 which contains reverse(15-10) = reverse(5) = 5 as a substring.
a(22) = 9 as 22+9 = 31 which contains reverse(22-9) = reverse(13) = 31 as a substring.

Scott R. Shannon, Image of the terms for n = 5..300000. It is possible the frame-like pattern continues to larger and larger scales resulting in a fractal structure.

Cf. A341034 (forward), A341035 (forward and reverse), A339403, A339144, A328095, A333410, A332703.

A341034 a(n) is the smallest positive integer such that n+a(n) contains the string n-a(n) as a substring. If no such number exists then a(n) = -1.

Original entry on oeis.org

-1, -1, -1, -1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 10, 10, 10, 10, 10, 15, 15, 15, 15, 15, 20, 20, 20, 20, 20, 25, 25, 25, 25, 25, 29, 30, 30, 30, 30, 33, 34, 35, 35, 35, 37, 38, 39, 40, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48, 49, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50

Offset: 1

Scott R. Shannon, Feb 03 2021

Based on a search limit of 5*10^9 up to n = 200000 the values of n for which no a(n) is found are n = 1,2,3,4. This is likely the complete list of values for which no a(n) exists.

The sequence contains long runs of consecutive terms with the same value, resulting in the image for the values having a staircase-like pattern. In the first 200000 terms the longest run is 88890 terms, starting from a(61110), all of which have a(n) = 50000.

			a(5) = 5 as 5+5 = 10 which contains 5-5 = 0 as a substring.
a(6) = 5 as 6+5 = 11 which contains 6-5 = 1 as a substring.
a(15) = 10 as 15+10 = 25 which contains 15-10 = 5 as a substring.
a(35) = 29 as 35+29 = 64 which contains 35-29 = 6 as a substring.

Scott R. Shannon, Line graph of the terms for n=5..6111.

Cf. A341028 (reverse), A341035 (forward and reverse), A339403, A339144, A328095, A333410, A332703.

A329271 Numbers k such that k multiplied by the product of its divisors contains k as a substring.

Original entry on oeis.org

1, 5, 6, 10, 16, 24, 25, 30, 36, 40, 50, 51, 60, 70, 76, 90, 92, 100, 125, 176, 195, 240, 249, 250, 363, 375, 376, 430, 490, 500, 501, 510, 546, 556, 560, 568, 570, 600, 620, 624, 625, 648, 680, 730, 749, 750, 760, 810, 875, 909, 930, 972, 975, 976, 990, 999, 1000, 1001, 1010, 1636, 1680, 1930, 2354, 2400, 2490, 2500, 2510, 2512, 2943, 3000

Offset: 1

Scott R. Shannon, Nov 10 2019

Inspired by A328095. To avoid all primes being in the sequence the divisors of k includes k itself.

Contains 10^k, 5*10^k and 6*10^k for all k, 3*10^k, 4*10^k, 7*10^k and 9*10^k for all odd k. - Robert Israel, Nov 11 2019

			16 is in the sequence as the divisors of 16 are 1,2,4,8,16, and 16*(1*2*4*8*16) = 16*1024 = 16384, and '16384' contains '16' as a substring.
30 is in the sequence as the divisors of 30 are 1,2,3,5,6,10,15,30, and 30*(1*2*3*5*6*10*15*30) = 30*810000 = 24300000, and '24300000' contains '30' as a substring.

Scott R. Shannon, Table of n, a(n) for n = 1..1000. Note when searching for these numbers one needs to use arbitrary precision packages; the product for 24570000 has 1486 digits.

Cf. A007955, A328095, A326806, A328544.

The sequence of primes contained in their squares is A115738.

a:=[]; for k in [1..3000] do t:=IntegerToString(k*(&*Divisors(k))); s:=IntegerToString(k); if s in t then Append(~a,k); end if; end for; a; // Marius A. Burtea, Nov 10 2019

f[n_] := n^(1+DivisorSigma[0, n]/2); aQ[n_] := SequenceCount[IntegerDigits[f[n]], IntegerDigits[n]] > 0; Select[Range[3000], aQ] (* Amiram Eldar, Nov 10 2019 *)

cp's OEIS Frontend

A328561 Numbers in A328095 whose product of digits is not a power of 10.

Views

Author

Keywords

Comments

Crossrefs

Extensions

A339144 a(n) is the smallest positive integer such that n*a(n) contains n+a(n) as a substring. If no such number exists then a(n) = -1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A328560 Numbers whose product of digits is a power of 10.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

A326806 Numbers k such that k multiplied by the sum of all its digits contains k as a substring.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Python

A339403 a(n) is the smallest positive integer such that n*a(n) contains the string n+a(n) in reverse as a substring. If no such number exists then a(n) = -1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A341035 a(n) is the smallest positive integer such that n+a(n) contains the string n-a(n), in both forward and reverse directions, as a substring. If no such number exists then a(n) = -1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A328544 Numbers k such that k multiplied by any of its digits contains k as a substring.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

A341028 a(n) is the smallest positive integer such that n+a(n) contains the string n-a(n) in reverse as a substring. If no such number exists then a(n) = -1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A341034 a(n) is the smallest positive integer such that n+a(n) contains the string n-a(n) as a substring. If no such number exists then a(n) = -1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A329271 Numbers k such that k multiplied by the product of its divisors contains k as a substring.