Graham Holmes - cp's OEIS frontend

User: Graham Holmes

Graham Holmes has authored 3 sequences.

A373641 The number of positive n-digit integers whose digit product is n.

1, 2, 3, 10, 5, 36, 7, 120, 45, 90, 0, 924, 0, 182, 210, 3860, 0, 3060, 0, 3800, 420, 0, 0, 61824, 300, 0, 3627, 10584, 0, 25230, 0, 375968, 0, 0, 1190, 441000, 0, 0, 0, 426400, 0, 70602, 0, 0, 44550, 0, 0, 11936496, 1176, 58800, 0, 0, 0, 1491102, 0, 1638560

Offset: 1

Graham Holmes, Jun 12 2024

Trivially, for the four single-digit primes p, a(p)=p.

It's not possible by definition to have a digit product equal to a prime number greater than 10, so a(p)=0 for prime p > 10.

			a(4) = 10: 1114, 1122, 1141, 1212, 1221, 1411, 2112, 2121, 2211, 4111.

Graham Holmes, Table of n, a(n) for n = 1..10000

Cf. A002033, A002473, A068191, A074206, A163767.

b:= proc(n, t, i) option remember; `if`(n=1, 1/t!, `if`(i<2, 0,
      add(b(n/i^j, t-j, i-1)/j!, j=0..padic[ordp](n, i))))
    end:
a:= n-> n!*b(n$2, 9):
seq(a(n), n=1..56);  # Alois P. Heinz, Jun 12 2024

a(n) = 0 <=> n in { A068191 }.
a(n) > 0 <=> n in { A002473 }.
a(n) = A163767(n) for n <= 9.

A339700 a(n) is the n-th nonnegative number to light exactly n segments when displayed on a calculator.

Original entry on oeis.org

71, 77, 47, 61, 70, 52, 62, 99, 136, 190, 246, 263, 306, 589, 882, 1085, 1838, 2059, 2308, 2869, 5886, 8689, 10800, 18098, 20268, 20896, 28608, 58880, 86886, 106898, 180889, 200858, 208698, 283888, 588868, 868880, 1068889, 1808886, 2008086, 2086868, 2809888, 5888808, 8688868, 10688886, 18088880

Offset: 5

Graham Holmes, Dec 13 2020

a(n) is undefined for n<5, as there are no numbers with 1 segment, 1 with 2 segments, 1 with 3 segments, and 2 with 4 segments. If 0 is excluded as a valid input - so the series would refer to "the n-th positive number" - then a(6) would be 111 rather than 77.

			For n=7, 47 is the 7th positive number to light 7 segments, after 8, 12, 13, 15, 21, and 31.

Graham Holmes, Table of n, a(n) for n = 5..82
Index entries for sequences related to calculator display

Cf. A006942 (segments lit), A216261, A331529.

s=[6,2,5,5,4,5,6,3,7,6];p=[];a=[];for(i=2;i<=100;i++)p[i]=0;for(i=1;i<=1000000;i++){d=i;n=0;do{x=d%10;n+=s[x];d=(d-x)/10;}while(d>0)p[n]++;if(p[n]==n)a[n]=i;}for(c=2;c<=40;c++)document.write(c+": "+a[c]+"
");

A336733 Positive integers which can be written in two bases smaller than 10 as mutually-reversed strings of digit(s).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 11, 17, 22, 31, 51, 87, 91, 102, 121, 212, 220, 248, 2601, 5258, 7491, 8283, 9831, 10516, 13541, 15774, 16566, 71500, 644765, 731445, 811518, 3552340, 314767045, 1427310725, 1848187230, 1916060910, 47124212513, 455075911977

Offset: 1

Graham Holmes, Aug 02 2020

Base conversion yields a string of digits which by convention has any leading zeros suppressed. However, a conversion which yields a low zero (e.g., 96_10 = 240_6) will see that zero preserved when the string of digits is reversed (e.g., into "042"), so it can never match any base-converted strings before reversal. It's therefore not possible to have a solution involving a base which exhibits a low zero for any input x. A consequence of this is that any solution will require both base-converted strings to be of the same length; considering that any solution for a sufficiently-high x will involve only bases 8 and 9 (these having the slowest rate of change with respect to x), we can deduce that the upper limit for valid solutions occurs at the point beyond which length(x base 8) - length(x base 9) is permanently greater than unity, and this can be shown to occur at 8^18, or approximately 1.80*10^16.
40 terms are known up to 4.7*10^13.
It's worthy of note that 22 has two distinct nontrivial solutions as 22_10 = 211_3 = 112_4, and 22_10 = 42_5 = 24_9.
As 1 through 6 have one digit in at least two distinct bases each less than 10 they are trivially included in the sequence. - David A. Corneth, Aug 03 2020
No more terms beyond a(40). - Bert Dobbelaere, Sep 26 2020

			7 is a term since 7 = 21 (base 3) = 12 (base 5).
9 is a term since 9 = 21 (base 4) = 12 (base 7).
...
1916060910 is a term since it is = 65324151261 (base 7) = 16215142356 (base 8).

David A. Corneth, PARI program

Cf. A336768 (for bases >= 4).

n=[]; rev=[]; incl=[]; for (i=1; i<=1000; i++) { for (j=2; j<=9; j++) { n[j]=i.toString(j); rev[j]=n[j].split("").reverse().join(""); } for (j=2; j<=8; j++) for (k=j+1; k<=9; k++) if (n[j]==rev[k]) if (incl.indexOf(i)==-1) incl.push(i); } document.write(incl);

seqQ[n_] := Module[{dig = IntegerDigits[n, Range[2, 9]]}, dig = Select[dig, ! PalindromeQ[#] &]; n < 7 || Intersection[dig, Reverse /@ dig] != {}]; Select[Range[10^6], seqQ] (* Amiram Eldar, Aug 04 2020 *)

isok(m) = {for (b=2, 8, my(db = digits(m, b)); for(c=b+1, 9, my(dc = digits(m, c)); if (Vecrev(dc) == db, return (1));););} \\ Michel Marcus, Aug 03 2020

is(n) = {my(v = vecsort(vector(8, i, d = digits(n, i+1); if(d[1] < d[#d], Vecrev(d), d)))); for(i = 1, 7, if(v[i] == v[i+1], return(1))); 0} \\ David A. Corneth, Aug 03 2020

a(40) from David A. Corneth, Aug 07 2020

cp's OEIS Frontend

User: Graham Holmes

A373641 The number of positive n-digit integers whose digit product is n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A339700 a(n) is the n-th nonnegative number to light exactly n segments when displayed on a calculator.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

JavaScript

A336733 Positive integers which can be written in two bases smaller than 10 as mutually-reversed strings of digit(s).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

JavaScript

Mathematica

PARI

PARI

Extensions