A061486 -id:A061486 - cp's OEIS frontend

A137923 Zerofree numbers k such that A061486(k) is prime.

2, 3, 5, 7, 11, 12, 13, 15, 16, 18, 19, 21, 23, 25, 27, 29, 31, 32, 34, 35, 37, 43, 45, 51, 52, 53, 54, 56, 57, 58, 59, 61, 65, 72, 73, 75, 78, 79, 81, 85, 87, 89, 91, 92, 95, 97, 98, 212, 213, 216, 218, 219, 223, 225, 229, 232, 233, 235, 236, 239, 243, 245, 249, 255, 256, 269, 272, 273, 278, 283

Offset: 1

Ctibor O. Zizka, Apr 30 2008

			2-digit numbers are of the form X(1)X(2).
The equation is then X(1) + X(2) + X(1)*X(2) = p, where p is prime and both digits are nonzero. Power set is {();(1);(2);(1,2)}, so indices of digits in the equation are running through the power set.
Following numbers n are solutions of the equation:
11 because 1 + 1 + 1*1 = 3;
12 (and its reverse, 21) because 1 + 2 + 1*2 = 5;
13 (and its reverse, 31) because 1 + 3 + 1*3 = 7;
15 (and its reverse, 51) because 1 + 5 + 1*5 = 11;
16 (and its reverse, 61) because 1 + 6 + 1*6 = 13;
18 (and its reverse, 81) because 1 + 8 + 1*8 = 17;
19 (and its reverse, 91) because 1 + 9 + 1*9 = 19;
23 (and its reverse, 32) because 2 + 3 + 2*3 = 11;
25 (and its reverse, 52) because 2 + 5 + 2*5 = 17;
...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A052382, A061486.

filter := proc (n) local L; L := convert(n, base, 10); not has(L, 0) and isprime(add(add(convert(L[i .. j], `*`), i = 1 .. j), j = 1 .. nops(L))) end proc:
select(filter, [$1..1000]); # Robert Israel, Feb 11 2018

Members of the sequence are numbers n = X(1)...X(r) for which digits the following equation holds: (X(1) + ... + X(r)) + (X(1)*X(2) + ... + X(r-1)*X(r)) + ... + (X(1)*...*X(r)) = p, where p is a prime number, X(i) is the i-th digit of n, and every digit is nonzero.

More terms from Robert Israel, Feb 11 2018

A264668 a(n) = A264600(n) - A061486(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 0, 7, 14, 21, 28, 35, 42, 49, 56, 63

Offset: 0

Alois P. Heinz, Nov 20 2015

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A061486, A264600.

with(combinat):
f:= n-> (l-> add(mul(l[i], i=w), w=choose(
         nops(l)))-1)(convert(n, base, 10)):
g:= proc(n) option remember; `if`(n=0, 0,
      irem(n, 10, 'r')*(r+1)+g(r))
    end:
a:= n-> g(n)-f(n):
seq(a(n), n=0..150);

A343132 a(n) is the quotient obtained when integer A343131(n) = k is divided by A061486(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 10, 1, 10, 1, 10, 3, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 100, 6, 5, 24, 11, 10, 1, 100, 9, 8, 42, 13, 13, 10, 1, 100, 43, 16, 22, 10, 1, 100, 30, 9, 2, 10, 1, 100, 4, 3, 10, 1, 100, 31, 6, 5, 10, 1, 100, 15, 10, 1, 100, 13, 11, 10, 1, 100, 10, 1, 1000

Offset: 1

Bernard Schott, Apr 07 2021

The first 9 terms corresponding to the 1-digit numbers k = u are the quotients u/u = 1.
The next 19 terms from a(10) = 10 to a(28) = 1 corresponding to 2-digit numbers k = du are the quotients du/(d+u + d*u).
The next 50 terms from a(29) = 100 to a(78) = 1 corresponding to 3-digit numbers k = hdu (in A328864) are the quotients hdu/f_3(h,d,u) where f_3(h,d,u) = (h+d+u) + (h*d+d*u+u*h) + (h*d*u).
The next 87 terms, from a(79) = 1000 to a(165) = 1, corresponding to 4-digit numbers k = thdu are the quotients thdu/f_4(t,h,d,u) where f_4(t,h,d,u) = (t+h+d+u) + (t*h+t*d+t*u+h*d+h*u+d*u) + (t*h*d+t*h*u+t*d*u+h*d*u) + (t*h*d*u).
When A343131(n) = z*10^q = A037124(r) is a number that contains only one nonzero digit z, then A061486(A037124(r)) = this nonzero digit z and a(n) = 10^q.

			For A343131(7) = 7, A061486(7) = 7 and a(7) = 7/7 = 1.
For A343131(17) = 42, A061486(42) = 4+2 + 4*2 = 14 and a(17) = 42/14 = 3.
For A343131(58) = 573, A061486(573) = 5+7+3 + 5*7+7*3+3*5 + 5*7*3 = 191 and a(58) = 573/191 = 3.

Eric Weisstein's World of Mathematics, Symmetric polynomial.
Wikipedia, Elementary symmetric polynomial.

Cf. A005349, A007602, A038366, A061486, A328864, A343131.

sympol(X, n) = my(s=0); forvec(i=vector(n, j, [1, #X]), s+=prod(k=1, n, X[i[k]]), 2); s ;
f(n) = my(d=digits(n)); sum(k=1, #d, sympol(d, k));
lista(nn) = {for (n=1, nn, my(q = n/f(n)); if (denominator(q) == 1, print1(q, ", ")););} \\ Michel Marcus, Apr 08 2021

a(n) = A343131(n)/A061486(A343131(n)).

A051885 Smallest number whose sum of digits is n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 29, 39, 49, 59, 69, 79, 89, 99, 199, 299, 399, 499, 599, 699, 799, 899, 999, 1999, 2999, 3999, 4999, 5999, 6999, 7999, 8999, 9999, 19999, 29999, 39999, 49999, 59999, 69999, 79999, 89999, 99999, 199999, 299999, 399999, 499999

Offset: 0

Felice Russo, Dec 15 1999

This is also the list of lunar triangular numbers: A087052 with duplicates removed. - N. J. A. Sloane, Jan 25 2011
Numbers n such that A061486(n) = n. - Amarnath Murthy, May 06 2001
The product of digits incremented by 1 is the same as the number incremented by 1. If a(n) = abcd...(a,b,c,d, etc. are digits of a(n)) {a(n) + 1} = (a+1)*(b+1)(c+1)*(d+1)*..., e.g., 299 + 1 = (2+1)*(9+1)*(9+1) = 300. - Amarnath Murthy, Jul 29 2003
A138471(a(n)) = 0. - Reinhard Zumkeller, Mar 19 2008
a(n+1) = A108971(A179988(n)). - Reinhard Zumkeller, Aug 09 2010, Jul 10 2011
Positions of records in A003132: A080151(n) = A003132(a(n)). - Reinhard Zumkeller, Jul 10 2011
a(n) = A242614(n,1). - Reinhard Zumkeller, Jul 16 2014
A254524(a(n)) = 1. - Reinhard Zumkeller, Oct 09 2015
The slowest strictly increasing sequence of nonnegative integers such that, for any two terms, calculating the difference of their decimal representations requires no borrowing. - Rick L. Shepherd, Aug 11 2017

Iain Fox, Table of n, a(n) for n = 0..9000 (first 101 terms from Reinhard Zumkeller)
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
A. Murthy, Exploring some new ideas on Smarandache type sets, functions and sequences, Smarandache Notions Journal Vol. 11 N. 1-2-3 Spring 2000.
Index entries for sequences related to dismal (or lunar) arithmetic
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,10,-10).

Cf. A061104, A061105, A061486, A007953, A067043, A087052.
Numbers of form i*b^j-1 (i=1..b-1, j >= 0) for bases b = 2 through 9: A000225, A062318, A180516, A181287, A181288, A181303, A165804, A140576. - N. J. A. Sloane, Jan 25 2011
Cf. A002283.
Cf. A254524.

a051885 n = (m + 1) * 10^n' - 1 where (n',m) = divMod n 9
-- Reinhard Zumkeller, Jul 10 2011

Magma
```
[i*10^j-1: i in [1..9], j in [0..5]];
```

b:=10; t1:=[]; for j from 0 to 15 do for i from 1 to b-1 do t1:=[op(t1), i*b^j-1]; od: od: t1; # N. J. A. Sloane, Jan 25 2011

a[n_] := (Mod[n, 9] + 1)*10^Floor[n/9] - 1; Table[a[n], {n, 0, 49}](* Jean-François Alcover, Dec 01 2011, after Henry Bottomley *)

A051885(n) = (n%9+1)*10^(n\9)-1  \\ M. F. Hasler, Jun 17 2012

first(n) = Vec(x*(x^2 + x + 1)*(x^6 + x^3 + 1)/((x - 1)*(10*x^9 - 1)) + O(x^n), -n) \\ Iain Fox, Dec 30 2017

def A051885(n): return ((n % 9)+1)*10**(n//9)-1 # Chai Wah Wu, Apr 04 2021

These are the numbers i*10^j-1 (i=1..9, j >= 0). - N. J. A. Sloane, Jan 25 2011
a(n) = ((n mod 9) + 1)*10^floor(n/9) - 1 = a(n-1) + 10^floor((n-1)/9). - Henry Bottomley, Apr 24 2001
a(n) = A037124(n+1) - 1. - Reinhard Zumkeller, Jan 03 2008, Jul 10 2011
G.f.: x*(x^2+x+1)*(x^6+x^3+1) / ((x-1)*(10*x^9-1)). - Colin Barker, Feb 01 2013

More terms from James Sellers, Dec 16 1999

Offset fixed by Reinhard Zumkeller, Jul 10 2011

A264600 Let S_n denote the list of decimal numbers 0 to n, written backwards (allowing leading zeros) and arranged in lexicographic order; a(n) = position where backwards-n appears, starting indexing at 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 4, 9, 14, 19, 24, 29, 34, 39, 44, 49, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 6, 13, 20, 27, 34, 41, 48, 55, 62, 69, 7, 15, 23, 31, 39, 47, 55, 63, 71, 79, 8, 17, 26, 35, 44, 53, 62, 71, 80, 89, 9, 19, 29, 39, 49, 59, 69, 79, 89, 99, 1, 12

Offset: 0

N. J. A. Sloane, Nov 20 2015

			S_0 = [0], so a(0)=0,
...
S_9 = [0,1,2,3,4,5,6,7,8,9], so a(9) = 9,
S_10 = [0,01,1,2,3,4,5,6,7,8,9], so a(10) = 1,
S_11 = [0,01,1,11,2,3,4,5,6,7,8,9], so a(11) = 3,
...
S_20 = [0,01,02,1,11,2,21,3,31,4,41,5,51,6,61,7,71,8,81,9,91], so a(20) = 2, and so on

Alois P. Heinz, Table of n, a(n) for n = 0..20000 (first 10000 terms from Chai Wah Wu)

Decimal analog of A264596.

Has same beginning as A061486 but is ultimately different: see A264668.

a:= proc(n) option remember; `if`(n=0, 0,
      irem(n, 10, 'r')*(r+1)+a(r))
    end:
seq(a(n), n=0..101);  # Alois P. Heinz, Nov 20 2015

A264600[0]=0;A264600[n_]:=A264600[n]=Mod[n,10](Floor[n/10]+1)+A264600[Floor[n/10]];Array[A264600,100,0] (* Paolo Xausa, Nov 04 2023, after Alois P. Heinz *)

def A264600(n):
    return sorted(str(i)[::-1] for i in range(n+1)).index(str(n)[::-1]) # Chai Wah Wu, Nov 20 2015

a(0) = 0, a(10n+m) = a(n) + m*(n+1) for m in {0,...,9}. - Alois P. Heinz, Nov 20 2015

More terms from Alois P. Heinz, Nov 20 2015

A343131 For m >= 1, the m-digit number k = d_m||...||d_2||d_1 is a term if it is divisible by f_m(k) that is the sum of the m elementary symmetric polynomials in m variables e_i(k): f_m(k) = Sum_{i=1..m} e_i(d_1, ..., d_m).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 19, 20, 29, 30, 39, 40, 42, 49, 50, 59, 60, 69, 70, 79, 80, 89, 90, 99, 100, 114, 115, 120, 121, 190, 199, 200, 207, 208, 210, 221, 260, 290, 299, 300, 301, 304, 330, 390, 399, 400, 420, 441, 448, 490, 499, 500, 572, 573, 590, 599, 600, 620

Offset: 1

Bernard Schott, Apr 06 2021

Equivalently, integers k that are divisible by A061486(k); the quotients obtained when these terms k are divided by A061486(k) are in A343132.
The idea of this sequence comes from A328864 (1st problem of the 30th British Mathematical Olympiad in 1994) and also from the elementary symmetric polynomials in m variables (see links) that allow the generalization of this Olympiad problem to m-digit numbers.
-> When k = u has only one digit, then f_1(k) = e_1(u) = u and the numbers u from 1 to 9 satisfy u/e_1(u) = 1, so those are the first nine terms of this sequence.
-> When k = du has two digits, the numbers that are divisible by f_2(k) = (e_1(d,u) + e_2(d,u)) = (d+u) + (d*u) are the first 19 terms of A038366, from a(10) = 10 to a(28) = 99.
-> when k = hdu is a 3-digit number, then e_1(h,d,u) = h+d+u, e_2(h,d,u) = h*d+d*u+u*h and e_3(h,d,u) = h*d*u so f_3(k) = (h+d+u) + (hd+du+uh) + (hdu). This subsequence with 3-digit numbers A328864 has 50 terms that are here from a(29) = 100 to a(78) = 999.
-> When k = thdu is a 4-digit number, then e_1(t,h,d,u) = e_1(k) = t+h+d+u, e_2(t,h,d,u)= t*h+t*d+t*u+h*d+h*u+d*u, e_3(t,h,d,u) = t*h*d+t*h*u+t*d*u+h*d*u and e_4(t,h,d,u) = t*h*d*u with f_4(k) = e_1(t,h,d,u) + e_2(t,h,d,u) + e_3(t,h,d,u) + e_4(t,h,d,u). Numbers k such that k/f(k) is an integer form another subsequence with 87 terms. This subsequence with 4-digit numbers goes from a(79) = 1000 to a(165) = 9999.

			For k = 7, f_1(7) = 7, and 7/7 = 1, hence 7 is a term.
For k = 42, f_2(4,2) = 4+2 + 4*2 = 14 and 42/14 = 3, hence 42 is a term.
For k = 572, f_3(5,7,2) = 5+7+2 + 5*7+7*2+2*5 + 5*7*2 = 14 + 59 + 70 = 143 and 572/143 = 4, hence 572 is a term.
For k = 3225, f_4(3,2,2,5) = 3+2+2+5 + 3*2+3*2+3*5+2*2+2*5+2*5 + 3*2*2+3*2*5+3*2*5+2*2*5 + 3*2*2*5 = 215 and 3225/215 = 15, hence 3225 is a term.

Eric Weisstein's World of Mathematics, Symmetric polynomial.
Wikipedia, Elementary symmetric polynomial.

Cf. A005349, A007602, A038366, A061486, A328864, A343132.

sympol(X, n)=my(s=0); forvec(i=vector(n, j, [1, #X]), s+=prod(k=1, n, X[i[k]]), 2); s ;
f(n) = my(d=digits(n)); sum(k=1, #d, sympol(d, k)); \\ A061486
isok(m) = (m % f(m)) == 0; \\ Michel Marcus, Apr 06 2021

cp's OEIS Frontend

A137923 Zerofree numbers k such that A061486(k) is prime.

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A264668 a(n) = A264600(n) - A061486(n).

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A343132 a(n) is the quotient obtained when integer A343131(n) = k is divided by A061486(k).

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A051885 Smallest number whose sum of digits is n.

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A264600 Let S_n denote the list of decimal numbers 0 to n, written backwards (allowing leading zeros) and arranged in lexicographic order; a(n) = position where backwards-n appears, starting indexing at 0.

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A343131 For m >= 1, the m-digit number k = d_m||...||d_2||d_1 is a term if it is divisible by f_m(k) that is the sum of the m elementary symmetric polynomials in m variables e_i(k): f_m(k) = Sum_{i=1..m} e_i(d_1, ..., d_m).

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