A241754 -id:A241754 - cp's OEIS frontend

A245438 a(n) = the number of ways in which n is equal to the sum of digits > 0 taken from numbers <= n.

1, 1, 2, 2, 3, 4, 5, 6, 8, 17, 53, 158, 450, 1224, 3195, 8036, 19585, 46549, 108541, 219677, 664149, 1891075, 5091680, 13004347, 31632641, 73745789, 166055768, 364027232, 782374631, 1462836178, 4198493416, 11171538552, 27755958012

Offset: 1

Anthony Sand, Jul 22 2014

Let the range (1,n) in base 10 be represented in the form (1.A,n.A) = (1.A, 2.A, 3.A...n.A), where digit A = 10 in bases >= 11. Let samplesum(d(i),i=1..n) sum single digits d(i) sampled from each member of (1.A,n.A). The list above is the number of ways in which n = samplesum(d(i),i=1..n) when 0 < d(i) < A, for all i. Because d(i) is not permitted to equal 0, sums like these are not counted separately:
= 1 + 2 + 3 + 4.
= 1 + 2 + 3 + 4 + 0 (of 10).
= 1 + 2 + 3 + 4 + 0 (of 10) + 1 (of 11).
= 1 + 2 + 3 + 4 + 1 (of 11).
= 1 + 2 + 3 + 6.
= 1 + 2 + 3 + 6 + 0 (of 10).
However, these two sums are counted separately:
= 1 + 2 + 3 + 4 + 1 (first digit of 11).
= 1 + 2 + 3 + 4 + 1 (second digit of 11).

			1 = 1 (sum=1).
2 = 2 (s=1).
3 = 1 + 2; 3 (s=2).
4 = 1 + 3; 4 (s=2).
5 = 2 + 3; 1 + 4; 5 (s=3).
6 = 1 + 2 + 3; 2 + 4; 1 + 5; 6 (s=4).
7 = 1 + 2 + 4; 3 + 4; 2 + 5; 1 + 6; 7 (s=5).
8 = 1 + 3 + 4; 1 + 2 + 5; 3 + 5; 2 + 6; 1 + 7; 8 (s=6).
9 = 2 + 3 + 4; 1 + 3 + 5; 4 + 5; 1 + 2 + 6; 3 + 6; 2 + 7; 1 + 8; 9 (s=8).
10 = 1 + 2 + 3 + 4; 2 + 3 + 5; 1 + 4 + 5; 1 + 3 + 6; 4 + 6; 1 + 2 + 7; 3 + 7; 2 + 8; 1 + 9; 2 + 3 + 4 + 1 (of 10); 1 + 3 + 5 + 1 (of 10); 4 + 5 + 1 (of 10); 1 + 2 + 6 + 1 (of 10); 3 + 6 + 1 (of 10); 2 + 7 + 1 (of 10); 1 + 8 + 1 (of 10); 9 + 1 (of 10) (s=17).
11 = 3 + 4 + 5 + 1 (of 10).
12 = 1 + 2 + 5 + 1 (of 10) + 1 (of 11) + 2 (of 12).
13 = 1 + 2 + 6 + 1 (of 11) + 2 (of 12).
14 = 3 + 4 + 1 (of 10) + 1 (of 11) + 2 (of 12) + 3 (of 13).
15 = 3 + 5 + 1 (of 10) + 2 (of 12) + 3 (of 13) + 1 (of 14).

Cf. A241754, A245439.

/* To include 0 in sums, change "dn[i]>0" to "dn[i]>=0" */
{ nmx=20; b=10; d = vector(nmx+1); s = vector(nmx+1); for(n=1,nmx+1, dn=digits(n,b); nn=0; for(i=1,#dn,if(dn[i]>0,nn=nn*b+dn[i])); d[n]=nn; ); for(n=1,nmx, si=1; c=0; until(si>n, nn=0; for(i=1,si,if(s[i]>0,nn+=(d[i]\b^(s[i]-1))%b);if(nn>n,i=si)); if(nn==n,c++); incs(); ); s[si]=0; print1(c,", ")); break; }
{incs() = s[1]++; i=1; while(d[i]\b^(s[i]-1)==0, s[i]=0; i++; s[i]++; ); if(i>si,si=i); } \\ Anthony Sand, Aug 15 2014
A245438(n) = my(X = 'x + O('x^(n+1))); polcoef( prod(i=1,n, 1 + vecsum(apply(t->(t>0)*X^t,digits(i))) ), n); \\ Max Alekseyev, Sep 04 2023

More terms from Max Alekseyev, Sep 04 2023

A245439 The number of ways in which n is equal to the sum of digits taken from the numbers <= n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 8, 26, 81, 243, 693, 1887, 4932, 12418, 30288, 72026, 167989, 541500, 1635975, 4662579, 12580587, 32228307, 78662108, 183988734, 415466897, 912816164, 1965020012, 6121555788, 17573354640, 46896718806

Offset: 1

Anthony Sand, Jul 22 2014

Let the range (1,n) in base 10 be represented in the form (1.A,n.A) = (1.A, 2.A, 3.A...n.A), where digit A = 10 in bases >= 11. Let samplesum(d(i),i=1..n) sum solitary digits d(i) sampled from each member of (1.A,n.A) (i.e., only one digit at a time is sampled from a particular number). The list above is the number of ways in which n = samplesum(d(i),i=1..n) when 0 <= d(i) < A for all i. Because d(i) is permitted to equal 0, sums like these are counted separately:
= 1 + 2 + 3 + 4
= 1 + 2 + 3 + 4 + 0 (of 10)
= 1 + 2 + 3 + 4 + 1 (of 11)
= 1 + 2 + 3 + 4 + 0 (of 10) + 1 (of 11)
= 1 + 2 + 3 + 6
= 1 + 2 + 3 + 6 + 0 (of 10)
These two sums are also counted separately:
= 1 + 2 + 3 + 4 + 1 (first digit of 11)
= 1 + 2 + 3 + 4 + 1 (second digit of 11)
However, this sum is excluded:
= 2 + 3 + 4 + 1 (first digit of 11) + 1 (second digit of 11)

			1 = 1 (sum=1)
2 = 2 (s=1)
3 = 1 + 2; 3 (s=2)
4 = 1 + 3; 4 (s=2)
5 = 2 + 3; 1 + 4; 5 (s=3)
6 = 1 + 2 + 3; 2 + 4; 1 + 5; 6 (s=4)
7 = 1 + 2 + 4; 3 + 4; 2 + 5; 1 + 6; 7 (s=5)
8 = 1 + 3 + 4; 1 + 2 + 5; 3 + 5; 2 + 6; 1 + 7; 8 (s=6)
9 = 2 + 3 + 4; 1 + 3 + 5; 4 + 5; 1 + 2 + 6; 3 + 6; 2 + 7; 1 + 8; 9 (s=8)
10 = 1 + 2 + 3 + 4; 2 + 3 + 5; 1 + 4 + 5; 1 + 3 + 6; 4 + 6; 1 + 2 + 7; 3 + 7; 2 + 8; 1 + 9; 1 + 2 + 3 + 4 + 0 (of 10); 2 + 3 + 5 + 0 (of 10); 1 + 4 + 5 + 0 (of 10); 1 + 3 + 6 + 0 (of 10); 4 + 6 + 0 (of 10); 1 + 2 + 7 + 0 (of 10); 3 + 7 + 0 (of 10); 2 + 8 + 0 (of 10); 1 + 9 + 0 (of 10); 2 + 3 + 4 + 1 (of 10); 1 + 3 + 5 + 1 (of 10); 4 + 5 + 1 (of 10); 1 + 2 + 6 + 1 (of 10); 3 + 6 + 1 (of 10); 2 + 7 + 1 (of 10); 1 + 8 + 1 (of 10); 9 + 1 (of 10) (s=26)

Cf. A000009, A000041, A245438, A241754.

/* To exclude 0 from sums, change "dn[i]>=0" to "dn[i]>0" */
{ nmx=20; b=10; d = vector(nmx+1); s = vector(nmx+1); for(n=1,nmx+1, dn=digits(n,b); nn=0; for(i=1,#dn,if(dn[i]>=0,nn=nn*b+dn[i])); d[n]=nn; ); for(n=1,nmx, si=1; c=0; until(si>n, nn=0; for(i=1,si,if(s[i]>0,nn+=(d[i]\b^(s[i]-1))%b);if(nn>n,i=si)); if(nn==n,c++); incs();
); s[si]=0; print1(c,", ")); break; }
{incs() = s[1]++; i=1; while(d[i]\b^(s[i]-1)==0, s[i]=0; i++; s[i]++; ); if(i>si,si=i); } \\ Anthony Sand, Aug 19 2014
A245439(n) = my(X = 'x + O('x^(n+1))); polcoef( prod(i=1,n, 1 + vecsum( apply(t->X^t, digits(i)) ) ), n); \\ Max Alekseyev, Sep 04 2023

More terms from Max Alekseyev, Sep 04 2023

A319274 Osiris or Digit re-assembly numbers: numbers that are equal to the sum of permutations of subsamples of their own digits.

Original entry on oeis.org

132, 264, 396, 8991, 10545, 35964, 255530, 1559844, 9299907, 47755078, 89599104, 167264994, 283797162, 473995260, 3929996070, 6379993620, 10009998999, 11111111110, 22222222220, 33333333330, 44444444440, 55555555550, 66666666660, 77777777770, 88888888880, 99999999990

Offset: 1

Pieter Post, Sep 16 2018

This sequence differs from A241754 because this sequence uses permutations only once.
Permutations are of the same length k, leading zeros are allowed.
The k's in the sequence are: 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 7, 10, 10, 10, 10, 10, 10, 10, 10, 10, 6, 7, 7, 8, 7, 9, 9.

			10545 = 014 + 015 + 041 + 045 + 051 + 054 + 055 + 104 + 105 + 140 + 145 + 150 + 154 + 155 + 401 + 405 + 410 + 415 + 450 + 451 + 455 + 501 + 504 + 505 + 510 + 514 + 515 + 540 + 541 + 545 + 550 + 551 + 554.

Giovanni Resta, Table of n, a(n) for n = 1..33 (terms < 10^16)

Cf. A047726, A179239, A241754, A241899.

import itertools
def getData(a, b):
    dig = (itertools.permutations(str(a), b))
    for d in dig:
        yield d
for w in range(2, 6):
    kk=int(w*'1')
    for i in range (kk, 10**(w+3), kk):
        m=[]
        get = getData(i, w)
        while True:
            try:
                n = next(get)
                ee=int("".join((n)))
                if ee not in m:
                    m.append(ee)
            except StopIteration:
                if sum (m)==i and len(m)>1:
                    m.sort()
                    print (sum(m), len(m), m, i)
                break

a(12)-a(26) from Giovanni Resta, Sep 16 2018

cp's OEIS Frontend

A245438 a(n) = the number of ways in which n is equal to the sum of digits > 0 taken from numbers <= n.

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A245439 The number of ways in which n is equal to the sum of digits taken from the numbers <= n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

A319274 Osiris or Digit re-assembly numbers: numbers that are equal to the sum of permutations of subsamples of their own digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Extensions