A125746 -id:A125746 - cp's OEIS frontend

A300826 a(n) = n/A125746(n), where A125746(n) gives the smallest divisor d of n such that the sum which includes d and all smaller divisors is >= n.

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1

Offset: 1

Antti Karttunen, Mar 21 2018

nonn

Records occur at 1, 6, 24, 120, 240, 504, 1260, 2520, 5040, 15120, 50400, 55440, ... and they are 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, ...

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A125746.

Cf. A005100 (positions of ones), A023196 (positions of terms > 1).

A300826(n) = { my(k=0,s=0); fordiv(n,d, k++; s += d; if(s>=n,return(n/d))); };

a(n) = n/A125746(n).

A117553 When adding some positive divisors of n in order from lowest divisor to highest divisor, a(n) is lowest sum achievable which is >= n.

Original entry on oeis.org

1, 3, 4, 7, 6, 6, 8, 15, 13, 18, 12, 16, 14, 24, 24, 31, 18, 21, 20, 22, 32, 36, 24, 24, 31, 42, 40, 28, 30, 42, 32, 63, 48, 54, 48, 37, 38, 60, 56, 50, 42, 54, 44, 84, 78, 72, 48, 52, 57, 93, 72, 98, 54, 66, 72, 64, 80, 90, 60, 78, 62, 96, 104, 127, 84, 78, 68, 126, 96, 74, 72

Offset: 1

Leroy Quet, Mar 28 2006

nonn

Often, but not always, a(n)=n+A054024(n). The exceptions to this rule are at n=24, 36, 48, 60, 72, 84,90, 96, 108, ... - R. J. Mathar, Mar 14 2007

			12's divisors are 1,2,3,4,6 and 12. Adding the divisors in order we have:
1 = 1, 1+2 = 3, 1+2+3 = 6, 1+2+3+4 = 10, 1+2+3+4+6 = 16 and 1+2+3+4+6+12 = 28.
Of these sums, 1+2+3+4+6 = 16 is the lowest which is >= 12. So a(12) = 16.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A117552, A125746, A125747.

A117553 := proc(n) local divs,a,i ; divs := numtheory[divisors](n) ; a := op(1,divs) ; i := 1 ; while a < n do i := i+1 ; a := a+op(i,divs) ; od ; RETURN(a) ; end: for n from 1 to 80 do printf("%d, ",A117553(n)) ; od ; # R. J. Mathar, Mar 14 2007

Table[Select[Accumulate[Divisors[n]],#>=n&,1],{n,80}]//Flatten (* Harvey P. Dale, Apr 05 2017 *)

More terms from R. J. Mathar, Mar 14 2007

A125747 a(n) is the smallest positive integer such that (Sum_{t(k)|n, 1 <= k <= a(n)} t(k)) >= n, where t(k) is the k-th positive divisor of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 4, 2, 5, 2, 4, 4, 5, 2, 5, 2, 5, 4, 4, 2, 6, 3, 4, 4, 5, 2, 7, 2, 6, 4, 4, 4, 7, 2, 4, 4, 7, 2, 7, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 7, 4, 7, 4, 4, 2, 10, 2, 4, 6, 7, 4, 7, 2, 6, 4, 7, 2, 10, 2, 4, 6, 6, 4, 7, 2, 9, 5, 4, 2, 10, 4, 4, 4, 7, 2, 10, 4, 6, 4, 4, 4, 10, 2, 6, 6, 8, 2, 7, 2

Offset: 1

Leroy Quet, Dec 05 2006

nonn

a(n) = the least number of divisors of n, taken in increasing order as 1, A020639(n), A292269(n), etc. needed so that their sum is >= n. - Antti Karttunen, Mar 21 2018

			The divisors of 12 are 1,2,3,4,6,12. 1+2+3+4 = 10, which is smaller than 12; but 1+2+3+4+6 = 16, which is >= 12. 6 is the 5th divisor of 12, so a(12) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A020639, A292269, A125746, A117553.

f[n_] := Block[{k = 1, d = Divisors[n]},While[Sum[d[[i]], {i, k}] < n, k++ ];k];Table[f[n], {n, 105}] (* Ray Chandler, Dec 06 2006 *)

A125747(n) = { my(k=0,s=0); fordiv(n,d, k++; s += d; if(s>=n,return(k))); }; \\ Antti Karttunen, Mar 21 2018

Extended by Ray Chandler, Dec 06 2006

A302110 Let d be the list of A000005(n) = tau(n) divisors of n. Then a(n) is the largest k such that Sum_{i=1..#d-k} d_i > n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1

Offset: 1

David A. Corneth, Apr 01 2018

Records (0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, ...) occur at 1, 6, 24, 120, 240, 720, 1260, 2520, 5040, 15120, 27720, 55440, ... - Antti Karttunen, Apr 02 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000005, A023196, A125746, A125747, A300826.

A125747(n) = { my(k=0,s=0); fordiv(n,d, k++; s += d; if(s>=n,return(k))); };
A302110(n) = (numdiv(n) - A125747(n)); \\ Antti Karttunen, Apr 02 2018

a(n) = A000005(n) - A125747(n).

a(n) > 0 if and only if n is in A023196.

cp's OEIS Frontend

A300826 a(n) = n/A125746(n), where A125746(n) gives the smallest divisor d of n such that the sum which includes d and all smaller divisors is >= n.

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A117553 When adding some positive divisors of n in order from lowest divisor to highest divisor, a(n) is lowest sum achievable which is >= n.

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A125747 a(n) is the smallest positive integer such that (Sum_{t(k)|n, 1 <= k <= a(n)} t(k)) >= n, where t(k) is the k-th positive divisor of n.

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A302110 Let d be the list of A000005(n) = tau(n) divisors of n. Then a(n) is the largest k such that Sum_{i=1..#d-k} d_i > n.

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