A319712 -id:A319712 - cp's OEIS frontend

A333622 Numbers k such that k is divisible by the sum of digits of all the divisors of k in factorial base (A319712).

1, 2, 3, 4, 14, 22, 24, 27, 33, 36, 52, 72, 91, 92, 100, 135, 150, 187, 221, 231, 310, 323, 448, 481, 493, 494, 589, 663, 708, 754, 816, 884, 893, 897, 946, 1080, 1155, 1159, 1178, 1200, 1357, 1462, 1475, 1518, 1530, 1536, 1550, 1702, 1710, 1836, 1972, 1978, 2231

Offset: 1

Amiram Eldar, Mar 29 2020

			14 is a term since its divisors are {1, 2, 7, 14}, their representations in factorial base (A007623) are {1, 10, 101, 210}, and their sum of sums of digits is 1 + (1 + 0) + (1 + 0 + 1) + (2 + 1 + 0) = 7 which is a divisor of 14.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007623, A034968, A093705, A319712, A333617, A333619, A333620, A333623.

fctDigSum[n_] := Module[{s=0, i=2, k=n}, While[k > 0, k = Floor[n/i!]; s = s + (i-1)*k; i++]; n-s]; fctDivDigDum[n_] := DivisorSum[n, fctDigSum[#] &]; Select[Range[10^3], Divisible[#, fctDivDigDum[#]] &] (* after Jean-François Alcover at A034968 *)

A319715 Sum of A276150(d) over divisors d of n, where A276150 gives the sum of digits in primorial base.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 3, 6, 6, 8, 5, 9, 4, 7, 10, 10, 6, 11, 5, 14, 10, 11, 7, 15, 9, 10, 12, 15, 8, 16, 3, 12, 10, 10, 10, 17, 4, 9, 10, 20, 6, 18, 5, 17, 18, 13, 7, 23, 8, 18, 14, 18, 8, 22, 14, 23, 14, 16, 9, 26, 4, 7, 17, 16, 12, 20, 5, 16, 14, 22, 7, 27, 6, 10, 21, 17, 14, 22, 7, 30, 19, 14, 9, 34, 16, 13, 18, 27, 10, 30, 10, 19, 10

Offset: 1

Antti Karttunen, Oct 02 2018

Inverse Möbius transform of A276150.

Antti Karttunen, Table of n, a(n) for n = 1..30030
Index entries for sequences related to primorial base.

Cf. A276150, A319712, A319713.

d[n_] := Module[{k = n, p = 2, s = 0, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, s += r; p = NextPrime[p]]; s]; a[n_] := DivisorSum[n, d[#] &]; Array[a, 100] (* Amiram Eldar, Mar 05 2024 *)

A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A319715(n) = sumdiv(n,d,A276150(d));

a(n) = Sum_{d|n} A276150(d).

a(n) = A319713(n) + A276150(n).

A319711 Sum of A034968(d) over proper divisors d of n, where A034968 gives the sum of digits in factorial base.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 4, 3, 5, 1, 7, 1, 4, 6, 6, 1, 8, 1, 10, 5, 6, 1, 11, 4, 5, 6, 9, 1, 15, 1, 10, 7, 7, 6, 15, 1, 6, 6, 16, 1, 15, 1, 13, 13, 8, 1, 16, 3, 10, 8, 9, 1, 14, 8, 14, 7, 6, 1, 25, 1, 5, 13, 13, 7, 18, 1, 13, 9, 18, 1, 21, 1, 6, 12, 12, 7, 15, 1, 25, 9, 8, 1, 26, 9, 7, 7, 20, 1, 29, 6, 16, 6, 9, 8, 21, 1, 10, 14, 19, 1, 18, 1, 15, 22

Offset: 1

Antti Karttunen, Oct 02 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to factorial base representation.

Cf. A034968, A319712, A319713.

d[n_] := Module[{k = n, m = 2, s = 0, r}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, s += r; m++]; s]; a[n_] := DivisorSum[n, d[#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Mar 05 2024 *)

A034968(n) = { my(s=0, b=2, d); while(n, d = (n%b); s += d; n = (n-d)/b; b++); (s); };
A319711(n) = sumdiv(n,d,(dA034968(d));

a(n) = Sum_{d|n, dA034968(d).

a(n) = A319712(n) - A034968(n).

cp's OEIS Frontend

A333622 Numbers k such that k is divisible by the sum of digits of all the divisors of k in factorial base (A319712).

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A319715 Sum of A276150(d) over divisors d of n, where A276150 gives the sum of digits in primorial base.

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A319711 Sum of A034968(d) over proper divisors d of n, where A034968 gives the sum of digits in factorial base.

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