A034968 -id:A034968 - cp's OEIS frontend

A219650 The nonnegative integers n such that there exists a number k with A034968(n+k)=k.

0, 1, 2, 5, 6, 7, 10, 11, 12, 15, 16, 17, 23, 24, 25, 28, 29, 30, 33, 34, 35, 38, 39, 40, 46, 47, 48, 51, 52, 53, 56, 57, 58, 61, 62, 63, 69, 70, 71, 74, 75, 76, 79, 80, 81, 84, 85, 86, 92, 93, 94, 97, 98, 99, 102, 103, 104, 107, 108, 109, 119, 120, 121, 124

Offset: 0

Antti Karttunen, Nov 25 2012

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10080

Inverse: A230414. (In a sense that A230414(a(n)) = n for all n).
First differences: A230405. Bisection of A219651. Complement: A219658. Characteristic function: A230412. Cf. also A230423 and A230424.
Analogous sequence for binary system: A005187, for Zeckendorf expansion: A219640.

a(0) = 0; and for n>0, a(n) = a(n-1) + A230405(n-1).
a(n) = A219651(2*n).
a(n) ~ 2*n. - Amiram Eldar, Jan 21 2024

Name changed by Antti Karttunen, Nov 01 2013

A230423 a(n) = smallest natural number x such that x=n+A034968(x), or zero if no such number exists.

Original entry on oeis.org

0, 2, 4, 0, 0, 6, 8, 10, 0, 0, 12, 14, 16, 0, 0, 18, 20, 22, 0, 0, 0, 0, 0, 24, 26, 28, 0, 0, 30, 32, 34, 0, 0, 36, 38, 40, 0, 0, 42, 44, 46, 0, 0, 0, 0, 0, 48, 50, 52, 0, 0, 54, 56, 58, 0, 0, 60, 62, 64, 0, 0, 66, 68, 70, 0, 0, 0, 0, 0, 72, 74, 76, 0, 0, 78

Offset: 0

Antti Karttunen, Oct 31 2013

nonn

Also, if n can be partitioned into sum d1*(k1!-1) + d2*(k2!-1) + ... + dj*(kj!-1), where all k's are distinct and greater than one and each di is in range [1,ki] (in other words, if A230412(n)=1), then a(n) = d1*k1! + d2*k2! + ... + dj*kj!. If this is not possible, then n is one of the terms of A219658, and a(n)=0.

Antti Karttunen, Table of n, a(n) for n = 0..10079

a(A219650(n)) = A005843(n) = 2n. Cf. also A230414, A230424.
Can be used to compute A230425-A230427.
This sequence relates to the factorial base representation (A007623) in a similar way as A213723 relates to the binary system.

(define (A230423 n) (let loop ((k n)) (cond ((= (A219651 k) n) k) ((> k (+ n n)) 0) (else (loop (+ 1 k))))))

a(n) = 2*A230414(n).

A230424 a(n) = largest natural number x such that x=n+A034968(x), or zero if no such number exists.

Original entry on oeis.org

1, 3, 5, 0, 0, 7, 9, 11, 0, 0, 13, 15, 17, 0, 0, 19, 21, 23, 0, 0, 0, 0, 0, 25, 27, 29, 0, 0, 31, 33, 35, 0, 0, 37, 39, 41, 0, 0, 43, 45, 47, 0, 0, 0, 0, 0, 49, 51, 53, 0, 0, 55, 57, 59, 0, 0, 61, 63, 65, 0, 0, 67, 69, 71, 0, 0, 0, 0, 0, 73, 75, 77, 0, 0, 79

Offset: 0

Antti Karttunen, Oct 31 2013

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10079

Cf. A034968, A230414. a(A219650(n)) = A005408(n) = 2n+1.
One more than A230423 at its nonzero points. A219658 gives the positions of zeros. Can be used to compute A230425-A230427.
Analogous sequence for binary system: A213724.

(define (A230424 n) (if (zero? n) 1 (let ((v (A230423 n))) (if (zero? v) v (+ 1 v)))))

a(n) = A230412(n)*(A230423(n)+1).

A230406 a(n) = A034968(A219666(n)); after zero, the differences between successive nodes in the infinite trunk of the factorial beanstalk (A219666).

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 2, 5, 6, 2, 3, 2, 5, 5, 6, 2, 4, 5, 6, 7, 4, 5, 6, 7, 5, 5, 7, 10, 2, 3, 2, 5, 5, 6, 2, 4, 5, 6, 7, 4, 5, 6, 7, 5, 5, 6, 9, 8, 7, 10, 2, 4, 5, 6, 7, 4, 5, 6, 7, 5, 5, 6, 8, 6, 8, 8, 7, 10, 11, 4, 5, 6, 7, 5, 5, 6, 8, 6, 8, 8, 7, 10, 12, 10, 11

Offset: 0

Antti Karttunen, Nov 09 2013

nonn

Also the first differences of A219666, shifted once right and prepended with zero.

This sequence relates to the factorial base representation (A007623) in the same way as A213712 relates to the binary system.

Antti Karttunen, Table of n, a(n) for n = 0..3149

Cf. also A230418, A230410.

(define (A230406 n) (A034968 (A219666 n)))
;; Alternative definition:
(define (A230406 n) (if (zero? n) n (- (A219666 n) (A219666 (- n 1)))))

a(n) = A034968(A219666(n)).
a(0) = 0, and for n>=1, a(n) = A219666(n) - A219666(n-1).
a(A226061(n)) = A000217(n-1) for all n.

A339214 Factorial-base self numbers: numbers not of the form k + A034968(k).

Original entry on oeis.org

1, 4, 11, 18, 36, 43, 61, 68, 86, 93, 111, 118, 125, 132, 139, 157, 164, 182, 189, 207, 214, 232, 239, 246, 253, 260, 278, 285, 303, 310, 328, 335, 353, 360, 367, 374, 381, 399, 406, 424, 431, 449, 456, 474, 481, 488, 495, 502, 520

Offset: 1

Amiram Eldar, Nov 27 2020

Analogous to self numbers (A003052) using factorial base representation (A007623) instead of decimal expansion.

The numbers of terms that do not exceed 10^k, for k = 0, 1, ..., are 1, 2, 10, 90, 878, 8749, 87455, 874499, 8744934, 87449296, 874492907, ... . Apparently, the asymptotic density of this sequence exists and equals 0.08744929... . - Amiram Eldar, Aug 08 2025

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rosalind Guaraldo, On the Density of the Image Sets of Certain Arithmetic Functions - III, The Fibonacci Quarterly, Vol. 16, No. 6 (1978), pp. 481-488.
Eric Weisstein's World of Mathematics, Self Number.
Wikipedia, Factorial number system.
Wikipedia, Self number.
Index entries for Colombian or self numbers and related sequences

Cf. A003052, A007623, A010061, A010064, A010067, A010070, A034968, A118363, A339211, A339212, A339213, A339215.

Cf. also A219650, A219658, A230412.

max = 6; s[n_] := n + Plus @@ IntegerDigits[n, MixedRadix[Range[max, 2, -1]]]; m = max!; Complement[Range[m], Array[s, m]]

A377385 Factorial-base Niven numbers (A118363) k such that k/f(k) is also a factorial-base Niven number, where f(k) = A034968(k) is the sum of digits in the factorial-base representation of k.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 24, 27, 36, 40, 48, 54, 72, 80, 96, 108, 120, 135, 144, 168, 175, 180, 192, 208, 210, 240, 280, 288, 336, 360, 384, 420, 432, 468, 480, 490, 572, 576, 594, 600, 630, 720, 732, 740, 750, 780, 784, 819, 840, 846, 861, 864, 888, 900, 924, 936, 945, 980, 984

Offset: 1

Amiram Eldar, Oct 27 2024

			8 is a term since 8/f(8) = 4 is an integer and also 4/f(4) = 2 is an integer.

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

Subsequence of A118363.
Subsequences: A000142, A377386.
Cf. A034968, A377384.
Analogous sequences: A376616 (binary), A377209 (Zeckendorf).

fdigsum[n_] := Module[{k = n, m = 2, r, s = 0}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, s += r; m++]; s]; q[k_] := Module[{f = fdigsum[k]}, Divisible[k, f] && Divisible[k/f, fdigsum[k/f]]]; Select[Range[1000], q]

fdigsum(n) = {my(k = n, m = 2, r, s = 0); while([k, r] = divrem(k, m); k != 0 || r != 0, s += r; m++); s;}
is(k) = {my(f = fdigsum(k)); !(k % f) && !((k/f) % fdigsum(k/f));}

A232095 Minimal number of factorials which add to 0+1+2+...+n; a(n) = A034968(A000217(n)).

Original entry on oeis.org

0, 1, 2, 1, 3, 4, 5, 3, 3, 6, 4, 5, 4, 7, 7, 1, 5, 5, 5, 8, 7, 9, 5, 5, 6, 8, 10, 6, 9, 8, 10, 8, 6, 10, 12, 7, 10, 11, 6, 5, 7, 7, 8, 9, 5, 8, 5, 6, 8, 7, 10, 7, 11, 14, 8, 8, 6, 11, 7, 10, 7, 12, 10, 10, 12, 14, 7, 12, 9, 9, 11, 9, 12, 12, 12, 14, 10, 7, 11, 11

Offset: 0

Antti Karttunen, Nov 18 2013

nonn

1's occur at positions n=1, n=3 and n=15 as they are such natural numbers that A000217(n) is also one of the factorial numbers (A000142), as we have A000217(1) = 1 = 1!, A000217(3) = 1+2+3 = 6 = 3! and A000217(15) = 1 + 2 + ... + 15 = 120 = 5!
On the other hand, a(2)=2, as A000217(2) = 1+2 = 3 = 2! + 1!. Is this the only occurrence of 2?
Are some numbers guaranteed to occur an infinite number of times?

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A232094, A232096.

(define (A232095 n) (A034968 (A000217 n)))

a(n) = A034968(A000217(n)).
a(n) = A231717(A226061(n+1)). [Not a practical way to compute this sequence. Please see comments at A231717.]
For all n, a(n) >= A232094(n).

A286478 Ordinal transform of A034968, factorial base digit sum.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 20 2017

Antti Karttunen, Table of n, a(n) for n = 0..10080
Index entries for sequences related to factorial base representation

Cf. A000142, A034968, A254524 (base-10 analog).

f[n_] := If[n == 0, 0, Module[{s = 0, i = 2, k = n},
     While[k>0, k = Floor[n/i!]; s += (i-1) k; i++]; n-s]];
b[_] = 1;
a[n_] := a[n] = With[{t = f[n]}, b[t]++];
a /@ Range[0, 100] (* Jean-François Alcover, Dec 19 2021 *)

For all n>= 1, a(A000142(n)) = n.

A286607 Numbers that are not divisible by the sum of their factorial base digits (A034968).

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 28, 29, 31, 32, 33, 34, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 103, 104, 106, 107

Offset: 1

Antti Karttunen, Jun 18 2017

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to factorial base representation.

Cf. A034968, A118363 (complement), A286604.

q[n_] := Module[{k = n, m = 2, r, s = 0}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, s += r; m++]; !Divisible[n, s]]; Select[Range[120], q] (* Amiram Eldar, Feb 21 2024 *)

def a007623(n, p=2): return n if nIndranil Ghosh, Jun 21 2017

A319712 Sum of A034968(d) over divisors d of n, where A034968 gives the sum of digits in factorial 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, 12, 6, 7, 9, 12, 5, 17, 4, 13, 11, 11, 11, 18, 5, 10, 11, 21, 7, 19, 6, 18, 19, 14, 8, 18, 6, 13, 12, 13, 6, 17, 12, 18, 12, 11, 7, 29, 6, 10, 19, 19, 14, 23, 7, 19, 16, 25, 9, 24, 5, 10, 17, 17, 13, 19, 6, 30, 15, 14, 8, 31, 15, 13, 14, 27, 9, 35, 13, 23, 14, 17, 17

Offset: 1

Antti Karttunen, Oct 02 2018

Inverse Möbius transform of A034968.

Antti Karttunen, Table of n, a(n) for n = 1..40320
Index entries for sequences related to factorial base representation.

Cf. A034968, A319711, A319715.

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[#] &]; Array[a, 100] (* Amiram Eldar, Feb 14 2024 *)

A034968(n) = { my(s=0, b=2, d); while(n, d = (n%b); s += d; n = (n-d)/b; b++); (s); };
A319712(n) = sumdiv(n,d,A034968(d));

a(n) = Sum_{d|n} A034968(d).

a(n) = A319711(n) + A034968(n).

cp's OEIS Frontend

A219650 The nonnegative integers n such that there exists a number k with A034968(n+k)=k.

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A230423 a(n) = smallest natural number x such that x=n+A034968(x), or zero if no such number exists.

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A230424 a(n) = largest natural number x such that x=n+A034968(x), or zero if no such number exists.

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A230406 a(n) = A034968(A219666(n)); after zero, the differences between successive nodes in the infinite trunk of the factorial beanstalk (A219666).

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A339214 Factorial-base self numbers: numbers not of the form k + A034968(k).

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A377385 Factorial-base Niven numbers (A118363) k such that k/f(k) is also a factorial-base Niven number, where f(k) = A034968(k) is the sum of digits in the factorial-base representation of k.

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A232095 Minimal number of factorials which add to 0+1+2+...+n; a(n) = A034968(A000217(n)).

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A286478 Ordinal transform of A034968, factorial base digit sum.

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A286607 Numbers that are not divisible by the sum of their factorial base digits (A034968).

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