A014080 -id:A014080 - cp's OEIS frontend

A345406 Integers k such that k = d1^(c) + d2^(c) + ... + dc^(c), where d^(c) denotes the rising factorial of d, c is the length of k and di is the i-th digit of k in base 10.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 90, 744, 840

Offset: 1

Andrzej Kukla, Jun 18 2021

The rising factorial d^(c) is defined as d*(d+1)*(d+2)*...*(d+c-1).

			7^(3) + 4^(3) + 4^(3) = 7*8*9 + 4*5*6 + 4*5*6 = 504 + 120 + 120 = 744, therefore 744 is in the list.

Cf. A014080 (factorions), A265609 (rising factorials), A345405.

q[n_] := Module[{dig = IntegerDigits[n], nd}, nd = Length[dig]; Sum[(d + nd - 1)!/(d - 1)!, {d, dig}] == n]; Select[Range[0, 1000], q] (* Amiram Eldar, Jun 18 2021 *)

A101702 Numbers m such that the sum of the factorials of their digits is equal to the reversal of m.

Original entry on oeis.org

1, 2, 541, 52100, 58504, 66410, 430000, 863180, 8601400, 17927300, 27927300, 31000000, 665100000, 3715000000, 6739630000, 11000000000, 21000000000, 53100000000, 70858000000, 79637300000, 451000000000, 1715000000000, 2715000000000, 48304000000000, 340000000000000, 5520000000000000

Offset: 1

Farideh Firoozbakht, Dec 24 2004

If s=sum of the factorials of digits of m & reversal(m) >= s then 10^(reversal(m) - s)*m is in the sequence. Example m=23; s = 2! + 3!; reversal(23) - s = 24 & 23*10^24 is in the sequence. So this sequence is infinite because there exist infinitely many numbers m such that reversal(m) > s. If m is a k-digit term of this sequence and the first digit of m is 1 then 10^(k-1) + m is also in the sequence. Examples: m=1 so 10^(1-1) + 1 = 2 is in the sequence, m=17927300 so 10^7 + 17927300 = 27927300 is in the sequence. If m > 5 then 10 divides a(m). If 10 doesn't divide a(m) then the reversal of m is in the sequence A014080, so all terms of A014080 are: reversal(1), reversal(2), reversal(541) & reversal(58504).

			665100000 is in the sequence because reversal(665100000) = 1566 = 6! + 6! + 5! + 1! + 0! + 0! + 0! + 0! + 0!.

Cf. A014080, A049529, A101697.

Do[h = IntegerDigits[n]; l = Length[h]; If[FromDigits[Reverse[IntegerDigits[n]]] == Sum[h[[k]]!, {k, l}], Print[n]], {n, 10^9}]

More terms from Donovan Johnson, Feb 26 2008

A140764 Numbers equal to the sum of the squares of their duodecimal digit factorials.

Original entry on oeis.org

1, 37, 613, 519018

Offset: 1

Lekraj Beedassy, Jul 13 2008

In other words, numbers equal to the sum of the squares of factorials of their base-12 representation digits.

			We have, for instance, 37 = 31_(12) = (3!)^2 + (1!)^2 and 613 = 431_(12) = (4!)^2 + (3!)^2 + (1!)^2.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 37, pp 15, Ellipses, Paris 2008.

Cf. A014080.

A163752 Generalized factorions: numbers which are equal to the sum of the factorials of some or all of their digits in base 10.

Original entry on oeis.org

1, 2, 24, 145, 5760, 5761, 5762, 40328, 40585, 362904, 367920, 367921, 367922, 367926, 367928, 367932, 367944

Offset: 1

Berend Jan van der Zwaag (b.j.vanderzwaag(AT)utwente.nl), Aug 03 2009

Generalized factorions include factorions (A014080), near-factorions (A163576), and "not-so-near-factorions" such as 362904, which is equal to 9! + 4!.

			367932 = 3! + 7! + 9! + 3!.

Superset of A014080 and A163576.

Select[Range[370000],MemberQ[Total[#!]&/@Rest[Subsets[ IntegerDigits[ #]]], #]&] (* Harvey P. Dale, Mar 20 2017 *)

A182478 Numbers that can be truncated in base 10 such that the sum of the factorials of the truncations equals that number.

Original entry on oeis.org

1, 2, 145, 40585, 6402374184741226, 121645100891988866, 121666023198802103, 121666023198802144, 2432902008177819519, 2432902008217006118, 2432902008656812499, 4872206390059820318

Offset: 1

Bodo Zinser, May 01 2012

			a(5)=6402374184741226=6!+4!+(02)!+3!+7!+4!+18!+4!+7!+4!+12!+2!+6!
a(6): 2-digit-truncations are 12,10,19
a(7): 2-digit-truncs are 16,19
a(8): 2-digit-truncs are 16,19
a(9): 2-digit-trunc is 20
a(10): 2-digit-truncs are 20,11
a(11): 2-digit-truncs are 20,12
a(12): 2-digit-truncs are 20,20,18

A014080 is a subsequence.

A260651 Number of factorions in base n.

Original entry on oeis.org

2, 2, 3, 3, 4, 2, 2, 3, 4, 5, 2, 3, 3, 4, 3, 5, 2, 2, 2, 3, 2, 3, 4, 2, 4, 4, 3, 2, 3, 2, 4, 2, 6, 3, 3, 3, 3, 2

Offset: 2

Eric M. Schmidt (based on data from A193163), Nov 16 2015

1 and 2 are factorions of every integer number base, since 1 = 1! and 2 = 2!. Thus every integer number base has at least 2 factorions. - Michael De Vlieger, Nov 23 2015

A factorion is an integer which is equal to the sum of factorials of its digits. See A193163 for the list of all factorions in base n. - M. F. Hasler, Nov 25 2015

			a(6) = 4 because base 6 has the factorions {1, 2, 25, 26}. Expressed in base 6 these are {1, 2, 41, 42}. 1! = 1 and 2! = 2 and are factorions in every integer base b >= 2. Additionally, 4! + 1! = 24 + 1 = 25 and 4! + 2! = 24 + 2 = 26. - _Michael De Vlieger_, Nov 23 2015
a(2) = 2 = #{ 1, 2 }, indeed 1 = 1! and 2 = 10[2] = 1! + 0! and there cannot be any other since the sum of factorials of the binary digits equals the number of these digits, and from 3 on all numbers are larger than the number of their binary digits. - _M. F. Hasler_, Nov 25 2015

Eric Weisstein's World of Mathematics, Factorion

Cf. A014080, A193163.

Table[Length@ Select[Range[n Factorial[n - 1]], Total@ Map[Factorial, #] &@ IntegerDigits[#, n] == # &], {n, 2, 10}] (* Michael De Vlieger, Nov 23 2015 *)

A335961 Alternating factorions: Numbers m such that m = S_af(m) = af(d_1)+af(d_2)+...+af(d_k) where d_1 d_2 ... d_n is the decimal expansion of m and af(m) = m!-(m-1)!+(m-2)!+...1! (alternating factorial) with af(0) = 0 (base 10).

Original entry on oeis.org

0, 1, 620, 621, 643

Offset: 1

Andrzej Kukla, Jul 01 2020

Largest k such that S_af(k) > k is 1599999. That's why there are only five numbers such that S_af(m) = m. Proved by computer calculations.

If m has eight or more digits then S_af(m) < m. Proved directly.

			For m = 620, S_af(620) = af(6)+af(2)+af(0) = 619+1+0 = 620.

Cf. A005165 (alternating factorial), A014080 (factorions).

af[0] = 0; af[n_] := af[n] = n! - af[n - 1]; Select[Range[1000], Total[af /@ IntegerDigits[#]] == # &] (* Amiram Eldar, Jul 02 2020 *)

A343269 a(n) is the smallest integer whose orbit length is n under iteration of the map r -> A061602(r).

Original entry on oeis.org

1, 0, 169, 78, 69, 26, 24, 4, 22, 5, 122, 25, 14, 127, 6, 3, 12, 33, 136, 256, 57, 247, 148, 38, 1478, 368, 79, 1458, 48, 44, 29, 7, 13, 34, 9, 8, 23, 234, 37, 337, 58, 46, 139, 138, 369, 239, 267, 36, 334, 289, 3555, 49, 144, 45, 229, 2569, 22888, 136789, 334479, 1479, 1233466

Offset: 1

Lamine Ngom, Apr 10 2021

A303935 provides the orbit's lengths, i.e., the number of needed steps, starting from a given number, to reach a value that already exists in trajectory.
This sequence is infinite. Actually, given a number x whose orbit's length is k, one can always build a number y whose orbit's length is (k+1).
For instance, just consider either the number 10^(x-1), or Rx (the repunit of length x), or any other x-digit binary string, all of them leading to the number x after application of the mapping function: A061602(y) = x.
Indeed, none of them will correspond to the smallest integer m such that A303935(m) = k + 1.
In fact, it becomes computationally hard to determine further terms since, as in the Collatz mapping function and other similar problems, there is no predictable way to define the exact complete path without calculating all intermediary orbit's components until one reaches a previously calculated or encountered number.
a(59) = 334479, a(60) = 1479, a(61) = 1233466, next terms = ?

			a(6) = 26 because A303935(26) = 6, and 26 is the smallest nonnegative integer m such that A303935(m) = 6.

Cf. A303935 (orbit's length), A061602 (sum of factorials of digits), A014080 (factorions).

Cf. A193163, A214285, A254499, A188283, A244090.

cp's OEIS Frontend

A345406 Integers k such that k = d1^(c) + d2^(c) + ... + dc^(c), where d^(c) denotes the rising factorial of d, c is the length of k and di is the i-th digit of k in base 10.

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A101702 Numbers m such that the sum of the factorials of their digits is equal to the reversal of m.

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A140764 Numbers equal to the sum of the squares of their duodecimal digit factorials.

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A163752 Generalized factorions: numbers which are equal to the sum of the factorials of some or all of their digits in base 10.

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A182478 Numbers that can be truncated in base 10 such that the sum of the factorials of the truncations equals that number.

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A260651 Number of factorions in base n.

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A335961 Alternating factorions: Numbers m such that m = S_af(m) = af(d_1)+af(d_2)+...+af(d_k) where d_1 d_2 ... d_n is the decimal expansion of m and af(m) = m!-(m-1)!+(m-2)!+...1! (alternating factorial) with af(0) = 0 (base 10).

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A343269 a(n) is the smallest integer whose orbit length is n under iteration of the map r -> A061602(r).

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