Triangular number AND sum of first m factorials

01092018, 04:31 PM
Post: #1




Triangular number AND sum of first m factorials
153 (my favorite number) is both a triangular number (the sum of the integers 1 through \(n\); in this case \(n=17\)) as well as the sum of the factorials \(1!\) through \(m!\) (in this case \(m=5\)).
The first three natural numbers which have both of those properties are 1, 3 (both trivial) and 153. Find the next number in this sequence. For extra credit, find the mathematical relationship between \(n\) and \(m\) for all members of this sequence (which apparently is not yet in OEIS). X<> c Joe 

01092018, 08:53 PM
(This post was last modified: 01092018 10:58 PM by Gerson W. Barbosa.)
Post: #2




RE: Triangular number AND sum of first m factorials
001 LBL A 002 DEC X 003 ENTER 004 INC X 005 LBL 00 006 INC X 007 RCL* Y 008 DSE Y 009 GTO 00 010 PSE 10 011 STO+ X 012 +/ 013 1 014 ENTER 015 ENTER 016 R/\ 017 SLVQ 018 X<>Y 019 FP 020 PSE 10 021 RCL L 022 RTN 2 A > 3 > 0 > 2 5 A > 153 > 0 > 17 The next m should be greater than 31, but I can’t find it using the wp34s, at least not with help of this simple program… Edited to fix a typo pointed out by Dieter below. 

01092018, 10:16 PM
Post: #3




RE: Triangular number AND sum of first m factorials  
01092018, 10:19 PM
Post: #4




RE: Triangular number AND sum of first m factorials
(01092018 08:53 PM)Gerson W. Barbosa Wrote: 001 LBL A ?!? – increment register C ? Is this possibly supposed to mean INC X ? (01092018 08:53 PM)Gerson W. Barbosa Wrote: 005 LBL 00 Ah, here is the missing X from line 004, and the "C" from there goes here. ;) Dieter 

01092018, 11:00 PM
Post: #5




RE: Triangular number AND sum of first m factorials  
01092018, 11:10 PM
Post: #6




RE: Triangular number AND sum of first m factorials
A quick test of sums of factorials up to 69! finds no triangle numbers larger than 153. If one does exist, it has to have well over 100 digits. I may try later on the emulator.
John 

01102018, 04:03 AM
Post: #7




RE: Triangular number AND sum of first m factorials
Tn&Sf:
« { } 3 ROT FOR n n Sfac 8 * 1 + ISPF? { √ 1  2 / + n I→R + } { DROP } IFTE NEXT » Sfac: « DUP 1  2 FOR m 1 + m * 1 STEP 1 + » ISPF? « 1 » 7 Tn&Sf > { '(√731)/2' 3. '(√2651)/2' 4. 17 5. '(√69851)/2' 6. '(√473051)/2' 7. } 49G or 50g in exact mode ISPF? (IsPerfectSquare?) has yet to be implemented. It should return 1 when the argument is a perfect square and 0 otherwise. Then the output would be a list of n and m pairs, separated by dots. Just in case someone wants to try. 

01102018, 04:58 AM
Post: #8




RE: Triangular number AND sum of first m factorials
(01102018 04:03 AM)Gerson W. Barbosa Wrote: ISPF? (IsPerfectSquare?) has yet to be implemented. It should return 1 when the argument is a perfect square and 0 otherwise. The HP 50g LongFloat library contains a function like the one you're looking for. It's called ZSqrt, and it returns IP(sqrt(x)) to level 2, and a 0 or 1 to level 1, with 1 meaning that x was a perfect square. It works on integers of any length. Gerald H also posted a program HERE which seems to do essentially the same thing. It returns IP(sqrt(x)) to level 2 of the stack, and on level 1 it leaves a SysRPL TRUE if x was a perfect square, otherwise a FALSE. That's the same idea as HP's FPTR2 ^ZSQRT, but Gerald's program is more accurate; see Gerald's posting for evidence of HP's function's inaccuracy. BTW, LongFloat's ZSqrt function gets the same result as Gerald's program when given the example in Gerald's posting, so I surmise that ZSqrt is trustworthy. X<> c Joe 

01102018, 06:35 AM
(This post was last modified: 01102018 10:59 AM by Paul Dale.)
Post: #9




RE: Triangular number AND sum of first m factorials
I've got a proof that there are only three such numbers.
Consider the last pair of digits in \( \sum_1^n i! \), from n=9 onwards these never change because subsequent factorial terms will always have a factor of 100 present. These digits are '13'. Note that n is triangular iff 8n+1 is a perfect square. For the sum of factorials to be triangular, the last two digits must therefore be '05'. Checking all possibilities shows that there are no square numbers that end '05'. Thus, numbers of the desired form must have n < 9. Checking all cases reveals that only 1, 3 and 153 have the desired properties. Pauli 

01112018, 03:01 AM
Post: #10




RE: Triangular number AND sum of first m factorials
Paul: Now THAT is beautiful! Thank you! I can stop my futile hunt now.
X<> c Joe 

01112018, 10:21 AM
Post: #11




RE: Triangular number AND sum of first m factorials
Thanks Don't let my proof stop your hunt, you'll be able to wile away many hours looking...
I hadn't realised that all square numbers that end in '5' actually end in '25'. I must have seen this before but never noticed or remembered it. Pauli 

01112018, 02:22 PM
Post: #12




RE: Triangular number AND sum of first m factorials
A really great thread Joe et. al. Thank you all.
John 

01112018, 06:29 PM
(This post was last modified: 01112018 06:31 PM by Gerson W. Barbosa.)
Post: #13




RE: Triangular number AND sum of first m factorials
(01112018 10:21 AM)Paul Dale Wrote: Don't let my proof stop your hunt, you'll be able to wile away many hours looking... At least I can do it a little more efficiently now :) 100 « { } SWAP 0 1 ROT 1 SWAP FOR m m * SWAP OVER + ROT OVER 8 * 1 + ZSqrt { 1  2 / + m I→R + } { DROP } IFTE SWAP ROT NEXT DROP2 » EVAL > { 1 1. 2 2. 17 5. } (about 17 seconds on the real 50g) ZSqrt from the LongFloat Library Yes, that's a consequence of the ever growing number of trailing zeros in factorials and the properties of perfect squares. Gerson. 

01112018, 10:30 PM
Post: #14




RE: Triangular number AND sum of first m factorials
(01112018 10:21 AM)Paul Dale Wrote: Thanks Don't let my proof stop your hunt, you'll be able to wile away many hours looking... I figured it was hopeless since I tested all sums of factorials up to 1000 (over 2500 digits) with no triangle numbers found. Took almost 20 min. on the emulator. An interesting and educational thread indeed! John 

01112018, 10:43 PM
Post: #15




RE: Triangular number AND sum of first m factorials
(01112018 06:29 PM)Gerson W. Barbosa Wrote: Yes, that's a consequence of the ever growing number of trailing zeros in factorials and the properties of perfect squares. Note, however, that due to the first four numbers, all of the sums of factorials end in 3. 

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