5, 10, 13, 17, 20, 25 ... are numbers that are the sum of 2 distinct nonzero squares (A004431-let's call it SEQ).
Now, some tasks for you:
1. 25,40,52,73,89 ...; add 2 more numbers in this sequence.
2. In the sequence SEQ find the 1st appearance of 3 consecutive numbers.
3. Find the index of the 1st 4-digit member in SEQ.
4. There is a "run" of 4 consecutive non members of SEQ between 45 and 50 .
a) find a longer "run".
b) find the longest "run" within the 1st 12000 members of SEQ.
5. Find the lowest pandigital member of SEQ.
6. Find a pandigital member(s?) of SEQ, equalling A2+ B2
such that the concatenation of the numbers A & B is also pandigital.
Rem: If you have solved 4 or more out of 6 listed tasks
please rate this post - I've spent about 2.5 hours to create it.
The program runs rapidly, as to test a given number, it merely tests a=1 through the square root of half n. Then subtracting its square from n the difference has to be a perfect square to be in SEQ:
DEFDBL A-Z
DIM seq(40), nseq(40)
OPEN "aboutseq.txt" FOR OUTPUT AS #2
CLS
FOR n = 1 TO 50000
flag = 0
FOR a = 1 TO SQR(n / 2)
n1 = a * a
n2 = n - n1
sr = INT(SQR(n2) + .5)
IF sr * sr = n2 AND n2 <> n1 THEN
index = index + 1
IF index = 7 THEN PRINT n; : PRINT #2, n;
IF index = 8 THEN PRINT n: PRINT #2, n
IF n > 999 AND flag4 = 0 THEN PRINT index; n: PRINT #2, index; n: flag4 = 1
flag = 1
EXIT FOR
END IF
NEXT a
IF flag THEN
IF nseqct >= 4 THEN
PRINT "ns"; nseqct;
PRINT #2, "ns"; nseqct;
FOR i = 1 TO nseqct: PRINT nseq(i); : PRINT #2, nseq(i); : NEXT: PRINT : PRINT #2,
IF nseqct > maxnseqct THEN
maxnseqct = nseqct
maxnseqstart = n - nseqct
END IF
END IF
seqct = seqct + 1: nseqct = 0
seq(seqct) = n
ELSE
IF seqct >= 3 THEN
PRINT "s"; seqct;
PRINT #2, "s"; seqct;
FOR i = 1 TO seqct: PRINT seq(i); : PRINT #2, seq(i); : NEXT: PRINT : PRINT #2,
IF seqct > maxseqct THEN
maxseqct = seqct
maxseqstart = n - seqct
maxseqstartindex = index - seqct + 1
END IF
END IF
nseqct = nseqct + 1: seqct = 0
nseq(nseqct) = n
END IF
NEXT n
PRINT index
PRINT maxseqstartindex, maxseqstart, maxseqct
PRINT maxnseqstart, maxnseqct
PRINT #2, index
PRINT #2, maxseqstartindex, maxseqstart, maxseqct
PRINT #2, maxnseqstart, maxnseqct
CLOSE
The start of the output is:
ns 4 1 2 3 4
ns 4 6 7 8 9
ns 4 21 22 23 24
26 29
ns 4 30 31 32 33
ns 4 46 47 48 49
ns 4 54 55 56 57
ns 4 69 70 71 72
ns 5 75 76 77 78 79
ns 6 91 92 93 94 95 96
ns 4 118 119 120 121
ns 4 126 127 128 129
ns 5 131 132 133 134 135
ns 7 138 139 140 141 142 143 144
ns 4 165 166 167 168
ns 4 174 175 176 177
ns 7 186 187 188 189 190 191 192
ns 5 213 214 215 216 217
s 3 232 233 234
ns 6 235 236 237 238 239 240
ns 4 246 247 248 249
ns 6 251 252 253 254 255 256
ns 7 282 283 284 285 286 287 288
ns 6 299 300 301 302 303 304
ns 6 307 308 309 310 311 312
ns 4 321 322 323 324
ns 4 329 330 331 332
ns 5 341 342 343 344 345
ns 8 378 379 380 381 382 383 384 385
ns 4 390 391 392 393
ns 5 411 412 413 414 415
ns 4 417 418 419 420
ns 7 426 427 428 429 430 431 432
ns 5 437 438 439 440 441
ns 4 453 454 455 456
ns 8 469 470 471 472 473 474 475 476
ns 6 494 495 496 497 498 499
ns 4 501 502 503 504
ns 4 510 511 512 513
ns 5 515 516 517 518 519
s 3 520 521 522
ns 7 523 524 525 526 527 528 529
showing:
1. The next two numbers in the sequnce are 26 and 29.
2. The first occurrence of 3 consecutive numbers in SEQ is
232 233 234
3. See below.
4a. Several runs of 7 and 8 non-members (marked ns, for non-sequence, with the length of each such sequence) appear above.
Later on, the line
293 1000
shows:
3. The first 4-digit number in SEQ, 1000, has index 293.
At the bottom of the listing:
70 232 3
31658 23
indicates:
4b. The longest run of members (marked with s) is only 3. ... of non-members is 23, beginning with 31658. (The highest index found among members was 12262 which in fact was the 50,000, the last number checked, so the non-member sequence starting at 31,658 was indeed within the 12,000 member set specified in the puzzle.)
For the pandigital questions we use another program:
DECLARE SUB permute (a$)
DEFDBL A-Z
CLS
a$ = "123456789": h$ = a$
DO
n = VAL(a$)
FOR a = 1 TO SQR(n / 2)
n1 = a * a
n2 = n - n1
sr = INT(SQR(n2) + .5)
IF sr * sr = n2 THEN PRINT a$, n1, n2: ct = ct + 1: EXIT FOR
NEXT a
IF ct > 5 THEN GOTO after
permute a$
LOOP UNTIL a$ = h$
after:
PRINT
ct = 0
a$ = "1234567890": h$ = a$
DO
n = VAL(a$)
FOR a = 1 TO SQR(n / 2)
n1 = a * a
n2 = n - n1
sr = INT(SQR(n2) + .5)
IF sr * sr = n2 THEN
PRINT a$, n1, n2: ct = ct + 1: EXIT FOR
END IF
NEXT a
IF ct > 5 THEN GOTO after2
permute a$
LOOP UNTIL a$ = h$
after2:
PRINT : ct = 0
a$ = "123456789": h$ = a$
DO
n = VAL(a$)
FOR a = 1 TO SQR(n / 2)
n1 = a * a
n2 = n - n1
sr = INT(SQR(n2) + .5): n1s = n1: n2s = n2
IF sr * sr = n2 THEN
tst$ = LTRIM$(STR$(a)) + LTRIM$(STR$(sr))
IF LEN(tst$) = 9 THEN
good = 1
FOR i = 1 TO 9
IF INSTR(tst$, MID$("123456789", i, 1)) = 0 THEN good = 0: EXIT FOR
NEXT i
IF good THEN PRINT n, a; sr: ct = ct + 1
END IF
END IF
NEXT a
IF ct > 5 THEN GOTO after3
permute a$
LOOP UNTIL a$ = h$
after3:
PRINT : ct = 0
a$ = "1234567890": h$ = a$
DO
n = VAL(a$)
FOR a = 1 TO SQR(n / 2)
n1 = a * a
n2 = n - n1
sr = INT(SQR(n2) + .5): n1s = n1: n2s = n2
IF sr * sr = n2 THEN
tst$ = LTRIM$(STR$(a)) + LTRIM$(STR$(sr))
IF LEN(tst$) = 10 THEN
good = 1
FOR i = 1 TO 10
IF INSTR(tst$, MID$("1234567890", i, 1)) = 0 THEN good = 0: EXIT FOR
NEXT i
IF good THEN PRINT n, a; sr: ct = ct + 1
END IF
END IF
NEXT a
IF ct > 5 THEN GOTO after4
permute a$
LOOP UNTIL a$ = h$
after4:
END
finds
The first few pandigital members of SEQ (without and with zeros):
a^2 + b^2 a^2 b^2
123456978 27279729 96177249
123457689 4473225 118984464
123457869 580644 122877225
123458976 55890576 67568400
123465789 4284900 119180889
123468957 5659641 117809316
1234569780 34363044 1200206736
1234570698 2350089 1232220609
1234570896 32400 1234538496
1234576089 595750464 638825625
1234576890 670761 1233906129
1234578609 125888400 1108690209
and the pandigital members whose a and b concatenate pandigitally:
a^2 + b^2 a b
175236849 3495 12768
179684325 3654 12897
189237465 4536 12987
197485632 5376 12984
218367945 5469 13728
231649785 6528 13749
1238497065 17628 30459
1239785640 16782 30954
1325479860 19482 30756
1328596740 19842 30576
1348657209 16947 32580
1352879460 17094 32568
These four lists would each probably continue further if we hadn't stopped at six each.
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Posted by Charlie
on 2012-12-20 01:18:32 |
solutions
