Number 64 – DigitSum() and DigitProduct()

 

 
Let   DigitSum \,(N)   be the sum of digits of   N,   and   DigitProduct \,(N),  
the product of digits of   N

To find a square number   x   such that

DigitSum \,(x) - 1     and     DigitProduct \,(x) + 1   are both squares

For example,     x = 64 = 8^2

DigitSum \,(64) = 6 + 4 = 10     and     10 - 1 = 3^2

DigitProduct \,(64) = 6 \times 4 = 24     and     24 + 1 = 5^2
 

 
Can you find another square like 64?

 

pipo found:

187489 = 433^2
1+8+7+4+8+9 = 37   ……….   37 - 1 = 6^2
1\times 8\times 7\times 4\times 8\times 9 = 16128   ….   16128 + 1 = 127^2

529984 = 728^2
5+2+9+9+8+4 = 37   ……….   37 - 1 = 6^2
5\times 2\times 9\times 9\times 8\times 4 = 25920   ….   25920 + 1 = 161^2

2982529 = 1727^2
2+9+8+2+5+2+9 = 37   ……….   37 - 1 = 6^2
2\times 9\times 8\times 2\times 5\times 2\times 9 = 25920   ….   25920 + 1 = 161^2

6165289 = 2483^2
6+1+6+5+2+8+9 = 37   ……….   37 - 1 = 6^2
6\times 1\times 6\times 5\times 2\times 8\times 9 = 25920   ….   25920 + 1 = 161^2

45819361 = 6769^2
4+5+8+1+9+3+6+1 = 37   ……….   37 - 1 = 6^2
4\times 5\times 8\times 1\times 9\times 3\times 6\times 1 = 25920   ….   25920 + 1 = 161^2

55279225 = 7435^2
5+5+2+7+9+2+2+5 = 37   ……….   37 - 1 = 6^2
5\times 5\times 2\times 7\times 9\times 2\times 2\times 5 = 63000   ….   63000 + 1 = 251^2

59613841 = 7721^2
5+9+6+1+3+8+4+1 = 37   ……….   37 - 1 = 6^2
5\times 9\times 6\times 1\times 3\times 8\times 4\times 1 = 25920   ….   25920 + 1 = 161^2

85914361 = 9269^2
8+5+9+1+4+3+6+1 = 37   ……….   37 - 1 = 6^2
8\times 5\times 9\times 1\times 4\times 3\times 6\times 1 = 25920   ….   25920 + 1 = 161^2

89151364 = 9442^2
8+9+1+5+1+3+6+4 = 37   ……….   37 - 1 = 6^2
8\times 9\times 1\times 5\times 1\times 3\times 6\times 4 = 25920   ….   25920 + 1 = 161^2

114297481 = 10691^2
1+1+4+2+9+7+4+8+1 = 37   ……….   37 - 1 = 6^2
1\times 1\times 4\times 2\times 9\times 7\times 4\times 8\times 1 = 16128   ….   16128 + 1 = 127^2

118417924 = 10882^2
1+1+8+4+1+7+9+2+4 = 37   ……….   37 - 1 = 6^2
1\times 1\times 8\times 4\times 1\times 7\times 9\times 2\times 4 = 16128   ….   16128 + 1 = 127^2

181252369 = 13463^2
1+8+1+2+5+2+3+6+9 = 37   ……….   37 - 1 = 6^2
1\times 8\times 1\times 2\times 5\times 2\times 3\times 6\times 9 = 25920   ….   25920 + 1 = 161^2

183196225 = 13535^2
1+8+3+1+9+6+2+2+5 = 37   ……….   37 - 1 = 6^2
1\times 8\times 3\times 1\times 9\times 6\times 2\times 2\times 5 = 25920   ….   25920 + 1 = 161^2

223981156 = 14966^2
2+2+3+9+8+1+1+5+6 = 37   ……….   37 - 1 = 6^2
2\times 2\times 3\times 9\times 8\times 1\times 1\times 5\times 6 = 25920   ….   25920 + 1 = 161^2

 

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged , . Bookmark the permalink.

2 Responses to Number 64 – DigitSum() and DigitProduct()

  1. paul says:

    Here are those with x<=10000.

    x = 46 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 64 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 113 … 5 – 1 = 2^2 & 3 + 1 = 2^2
    x = 118 … 10 – 1 = 3^2 & 8 + 1 = 3^2
    x = 131 … 5 – 1 = 2^2 & 3 + 1 = 2^2
    x = 179 … 17 – 1 = 4^2 & 63 + 1 = 8^2
    x = 181 … 10 – 1 = 3^2 & 8 + 1 = 3^2
    x = 197 … 17 – 1 = 4^2 & 63 + 1 = 8^2
    x = 226 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 262 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 311 … 5 – 1 = 2^2 & 3 + 1 = 2^2
    x = 467 … 17 – 1 = 4^2 & 168 + 1 = 13^2
    x = 476 … 17 – 1 = 4^2 & 168 + 1 = 13^2
    x = 622 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 647 … 17 – 1 = 4^2 & 168 + 1 = 13^2
    x = 674 … 17 – 1 = 4^2 & 168 + 1 = 13^2
    x = 719 … 17 – 1 = 4^2 & 63 + 1 = 8^2
    x = 746 … 17 – 1 = 4^2 & 168 + 1 = 13^2
    x = 764 … 17 – 1 = 4^2 & 168 + 1 = 13^2
    x = 791 … 17 – 1 = 4^2 & 63 + 1 = 8^2
    x = 811 … 10 – 1 = 3^2 & 8 + 1 = 3^2
    x = 917 … 17 – 1 = 4^2 & 63 + 1 = 8^2
    x = 971 … 17 – 1 = 4^2 & 63 + 1 = 8^2
    x = 1135 … 10 – 1 = 3^2 & 15 + 1 = 4^2
    x = 1153 … 10 – 1 = 3^2 & 15 + 1 = 4^2
    x = 1234 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 1243 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 1315 … 10 – 1 = 3^2 & 15 + 1 = 4^2
    x = 1324 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 1342 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 1351 … 10 – 1 = 3^2 & 15 + 1 = 4^2
    x = 1358 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 1385 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 1423 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 1432 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 1513 … 10 – 1 = 3^2 & 15 + 1 = 4^2
    x = 1531 … 10 – 1 = 3^2 & 15 + 1 = 4^2
    x = 1538 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 1583 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 1835 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 1853 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 2134 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 2143 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 2267 … 17 – 1 = 4^2 & 168 + 1 = 13^2
    x = 2276 … 17 – 1 = 4^2 & 168 + 1 = 13^2
    x = 2314 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 2341 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 2413 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 2431 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 2447 … 17 – 1 = 4^2 & 224 + 1 = 15^2
    x = 2474 … 17 – 1 = 4^2 & 224 + 1 = 15^2
    x = 2627 … 17 – 1 = 4^2 & 168 + 1 = 13^2
    x = 2672 … 17 – 1 = 4^2 & 168 + 1 = 13^2
    x = 2726 … 17 – 1 = 4^2 & 168 + 1 = 13^2
    x = 2744 … 17 – 1 = 4^2 & 224 + 1 = 15^2
    x = 2762 … 17 – 1 = 4^2 & 168 + 1 = 13^2
    x = 3115 … 10 – 1 = 3^2 & 15 + 1 = 4^2
    x = 3124 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 3142 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 3151 … 10 – 1 = 3^2 & 15 + 1 = 4^2
    x = 3158 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 3185 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 3214 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 3241 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 3412 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 3421 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 3446 … 17 – 1 = 4^2 & 288 + 1 = 17^2
    x = 3464 … 17 – 1 = 4^2 & 288 + 1 = 17^2
    x = 3511 … 10 – 1 = 3^2 & 15 + 1 = 4^2
    x = 3518 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 3581 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 3644 … 17 – 1 = 4^2 & 288 + 1 = 17^2
    x = 3815 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 3851 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 4123 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 4132 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 4213 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 4231 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 4247 … 17 – 1 = 4^2 & 224 + 1 = 15^2
    x = 4274 … 17 – 1 = 4^2 & 224 + 1 = 15^2
    x = 4312 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 4321 … 10 – 1 = 3^2 & 24 + 1 = 5^2
    x = 4346 … 17 – 1 = 4^2 & 288 + 1 = 17^2
    x = 4364 … 17 – 1 = 4^2 & 288 + 1 = 17^2
    x = 4427 … 17 – 1 = 4^2 & 224 + 1 = 15^2
    x = 4436 … 17 – 1 = 4^2 & 288 + 1 = 17^2
    x = 4463 … 17 – 1 = 4^2 & 288 + 1 = 17^2
    x = 4472 … 17 – 1 = 4^2 & 224 + 1 = 15^2
    x = 4634 … 17 – 1 = 4^2 & 288 + 1 = 17^2
    x = 4643 … 17 – 1 = 4^2 & 288 + 1 = 17^2
    x = 4724 … 17 – 1 = 4^2 & 224 + 1 = 15^2
    x = 4742 … 17 – 1 = 4^2 & 224 + 1 = 15^2
    x = 5113 … 10 – 1 = 3^2 & 15 + 1 = 4^2
    x = 5131 … 10 – 1 = 3^2 & 15 + 1 = 4^2
    x = 5138 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 5183 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 5311 … 10 – 1 = 3^2 & 15 + 1 = 4^2
    x = 5318 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 5381 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 5678 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
    x = 5687 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
    x = 5768 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
    x = 5786 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
    x = 5813 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 5831 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 5867 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
    x = 5876 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
    x = 6227 … 17 – 1 = 4^2 & 168 + 1 = 13^2
    x = 6272 … 17 – 1 = 4^2 & 168 + 1 = 13^2
    x = 6344 … 17 – 1 = 4^2 & 288 + 1 = 17^2
    x = 6434 … 17 – 1 = 4^2 & 288 + 1 = 17^2
    x = 6443 … 17 – 1 = 4^2 & 288 + 1 = 17^2
    x = 6578 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
    x = 6587 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
    x = 6722 … 17 – 1 = 4^2 & 168 + 1 = 13^2
    x = 6758 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
    x = 6785 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
    x = 6857 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
    x = 6875 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
    x = 7226 … 17 – 1 = 4^2 & 168 + 1 = 13^2
    x = 7244 … 17 – 1 = 4^2 & 224 + 1 = 15^2
    x = 7262 … 17 – 1 = 4^2 & 168 + 1 = 13^2
    x = 7424 … 17 – 1 = 4^2 & 224 + 1 = 15^2
    x = 7442 … 17 – 1 = 4^2 & 224 + 1 = 15^2
    x = 7568 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
    x = 7586 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
    x = 7622 … 17 – 1 = 4^2 & 168 + 1 = 13^2
    x = 7658 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
    x = 7685 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
    x = 7856 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
    x = 7865 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
    x = 8135 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 8153 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 8315 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 8351 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 8513 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 8531 … 17 – 1 = 4^2 & 120 + 1 = 11^2
    x = 8567 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
    x = 8576 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
    x = 8657 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
    x = 8675 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
    x = 8756 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
    x = 8765 … 26 – 1 = 5^2 & 1680 + 1 = 41^2

    Paul.

  2. pipo says:

    I looked at numbers, x, which are square, rest same conditions.
    Format (x, x^2, productofdigits+1, square root of (pod+1), sum of digits -1, square root of (sod -1).
    Strangely up till 1000000 all the square roots of (sod-1) are 6, except the first one.
    8 64 25 5 9 3
    433 187489 16129 127 36 6
    728 529984 25921 161 36 6
    1727 2982529 25921 161 36 6
    2483 6165289 25921 161 36 6
    6769 45819361 25921 161 36 6
    7435 55279225 63001 251 36 6
    7721 59613841 25921 161 36 6
    9269 85914361 25921 161 36 6
    9442 89151364 25921 161 36 6
    10691 114297481 16129 127 36 6
    10882 118417924 16129 127 36 6
    13463 181252369 25921 161 36 6
    13535 183196225 25921 161 36 6
    14966 223981156 25921 161 36 6
    15227 231861529 25921 161 36 6
    20278 411197284 16129 127 36 6
    21131 446519161 25921 161 36 6
    24139 582691321 25921 161 36 6
    24479 599221441 25921 161 36 6
    25381 644195161 25921 161 36 6
    29611 876811321 16129 127 36 6
    30689 941814721 16129 127 36 6
    34127 1164652129 25921 161 36 6
    35531 1262451961 25921 161 36 6
    38161 1456261921 25921 161 36 6
    40211 1616924521 25921 161 36 6
    40807 1665211249 25921 161 36 6
    42911 1841353921 25921 161 36 6
    47161 2224159921 25921 161 36 6
    50129 2512916641 25921 161 36 6
    52111 2715556321 63001 251 36 6
    57815 3342574225 201601 449 36 6
    59489 3538941121 25921 161 36 6
    74312 5522273344 201601 449 36 6
    90073 8113145329 25921 161 36 6
    97579 9521661241 25921 161 36 6
    105434 11116328356 25921 161 36 6
    105569 11144813761 16129 127 36 6
    111419 12414193561 25921 161 36 6
    118792 14111539264 25921 161 36 6
    127331 16213183561 25921 161 36 6
    134623 18123352129 25921 161 36 6
    138979 19315162441 25921 161 36 6
    146087 21341411569 25921 161 36 6
    152704 23318511616 25921 161 36 6
    154315 23813119225 25921 161 36 6
    178415 31831912225 25921 161 36 6
    270521 73181611441 16129 127 36 6
    279611 78182311321 16129 127 36 6
    334846 112121843716 16129 127 36 6
    337987 114235212169 25921 161 36 6
    339796 115461321616 25921 161 36 6
    343682 118117317124 9409 97 36 6
    355123 126112345129 25921 161 36 6
    363142 131872112164 16129 127 36 6
    368261 135616164121 25921 161 36 6
    380944 145118331136 25921 161 36 6
    383239 146872131121 16129 127 36 6
    388927 151264211329 25921 161 36 6
    391339 153146212921 25921 161 36 6
    426869 182217143161 16129 127 36 6
    462511 213916425121 25921 161 36 6
    558046 311415338116 25921 161 36 6
    561572 315363111184 25921 161 36 6
    567046 321541166116 25921 161 36 6
    575515 331217515225 63001 251 36 6
    584954 342171182116 16129 127 36 6
    721882 521113621924 25921 161 36 6
    788761 622143915121 25921 161 36 6
    901189 812141613721 16129 127 36 6

    pipo

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