D-POWERFUL NUMBERS

The D-powerful numbers are those that can be expressed as the sum of their individual digits raised to positive powers. For example, today I am 25430 days old and this can be written as \( 2^{13} + 5^3 + 4^7 + 3^6 + 0 \) where the 0 of course can be raised to any positive power. Whenever a d-powerful number ends in a 0, it follows that all the subsequent numbers in that decade will also be d-powerful. Thus:

\( 25431 = 2^{13} + 5^3 + 4^7 + 3^6 + 1 \)

\(25432 = 2^{13} + 5^3 + 4^7 + 3^6 + 2 \)

\(25433 = 2^{13} + 5^3 + 4^7 + 3^6 + 3 \)

\(25434 = 2^{13} + 5^3 + 4^7 + 3^6 + 4 \)

\(25435 = 2^{13} + 5^3 + 4^7 + 3^6 + 5 \)

\(25436 = 2^{13} + 5^3 + 4^7 + 3^6 + 6 \)

\(25437 = 2^{13} + 5^3 + 4^7 + 3^6 + 7 \)

\(25438 = 2^{13} + 5^3 + 4^7 + 3^6 + 8 \)

\(25439 = 2^{13} + 5^3 + 4^7 + 3^6 + 9 \)

The occurrences of such decades are relatively frequent. There are 600 d-powerful numbers in the range from 1 to 17463 inclusive. This represents a relative frequency of a little over 3.4%. D-powerful numbers can be written in more than one way. For example, 25432 can be written in five different ways:

\(25432 = 2^{13} + 5^3 + 4^7 + 3^6 + 2 \)

\(25432 = 2^{12} + 5^4 + 4^5 + 3^9 + 2^2 \)

\(25432 = 2^{10} + 5^4 + 4^6 + 3^9 + 2^2 \)

\(25432 = 2^{11} + 5^5 + 4^3 + 3^9 + 2^9 \)

\(25432 = 2^{12} + 5^4 + 4 + 3^9 + 2^{10} \)

Similarly, 25437 can be written in two ways:

\(25437 = 2^{13} + 5^3 + 4^7 + 3^6+ 7\)

\(25437 = 2^{11} + 5 + 4^2 +3^8 + 7^5\)

Finally, 25439 can be written in two ways:

\(25439 = 2^{13} + 5^3 + 4^7 + 3^6 +9 \)

\(25439 = 2^{13} + 5^3 + 4^7 +3^2 + 3^3 \)

OEIS A007532 refers to d-powerful numbers as simply powerful numbers and lists the following initial member of the sequence:

1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 43, 63, 89, 132, 135, 153, 175, 209, 224, 226, 262, 264, 267, 283, 332, 333, 334, 357, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 407, 445, 463, 518, 598, 629, 739, 794, 849, 935, 994, 1034

So I have another nine d-powerful days to go. This website gives some examples of interesting d-powerful numbers with additional special properties. For example, consider the case of d-powerful palindromic primes. There is only one such 3 digit number, 373, and it has two representations:

\(373 = 3^1 + 7^3 + 3^3 \)

\(373 = 3^4 + 7^2 + 3^5 \)

There is only one 5 digit d-powerful palindromic prime and that is 98389:

\(98389 = 9^4 + 8^1 + 3^1 + 8^5 + 9^5 \)

Another category are d-powerful numbers that a palindromic numbers and have a representation where the exponents are symmetric. The first such numbers are:

\(262 = 2^7 + 6^1 + 2^7 \)

\(4224 = 4^3 + 2^{11} + 2^{11} + 4^3 \)

\(39393 = 3^9 + 9^1 + 3^2 + 9^1 + 3^9 \)

\(79597 = 7^1 + 9^3 + 5^7 + 9^3 + 7^1 \)

The smallest d-powerful, pandigital number is 1023456879 with representation:

\(1023456879 = 1^1 + 0^1 + 2^{27} + 3^{18} + 4^6 + 5^1 + 6^{11} + 8^9 + 7^5 + 9^7\)

Here is the SageMath code to determine the indices of a number if it is d-powerful (25430 is used as the test number but any number can be substituted for it, provided it has five digits: