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WON plate
217 |

[ May 1, 2022 ]
Operations with nine- & pandigital factors creating palindromes
Alexandru Petrescu

Find all ninedigital/pandigital numbers having the following property :
Both sum and product of the prime factors (without multiplicity) are palindromic.

There are 8 solutions for the pandigitals and only one for the ninedigitals.
Note the two pandigital duplicates (#2 & #6) and (#7 & #8).

#NinedigitalSumProduct
17498135262+3+7+11+17+263 = 3032*3*7*11*17*263 = 2065602

#PandigitalSumProduct
118549062373+37+283 = 3233*37*283 = 31413
219578604322+3+11+499+2477 = 29922*3*11*499*2477 = 81577518
325893610473+7+13+37+313 = 3733*7*13*37*313 = 3161613
429438165073+43+61+197+211 = 5153*43*61*197*211 = 327090723
531695402873+37+101+283 = 4243*37*101*283 = 3172713
639157208642+3+11+499+2477 = 29922*3*11*499*2477 = 81577518
743578120962+3+11+17+613 = 6462*3*11*17*613 = 687786
871309652482+3+11+17+613 = 6462*3*11*17*613 = 687786

Here is some Pari/gp code to reproduce Alexandru's results.

```
{
for(n=1,10!, d=numtoperm(10, n+10!-1);
a=sum(i=1, #d, (d[i]-1)*10^(#d-i));
f=factor(a); sf=0; mf=1;
for(j=1,#f~, sf+=f[j,1]; mf*=f[j,1]);
dsf=digits(sf); dmf=digits(mf);
if(Vecrev(dsf)==dsf && Vecrev(dmf)==dmf,
print(a," ",f," sum = ",sf,", product = ",mf));
);
}

```

The pandigital 4357812096 (#7) appears also in wonplate 26.
It is the product of these two palindromes :
69696 * 62526 = 4357812096

A000217 Prime Curios! Prime Puzzle
Wikipedia 217 Le nombre 217
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