numbers of yet another particular type

[enzocreti]

Consider numbers N squarefree that is in the factorisation of N there is no prime facor raised to a power greater than 1.




56238 is square free


56238 has five digits


56238 has five prime factors




is there something in Oeis?


I mean square free numbers N such that the number of digits of N equals the number of prime factors of N?

[CRGreathouse]

A165256 is similar, but it contains 3460 terms with square factors in addition to the 4352 squarefree terms.

[CRGreathouse]

There are 10 terms with 10 prime factors:
6469693230, 6915878970, 8254436190, 8720021310, 9146807670, 9592993410

The largest prime factor among the terms is 467: 98070 = 2 * 3 * 5 * 7 * 467.

[enzocreti]

Pg(56238) is prime and also pg(75894) is prime

56238 and 75894 are multiple of 546.

56238 and 75894 are square free and belong to the Oeis sequence you mentioned


Pg(n) is the concatenation in base ten of 2^n-1 and 2^(n-1)-1

SO BECAuse the sequence oeis is finite, then pg(n) primes with n multiple of 546 should be not infinite

[enzocreti]

Originally Posted by CRGreathouse

A165256 is similar, but it contains 3460 terms with square factors in addition to the 4352 squarefree terms.

Are 56238 and 75894 the only multiple of 546 squarefree belonging to the sequence A165256 ?

[CRGreathouse]

Originally Posted by enzocreti

Are 56238 and 75894 the only multiple of 546 squarefree belonging to the sequence A165256 ?

There are 907 squarefree multiples of 546 in A165256. 10374 is the first and 9592993410 is the last.

[enzocreti]

Originally Posted by CRGreathouse

There are 907 squarefree multiples of 546 in A165256. 10374 is the first and 9592993410 is the last.

56238 is a not palindromic number such that the reverse 83265 (squarefree) also belongs to Oeis sequence A165256. Are there other examples of not palindromic numbers belonging to A165256 whose reverse also belongs to A165256? Are there other squarefree non palindromic numbers belonging to A165256 whose reverse is squarefree and belongs to A165256?

[Dr Sardonicus]

Originally Posted by enzocreti

56238 is a not palindromic number such that the reverse 83265 (squarefree) also belongs to Oeis sequence A165256. Are there other examples of not palindromic numbers belonging to A165256 whose reverse also belongs to A165256? Are there other squarefree non palindromic numbers belonging to A165256 whose reverse is squarefree and belongs to A165256?

Hey there, lazybones!

Using the plain-text exhaustive list provided at the OEIS page, I figured even a dunce at programming like me can tell Pari-GP to extract the answers. I tried it for practice. Pari-GP, in turn, yawned and rolled its eyes at being given such a trivial task, but delivered the results.

If my code was writ right, the list of the smaller of each such pair in the sequence, whether either number in the pair is square free or not, is

[12, 15, 26, 28, 36, 39, 45, 56, 57, 58, 68, 69, 132, 156, 165, 168, 204, 228, 246, 255, 258, 273, 276, 285, 286, 294, 366, 396, 408, 418, 426, 435, 438, 456, 465, 495, 498, 516, 528, 558, 588, 609, 618, 627, 638, 678, 759, 819, 1518, 2046, 2145, 2226, 2244, 2262, 2418, 2436, 2478, 2508, 2562, 2618, 2706, 2805, 2814, 2838, 2886, 2964, 3135, 3876, 3927, 4026, 4158, 4386, 4389, 4488, 4686, 4746, 4785, 4788, 4818, 4836, 4935, 4956, 5016, 5148, 5406, 5478, 5565, 5628, 5676, 5838, 5916, 5928, 6018, 6138, 6258, 6438, 6468, 6486, 6699, 6798, 7458, 8148, 8568, 15015, 23205, 24024, 24486, 24882, 26598, 26796, 27258, 42315, 45045, 45318, 48048, 54978, 55146, 56238, 57057, 58058, 58926, 59466, 60918, 62238, 64428, 66198, 68068, 68838, 69069, 78078, 80178, 82698, 88179, 204204, 228228, 246246, 255255, 258258, 285285, 408408, 435435, 438438, 456456, 465465, 491946, 498498, 516516, 558558, 585858, 618618, 678678, 686868]

The numbers in the preceding list for which both it and its reversal are square free are

[15, 26, 39, 58, 165, 246, 285, 286, 366, 418, 435, 438, 498, 609, 759, 1518, 2046, 2226, 2262, 2418, 2478, 2618, 2814, 2838, 2886, 3135, 3927, 4386, 4389, 4746, 4785, 4935, 5406, 5478, 5565, 5838, 6018, 6438, 6486, 6699, 7458, 15015, 24486, 24882, 26598, 45318, 55146, 56238, 58058, 58926, 59466, 60918, 62238, 66198, 68838, 246246, 285285, 435435, 438438, 491946, 498498, 585858]

[Batalov]

You guys are so easy to "snipe" (see this https://xkcd.com/356/ ).

When you deal with any enzocreti's posts, you have to remeber just 2-3 numbers (one of them "546"), and once you see any "science like" questions from him, always ask yourself "is this yet another inane attempt to predict the next member of his 'precious-s-s...' using small known members of the sequence, some of which happened to be multiples of 546?" If yes, simply check the tightness of the esteemed patient's garments and leave him alone.

[LaurV]

Originally Posted by Batalov

(see this https://xkcd.com/356/ ).

Reply to that: (from the "don't turn it on, take it apart!" guy)

[Batalov]

Don't feel bad -- I am easy to snipe too.

And I will dispense the single reason why periodicity easily enters the pg(n) sequence, and why it is apparent that it is a totally irrelevant distraction.

pg(k)=(2^k-1)*10^d+2^(k-1)-1, where d is the number of decimal digits of 2^(k-1)-1

pg(k)=(2^k-1)*10^ceil(L2*(k-1))+2^(k-1)-1, where L2 = log₁₀ 2

For an initial stretch of small k<104, one can write
pg_1(k)=(2^k-1)*10^((3*k+7)\10)+2^(k-1)-1
and for all PRP-testable testable numbers you can definitely write (...or something like that, I am typing on the fly. I will fix this later)
pg_2(k)=(2^k-1)*10^((3*k+7)\10+k\1000+k\33338)+2^(k-1)-1

pg_2(k) = pg(k) for all k<10^8 ...and then it isn't.
But you cannot test there for primality anyway, so one can spend their life mucking with "properties" of pg_2(k) (which are obvious after spending 10-15 minutes, modulos will be surely periodic and that will sieve out any possible candidates and the primes and PRPs will sit on invisible strings) - but it has nothing to do with "patterns" in real pg(k), which there will be none when k tends to infinity.

...And all talk (for a year, or is it more?) about "x less than a palindrome", "divisible by 546"... all vapors of a brain with high temperature.
"Vanity of vanities, it is all vanity." (Eccl. 1:2)