

A355713


Numbers k such that k and k+1 have the same sum of 5smooth divisors.


1



175, 2224, 2575, 4975, 7024, 9424, 9775, 11824, 12175, 14224, 14575, 16975, 19024, 21424, 21775, 23824, 24175, 26224, 26575, 28975, 31024, 33424, 33775, 35824, 36175, 38224, 38575, 40975, 43024, 45424, 45775, 47824, 48175, 50224, 50575, 52975, 55024, 57424, 57775
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OFFSET

1,1


COMMENTS

Equivalently, numbers k such that the largest 5smooth divisors of k and k+1, A355582(k) and A355582(k+1), have the same sum of divisors (A000203).
For all the terms k, both k and k+1 are not squarefree: each of the two largest 5smooth divisors, of k and k+1, cannot be squarefree, since the squarefree 5smooth numbers are the divisors of 30 = 2*3*5 (A018255) whose values of sigma (A000203), {1, 3, 4, 6, 12, 18, 24, 72}, are not shared with sigma of any other 5smooth number.
Apparently, all the terms are of only two types: numbers k such that A355582(k) = 16 and A355582(k+1) = 25, or numbers k such that A355582(k) = 25 and A355582(k+1) = 16. Both types are infinite sequences: The first type is the sequence of numbers of the form 2224 + 2400*m, where m is not congruent to 1 (mod 5), and the second type is the sequence of numbers of the form 175 + 2400*m, where m is not congruent to 3 (mod 5). If there are no other terms, then this sequence is a linear recurrence with a signature (1,0,0,0,0,0,0,1,1). The question of the existence of other types is equivalent to the question of the existence of two coprime 5smooth numbers other than 16 and 25 whose sums of divisors are equal.
Are there runs of 3 consecutive numbers with the same sum of 5smooth divisors? There are no such runs below 5*10^10.


LINKS



EXAMPLE



MATHEMATICA

f[p_, e_] := If[p > 5, 1, (p^(e + 1)  1)/(p  1)]; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^5], s[#] == s[# + 1] &]


PROG

(PARI) s(n) = (2^(valuation(n, 2) + 1)  1) * (3^(valuation(n, 3) + 1)  1) * (5^(valuation(n, 5) + 1)  1) / 8;
s1 = s(1); for(k = 2, 6e4, s2 = s(k); if(s1 == s2, print1(k1, ", ")); s1 = s2);


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



