A344826 Integers k such that k/A097621(k) is an integer.

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60, 891, 1584, 1782, 3564, 4032, 4455, 4752, 7920, 8910, 17820, 20160, 22275, 23760, 44550, 49896, 86400, 89100, 100800, 118800, 249480, 349272, 399168, 694008, 1097712, 1746360, 1778400, 1995840, 2181168, 2774304, 2794176, 3470040

Offset: 1

Author

Michel Marcus, May 29 2021, after a suggestion from Allan C. Wechsler

Keywords

• nonn

Comments

Allan C. Wechsler remarks that one can derive larger terms from existing terms. For instance, k = 5552064 has q = k/A097621(k) = 18. So multiplying 5552064 by 31 = A000961(18) will give a new term with q = 31.

More precisely, if k = a(n) has q = A343886(k) and m = A000961(q) such that gcd(k, m) = 1, then k*m is also a term. We could call "primitive" those terms not derived from a smaller term in this way. All the listed terms are primitive, but a({35, 36, 38, 42, 43}) allow the sequence to be extended to five larger non-primitive terms. The second and fourth one, having q = 17 resp. q = 23, both lead to a whole chain of many new terms. - M. F. Hasler, Jun 15 2021

Links

• Ray Chandler, Table of n, a(n) for n = 1..187 (terms 1..90 from Michel Marcus, terms 91..133 from Lars Blomberg)

Crossrefs

Cf. A000961, A095874, A097621, A127724, A343886 (the ratios k/A097621(k)).

Programs

PARI

f(n) = if(isprimepower(n), sum(i=1, logint(n, 2), primepi(sqrtnint(n, i)))+1, n==1); \\ A095874
ff(n) = my(fr=factor(n)); for (k=1, #fr~, fr[k,1] = f(fr[k,1]^fr[k,2]); fr[k,2] = 1); factorback(fr); \\ A097621
isok(k) = denominator(k/ff(k)) == 1;

PARI

mappp(nn) = {my(map = Map()); mapput(map, 1, 1); my(nb=1); for (n=2, nn, if (isprimepower(n), nb++; mapput(map, n, nb));); map;}
ff(n, map) = my(fr=factor(n)); for (k=1, #fr~, fr[k, 1] = mapget(map, fr[k, 1]^fr[k, 2]); fr[k, 2] = 1); factorback(fr); \\ A097621
wa(na, nb) = {my(map = mappp(nb)); for (k=na, nb, if (denominator(k/ff(k, map)) == 1, print1(k, ", ")););}
wa(1, 10^8)

PARI

is_A344826(n)=!(n%A097621(n))
extend(n)=n*if(gcd(n, n=A000961(n/A097621(n)))==1,n) \\ Return the larger non-primitive term "derived" from a term n = a(k) with gcd(n,q) = 1, cf. COMMENTS, or zero if gcd(n,q) > 1, i.e., it cannot be "extended" that way. This allows the production of (infinitely?) many new terms from the existing ones. - M. F. Hasler, Jun 15 2021