A336560 Numbers k at which points A336456(k) appears multiplicative, but A051027(k) does not.

15, 39, 51, 60, 78, 87, 95, 111, 123, 143, 159, 183, 204, 215, 219, 222, 231, 240, 247, 267, 291, 303, 312, 323, 327, 330, 335, 339, 348, 366, 380, 399, 407, 411, 438, 444, 447, 455, 471, 494, 506, 519, 543, 559, 579, 582, 591, 624, 636, 654, 671, 687, 695, 699, 703, 714, 723, 731, 732, 767, 771, 779, 798, 803, 807

Offset: 1

Author

Antti Karttunen, Jul 25 2020

Keywords

• nonn

Comments

Numbers in A336557 but not in A336547.

Note that if A051027(k) = Product_{p^e|k} A051027(p^e) then also A336456(n) = Product_{p^e|n} A336456(p^e), because A336456(n) = A335915(A051027(n)) and A335915 is fully multiplicative, thus A336547 is a subsequence of A336557.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..25000

Crossrefs

Cf. A051027, A335915, A336456.

Setwise difference of A336557 and A336547. Equally, setwise difference of A336559 and A336549. Subsequence of A336548.

Cf. also A336561.

Programs

PARI

is_fun_mult_on_n(fun,n) = { my(f=factor(n)); prod(k=1,#f~,fun(f[k,1]^f[k,2]))==fun(n); };
A051027(n) = sigma(sigma(n));
A000265(n) = (n>>valuation(n,2));
A335915(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265((f[k,1]^2)-1)^f[k,2]))); };
A336456(n) = A335915(A051027(n));
A336546(n) = is_fun_mult_on_n(A051027,n);
A336556(n) = is_fun_mult_on_n(A336456,n);
isA336560(n) = (A336546(n)<A336556(n));