A336561 -id:A336561

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A336548

Numbers k such that at least one pair sigma(p_i^e_i), sigma(p_j^e_j) [with i != j] share a prime factor, when k = p_1^e_1 * ... * p_h^e_h, where each p_i^e_i is the maximal power of prime p_i dividing k.

+10
16

10, 15, 21, 22, 30, 33, 34, 35, 39, 40, 42, 46, 51, 52, 55, 57, 58, 60, 65, 66, 69, 70, 77, 78, 82, 84, 85, 87, 88, 90, 91, 93, 94, 95, 98, 102, 105, 106, 110, 111, 114, 115, 118, 119, 120, 123, 129, 130, 132, 133, 135, 136, 138, 140, 141, 142, 143, 145, 152, 154, 155, 156, 159, 160, 161, 164, 165, 166, 168, 170

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Numbers k for which A353802(k) = Product_{p^e||k} A051027(p^e) > A051027(k), i.e. numbers at which points A051027 is not multiplicative. The notation p^e||k means that p^e divides k, but p^(1+e) does not.

If x is present, then also multiples y*x are present for all y for which gcd(x,y) = 1.

Also numbers at which points A062401 and A353750 are not multiplicative. - Antti Karttunen, May 09 2022

LINKS

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

FORMULA

{k | A336562(k) > 0}. - Antti Karttunen, May 09 2022

EXAMPLE

10 = 2*5 is present as sigma(2) = 3 and sigma(5) = 6, and 3 and 6 share a prime factor (gcd(3,6) = 3). Also we see that sigma(sigma(2))*sigma(sigma(5)) = 4*12 = 48 > sigma(sigma(10)) = 39.

PROG

(PARI) isA336548(n) = !A336546(n);

CROSSREFS

Cf. A051027, A062401, A336546, A336547 (complement), A336549, A353750, A353802.

Cf. A336357, A336558, A336560, A336561, A353807 (subsequences).

Positions of nonzero terms in A336562, in A353753 and in A353803.

Positions of terms larger than 1 in A353755, in A353784 and in A353806.

Subsequence of A024619.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jul 25 2020

EXTENSIONS

The old definition moved to comments and replaced with a more generic, but equivalent definition by Antti Karttunen, May 09 2022

STATUS

approved

A336560

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

+10
8

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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

PROG

(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));

CROSSREFS

Cf. A051027, A335915, A336456.

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

Cf. also A336561.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jul 25 2020

STATUS

approved

A353807

Numbers k such that A353802(k) / sigma(sigma(k)) is an integer > 1, where A353802(n) = Product_{p^e||n} sigma(sigma(p^e)).

+10
3

1819, 5088, 7215, 7276, 9487, 9523, 11895, 13303, 14235, 16371, 20179, 21079, 21255, 24531, 24751, 24931, 25824, 29104, 30615, 32224, 33855, 36199, 37635, 37948, 38092, 38664, 40443, 40515, 41847, 43831, 44655, 45475, 45695, 45883, 46995, 48043, 48399, 53835, 54015, 54568, 55747, 56899, 56928, 59599, 60495, 61035

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Numbers k such that A353805(k) = 1, but A353806(k) > 1.

LINKS

Table of n, a(n) for n=1..46.

Index entries for sequences related to sigma(n)

EXAMPLE

A353802(1819) = 10920 = 2*A051027(1819) = 2*5460, therefore 1819 is included as a term.

PROG

(PARI)

A051027(n) = sigma(sigma(n));

A353805(n) = { my(f = factor(n)); (A051027(n) / gcd(A051027(n), prod(k=1, #f~, A051027(f[k, 1]^f[k, 2])))); };

A353806(n) = { my(f = factor(n), u=prod(k=1, #f~, A051027(f[k, 1]^f[k, 2]))); (u / gcd(A051027(n), u)); };

isA353807(n) = ((1==A353805(n)) && (1!=A353806(n)));

CROSSREFS

Cf. A000203, A051027, A353802, A353805, A353806.

Cf. also A336561.

KEYWORD

nonn

AUTHOR

Antti Karttunen, May 08 2022

STATUS

approved

A336459

a(n) = A065330(sigma(sigma(n))), where A065330 is fully multiplicative with a(2) = a(3) = 1, and a(p) = p for primes p > 3.

+10
2

1, 1, 7, 1, 1, 7, 5, 1, 7, 13, 7, 7, 1, 5, 5, 1, 13, 7, 7, 1, 7, 91, 5, 7, 1, 1, 5, 5, 1, 65, 7, 13, 31, 5, 31, 7, 5, 7, 5, 13, 1, 7, 7, 7, 7, 65, 31, 7, 5, 1, 65, 19, 5, 5, 65, 5, 31, 13, 7, 5, 1, 7, 35, 1, 7, 403, 7, 13, 7, 403, 65, 7, 19, 5, 7, 7, 7, 5, 31, 1, 133, 13, 7, 7, 35, 7, 5, 91, 13, 91, 31, 5, 85, 403

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Sequence removes prime factors 2 and 3 from the prime factorization A051027(n) = sigma(sigma(n)).

Like A051027, neither this is multiplicative. For example, we have a(3) = 7, a(7) = 5, but a(21) = 7 <> 35. However, for example, a(10) = 13, and a(3*10) = a(3)*a(10) = 65.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Index entries for sequences related to sigma(n)

FORMULA

a(n) = A336457(A000203(n)) = A065330(A051027(n)).

PROG

(PARI)

A065330(n) = (n>>valuation(n, 2)/3^valuation(n, 3));

A336459(n) = A065330(sigma(sigma(n)));

CROSSREFS

Cf. A000203, A051027, A065330, A336456 (similar sequence), A336457.