A336561 -id:A336561 - cp's OEIS frontend

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, 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

Offset: 1

Antti Karttunen, Jul 25 2020

nonn

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

			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.

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

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.

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

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

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

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

Original entry on oeis.org

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

Antti Karttunen, Jul 25 2020

nonn

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.

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

Cf. A051027, A335915, A336456.
Setwise difference of A336557 and A336547. Equally, setwise difference of A336559 and A336549. Subsequence of A336548.
Cf. also A336561.

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));

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)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 08 2022

nonn

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

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

Index entries for sequences related to sigma(n)

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

Cf. also A336561.

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)));

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.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 25 2020

nonn

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.

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)

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

Cf. also A336561 (positions where this appears to be multiplicative but A051027 does not).

A065330(n) = (n>>valuation(n, 2)/3^valuation(n, 3));
A336459(n) = A065330(sigma(sigma(n)));

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

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

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)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula