A379217 -id:A379217 - cp's OEIS frontend

A326134 Numbers k such that A326057(k) is equal to A252748(k) and A252748(k) is not 1.

6, 28, 69, 91, 496, 2211, 4825, 8128, 12639, 22799825, 33550336, 60406599, 68258725, 569173299, 794579511, 984210266, 2830283326, 8589869056, 10759889913, 80295059913, 85871289682

Offset: 1

Antti Karttunen, Jun 11 2019

No other terms below 3221225472.
Numbers k such that A252748(k) [= A003961(k) - 2*k] <> 1 (i.e., k is not in A348514), and A286385(k) [= A003961(k) - A000203(k)] = m*A252748(k) for some positive integer m. Note that this entails that k is nonabundant (A000203(k) <= 2*k) and primeshift-abundant (A252748(k) > 2), thus this is a subsequence of A341614. - revised Dec 13 2024
This is a subsequence of A378980, see further comments there. - Antti Karttunen, Dec 13 2024

			28 is a term as A252748(28) = 43 > 1 and A286385(28) = 43, which is a multiple of 43.
69 is a term as A252748(69) = 7 > 1 and A286385(69) = 49 is a multiple of 7.
91 is a term as A252748(91) = 5 > 1 and A286385(91) = 75 is a multiple of 5.

Cf. A000396 (subsequence), A003961, A252748, A286385, A326057, A337345, A337372, A341611, A348514, A379216, A379217.
Subsequence of the following sequences: A246282, A341614, A378980.
Odd terms form a subsequence of A349753.

Select[Range[10^5], And[#3 - #1 != 1, GCD[#3 - #1, #3 - #2] == #3 - #1] & @@ {2 #, DivisorSigma[1, #], Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1]} &] (* Michael De Vlieger, Feb 22 2021 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
isA326134(n) = { my(s = A003961(n), t = (s-(2*n)), u = s-sigma(n)); ((1!=t)&&!(u%t)&&((u/t)>0)); };

a(18) from Antti Karttunen, Dec 14 2024

a(19)..a(21) from Antti Karttunen (from the b-file of A378980 computed by Amiram Eldar), Dec 20 2024

A379216 Difference 2*k - A003961(k) computed for k for which this difference divides difference (A003961(k)-sigma(k)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

Original entry on oeis.org

1, 1, 1, -1, -3, 3, -1, 1, 1, -43, 1, 5, 19, -1, -7, -5, 1, -2005, 1, -1, 149, -193, -1, -3, -79243, 1243, 1253, -7, 51, 581, -1, 3093, 1, 155491, 919, 1, -1, 15833, -877, -4295498497, 5129369, 31, 5779339, -69187, -29, 6745, 1, 181, 1, 69197, -397, -117433, -101, -1, 1, 2759, 1, -29479, 1, -5626288431709, 29669, -1, -132239, -1, -1, 14591, -2267959, -3187, 787250461

Offset: 1

Antti Karttunen, Dec 20 2024

sign

Among the initial 69 terms, there are eleven +1's and eleven -1's. The former correspond in A378980 with those of its terms that are in A048674 (1, 2, 3, 25, 26, 33, 93, 1034, ...), while the latter here correspond in A378980 with those of its terms that are in A348514 (4, 10, 57, 1054, 2626, ...).

Cf. A003961, A048674, A252748, A348514, A378980, A378981, A379217.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A378981(n) = ((A003961(n)-sigma(n))%((2*n)-A003961(n)));
is_A378980(n) = !A378981(n);
for(n=1,2^25,if(is_A378980(n),print1((2*n)-A003961(n),", ")));

a(n) = -A252748(A378980(n)).

cp's OEIS Frontend

A326134 Numbers k such that A326057(k) is equal to A252748(k) and A252748(k) is not 1.

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