A379216 -id:A379216 - cp's OEIS frontend

A048674 Fixed points of A048673 and A064216: Numbers n such that if n = product_{k >= 1} (p_k)^(c_k), then Product_{k >= 1} (p_{k+1})^(c_k) = (2*n)-1, where p_k indicates the k-th prime, A000040(k).

1, 2, 3, 25, 26, 33, 93, 1034, 970225, 8550146, 325422273, 414690595, 1864797542, 2438037206

Offset: 1

Antti Karttunen, Jul 14 1999

Equally: after 1, numbers n such that, if the prime factorization of 2n-1 = Product_{k >= 1} (p_k)^(c_k) then Product_{k >= 1} (p_{k-1})^(c_k) = n.
Factorization of the initial terms: 1, 2, 3, 5^2, 2*13, 3*11, 3*31, 2*11*47, 5^2*197^2, 2*11*47*8269, 3*11*797*12373, 5*11^2*433*1583, 2*23*59*101*6803, 2*11*53*1201*1741.
The only 3-cycle of permutation A048673 in range 1 .. 402653184 is (2821 3460 5639).
For 2-cycles, take setwise difference of A245449 and this sequence.
Numbers k for which A336853(k) = k-1. - Antti Karttunen, Nov 26 2021

			25 is present, as 2*25 - 1 = 49 = p_4^2, and p_3^2 = 5*5 = 25.
26 is present, as 2*26 - 1 = 51 = 3*17 = p_2 * p_8, and p_1 * p_7 = 2*13 = 26.
Alternatively, as 26 = 2*13 = p_1 * p_7, and ((p_2 * p_8)+1)/2 = ((3*17)+1)/2 = 26 also, thus 26 is present.

Fixed points of permutation pair A048673/A064216.
Positions of zeros in A349573.
Subsequence of the following sequences: A245449, A269860, A319630, A349622, A378980 (see also A379216).
This sequence is also obtained as a setwise difference of the following pairs of sequences: A246281 \ A246351, A246352 \ A246282, A246361 \ A246371, A246372 \ A246362.
Cf. A000040, A003961 (A045965), A064989, A285701, A336853.
Cf. also A348514 (fixed points of map A108228, similar to A048673).

A048673 := n -> (A003961(n)+1)/2;
A048674list := proc(upto_n) local b,i; b := [ ]; for i from 1 to upto_n do if(A048673(i) = i) then b := [ op(b), i ]; fi; od: RETURN(b); end;

Join[{1}, Reap[For[n = 1, n < 10^7, n++, ff = FactorInteger[n]; If[Times @@ Power @@@ (NextPrime[ff[[All, 1]]]^ff[[All, 2]]) == 2 n - 1, Print[n]; Sow[n]]]][[2, 1]]] (* Jean-François Alcover, Mar 04 2016 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA048674(n) = ((n+n)==(1+A003961(n))); \\ Antti Karttunen, Nov 26 2021

Entry revised and the names in Maple-code cleaned by Antti Karttunen, Aug 25 2014

Terms a(11) - a(14) added by Antti Karttunen, Sep 11-13 2014

A348514 Numbers k for which A003961(k) = 2k+1, where A003961 shifts the prime factorization of n one step towards larger primes.

Original entry on oeis.org

4, 10, 57, 1054, 2626, 68727, 12371554, 1673018314, 10475647197, 11154517557, 27594844918, 630178495917, 7239182861878

Offset: 1

Antti Karttunen, Oct 29 2021

Numbers k such that A064216(1+k) = k.
It seems that after 4, all other terms are squarefree. See conjecture in A348511.
a(9)..a(13) <= 10475647197, 11154517557, 27594844918, 630178495917, 7239182861878, which are also terms. - David A. Corneth, Oct 30 2021

Index entries for sequences computed from indices in prime factorization

Fixed points of map A108228. (Compare to A048674).
Positions of ones in A252748.
Subsequence of the following sequences: A246282, A319630, A348511, A378980 (see also A379216), A387411, A387414.
Cf. A000230, A003961, A064216, A326134.

f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^5], s[#] == 2*# + 1 &] (* Amiram Eldar, Oct 30 2021 *)

a(9)-a(11) verified by Amiram Eldar, Nov 01 2021

a(12)-a(13) verified by Martin Ehrenstein, Nov 08 2021

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

Original entry on oeis.org

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

A379217 Quotient (A003961(k)-sigma(k)) / (2*k-A003961(k)) computed for those k for which this quotient is an integer, where A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

Original entry on oeis.org

0, 0, 1, -2, -1, 1, -3, 18, 9, -1, 17, 3, 1, -35, -7, -15, 57, -1, 339, -381, 3, -7, -969, -1213, -1, 3, 3, -979, 419, 29, -42735, 21, 731232, 3, 1445, 2809731, -4566981, 557, -19691, -1, 5, 544371, 5, -475, -1784691, 9051, 176870849, 808683, 280791301, 1803, -891775, -3679, -3733533, -444406677, 731480523, 275091

Offset: 1

Antti Karttunen, Dec 20 2024

sign

Terms in A378980 that correspond here with -1's are perfect numbers (A000396).

Antti Karttunen, Table of n, a(n) for n = 1..69 (based on b-file of A378980 computed by _Amiram Eldar_)
Index entries for sequences related to prime indices in the factorization of n.
Index entries for sequences related to sigma(n).

Cf. A000396, A003961, A252748, A378980, A378981, A379216.

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(((A003961(n)-sigma(n))/((2*n)-A003961(n))),", ")));

a(n) = A286385(A378980(n)) / A379216(n) = A286385(A378980(n)) / -A252748(A378980(n)).

cp's OEIS Frontend

A048674 Fixed points of A048673 and A064216: Numbers n such that if n = product_{k >= 1} (p_k)^(c_k), then Product_{k >= 1} (p_{k+1})^(c_k) = (2*n)-1, where p_k indicates the k-th prime, A000040(k).

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A348514 Numbers k for which A003961(k) = 2k+1, where A003961 shifts the prime factorization of n one step towards larger primes.

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

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A379217 Quotient (A003961(k)-sigma(k)) / (2*k-A003961(k)) computed for those k for which this quotient is an integer, where A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

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