A064989 -id:A064989 - cp's OEIS frontend

A349164 a(n) = A064989(A003961(n) / gcd(sigma(n), A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes, while A064989 shifts it back towards smaller primes, and sigma is the sum of divisors function.

1, 1, 3, 4, 5, 3, 7, 4, 9, 5, 11, 12, 13, 7, 15, 16, 17, 9, 19, 2, 21, 11, 23, 4, 25, 13, 9, 28, 29, 15, 31, 8, 33, 17, 35, 36, 37, 19, 39, 10, 41, 21, 43, 22, 45, 23, 47, 48, 49, 25, 51, 52, 53, 9, 55, 28, 19, 29, 59, 6, 61, 31, 63, 64, 13, 33, 67, 17, 69, 35, 71, 12, 73, 37, 75, 76, 77, 39, 79, 40, 81, 41, 83, 84

Offset: 1

Antti Karttunen, Nov 09 2021

nonn

Cf. A000203, A003961, A319630, A326042, A336702, A342671, A348993, A349161, A349162, A349169, A349174.

Cf. A349144 and A349168 [positions where a(n) is / is not relatively prime with A349163(n) = n/a(n)].

Array[Times @@ Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger[#2/GCD[##]]] & @@ {DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &, 84] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(f=factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A349164(n) = { my(u=A003961(n)); A064989(u/gcd(u,sigma(n))); };

a(n) = A064989(A349161(n)).

a(n) = n / A349163(n).

A244319 Self-inverse permutation of natural numbers: a(1) = 1, a(2n) = A003961(1+a(A064989(2n-1))), a(2n+1) = 1+A003961(a(A064989(2n+1)-1)).

Original entry on oeis.org

1, 3, 2, 9, 6, 5, 26, 11, 4, 21, 8, 125, 56, 25, 16, 15, 344, 115, 36, 1015, 10, 39, 204, 41, 14, 7, 52, 45, 86, 301, 176, 155, 298, 51, 50, 19, 518, 305, 22, 189, 24, 895, 1376, 49, 28, 825, 1268, 11875, 44, 35, 34, 27, 3186, 6625, 2388, 13, 454, 153, 126, 3191, 476, 131

Offset: 1

Antti Karttunen, Jul 18 2014; description corrected and PARI code added Jul 30 2014

nonn

After 1, maps each even number to a unique odd number and vice versa, i.e., for all n > 1, A000035(a(n)) XOR A000035(n) = 1, where XOR is given in A003987.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A003987, A003961, A064989, A243501.
Related permutations: A048673, A064216, A245609-A245610.
Similar entanglement permutations: A245605-A245606, A235491, A236854, A243347, A244152.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ This function from Michel Marcus
A064989(n) = my(f = factor(n)); for(i=1, #f~, if((2 == f[i,1]),f[i,1] = 1,f[i,1] = precprime(f[i,1]-1))); factorback(f);
A244319(n) = if(1==n, 1, if(0==(n%2), A003961(1+A244319(A064989(n-1))), 1+A003961(A244319(A064989(n)-1))));
for(n=1, 10001, write("b244319.txt", n, " ", A244319(n)))
(Scheme, with Antti Karttunen's IntSeq-library for memoizing definec-macro)
(definec (A244319 n) (cond ((= 1 n) 1) ((even? n) (A003961 (+ 1 (A244319 (A064989 (- n 1)))))) (else (A243501 (A244319 (-1+ (A064989 n)))))))

a(1) = 1, a(2n) = A003961(1+a(A064989(2n-1))), a(2n+1) = A243501(a(A064989(2n+1)-1)).
As a composition of related permutations:
a(n) = A245609(A048673(n)) = A064216(A245610(n)).

A329044 a(n) = A064989(A324886(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 7, 6, 15, 25, 11, 81, 13, 49, 15625, 36, 17, 225, 19, 625, 117649, 121, 23, 135, 60025, 169, 21, 2401, 29, 21875, 31, 10, 1771561, 289, 697540921, 50625, 37, 361, 4826809, 35, 41, 77, 43, 14641, 84035, 529, 47, 375, 161212051, 3603000625, 24137569, 28561, 53, 441, 2474329, 5764801, 47045881, 841, 59, 42875, 61, 961

Offset: 1

Antti Karttunen, Nov 08 2019

nonn

Cf. A034386, A064989, A108951, A276086, A324886, A329045.

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324886(n) = A276086(A108951(n));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A329044(n) = A064989(A324886(n));

a(n) = A064989(A324886(n)).

a(A000040(n)) = A000040(n).

A348736 a(n) = n - A326042(n), where A326042(n) = A064989(sigma(A003961(n))).

Original entry on oeis.org

0, 1, 1, -7, 4, 4, 5, 5, -20, 9, 6, -10, 9, 12, 13, -33, 14, -11, 17, 9, 17, 17, 17, 18, -9, 22, 5, 6, 28, 28, 14, -23, 23, 31, 33, -283, 27, 36, 31, 37, 34, 38, 41, -11, 16, 40, 39, -50, -36, 16, 45, 8, 47, 32, 50, 50, 53, 57, 30, 38, 48, 45, 5, -1027, 61, 56, 63, 35, 57, 68, 40, -15, 70, 64, 7, 54, 67, 70, 69, 31

Offset: 1

Antti Karttunen, Nov 02 2021

sign

Cf. A000203, A003961, A064989, A326042, A348746 [= n - A161942(n)], A348940.

Cf. A348737, A348738 (positions of positive terms), A348739 (of negative terms).

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A326042(n) = A064989(sigma(A003961(n)));
A348736(n) = (n-A326042(n));

a(n) = n - A064989(A161942(A003961(n))).

A348738 Numbers k for which A326042(k) < k, where A326042(n) = A064989(sigma(A003961(n))).

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97

Offset: 1

Antti Karttunen, Nov 02 2021

nonn

Claim: If there is an odd term y of A336702 larger than one, and it is the least one of such terms, then it should satisfy condition that for all nontrivial unitary divisor pairs d and x/d of x = A064989(y) [with gcd(d,x/d) = 1, 1 < d < x], the other divisor should reside in this sequence, and the other divisor in A348739. Proof: Applying A064989 to the odd terms of A336702 gives the fixed points of A326042. Suppose there are other odd terms in A336702 in addition to its initial 1, and let y be the least of these odd terms > 1 and x = A064989(y). Because A326042 (from here on indicated with f) is multiplicative, it follows that if we take any two nontrivial unitary divisors a and b of x, with x = a*b, gcd(a,b) = 1, 1 < a,b < x, then f(a)*f(b) = f(x) = x. Because f(x)/x = 1, we must have f(a)/a * f(b)/b = 1, as also the ratio f(n)/n is multiplicative. But f(a)/a and f(b)/b cannot be equal to 1, because then a and b would also be fixed by f, which contradicts our assumption that x were the least such fixed point larger than one. Therefore f(a) < a and f(b) > b, or vice versa. See also the comments in A348930, A348933.
Moreover, all odd perfect numbers (a subsequence of A336702), if such numbers exist, should also satisfy the same condition, regardless of whether they are the least of such numbers or not, because having a non-deficient proper divisor will push the abundancy index (ratio sigma(n)/n) of any number over 2. That is, for any such pair of nontrivial unitary divisors d and x/d, both A003961(d) and A003961(x/d) should be deficient, i.e., neither one should be in A337386. See also the condition given in A347383.
Terms that occur also in A337386 are: 120, 240, 360, 420, 480, 504, 540, 600, 630, ...

Positions of positive terms in A348736, positions of 1's in A348737 (characteristic function).
Almost complement of A348739.
Cf. A000203, A003961, A003973, A006039, A027687, A064989, A326042, A336702, A337386, A347383, A348741, A348748 (corresponding odd numbers).
Subsequences: A000040, A374464 (after its initial 1).
Cf. also A348930, A348933.

f1[2, e_] := 1; f1[p_, e_] := NextPrime[p, -1]^e; s1[1] = 1; s1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := NextPrime[p]^e; s2[1] = 1; s2[n_] := Times @@ f2 @@@ FactorInteger[n]; Select[Range[100], s1[DivisorSigma[1, s2[#]]] < # &] (* Amiram Eldar, Nov 04 2021 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A326042(n) = A064989(sigma(A003961(n)));
isA348738(n) = (A326042(n)

A353750 a(n) = phi(sigma(n)) * A064989(sigma(n)), where A064989 shifts the prime factorization one step towards lower primes.

Original entry on oeis.org

1, 4, 2, 30, 4, 8, 4, 48, 132, 24, 8, 60, 30, 16, 16, 870, 24, 528, 24, 120, 16, 48, 16, 96, 870, 120, 48, 120, 48, 96, 16, 720, 32, 144, 32, 3960, 306, 96, 120, 288, 120, 64, 140, 240, 528, 96, 32, 1740, 1224, 3480, 96, 1050, 144, 192, 96, 192, 96, 288, 96, 480, 870, 64, 528, 14238, 240, 192, 416, 720, 64, 192, 96

Offset: 1

Antti Karttunen, May 07 2022

nonn

In contrast to A353749, this is not multiplicative, except on positions given by A336547.

It seems that a(n) = A353749(n) only on n=1. This would then imply that the intersection of A006872 and A336702 = {1}.

Cf. A000010, A000203, A006872, A062401, A064989, A336547, A336548, A336549, A336702, A350073, A353747, A353748, A353749.
Cf. A353757, A353758 (where a(n) < A353749(n)), A353759 (where a(n) >= A353749(n)), A353760, A353790 [= a(A003961(n))].
Cf. also A353792.

A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353750(n) = { my(s=sigma(n)); (eulerphi(s)*A064989(s)); };

a(n) = A353749(A000203(n)) = A062401(n) * A350073(n).

a(n) = A353749(n) + A353757(n).

Dubious comment deleted by Antti Karttunen, Jan 26 2023

A348749 Odd numbers k for which A064989(sigma(k)) > A064989(k), where A064989 shifts the prime factorization one step towards lower primes, and sigma is the sum of divisors function.

Original entry on oeis.org

9, 25, 45, 49, 75, 81, 117, 121, 225, 243, 289, 325, 333, 405, 441, 529, 549, 605, 625, 657, 675, 729, 841, 925, 1053, 1089, 1125, 1215, 1225, 1413, 1445, 1521, 1525, 1575, 1665, 1681, 1737, 1825, 1875, 2025, 2205, 2401, 2475, 2493, 2601, 2817, 2825, 2925, 2997, 3025, 3033, 3125, 3249, 3481, 3573, 3645, 3675, 3789

Offset: 1

Antti Karttunen, Nov 02 2021

nonn

Sequence obtained when A003961 is applied to A348739 and the terms are sorted into ascending order.
From Robert Israel, Nov 12 2024: (Start)
If a and b are terms and are coprime, then a * b is a term.
If p > 2 is in A053182, Legendre's conjecture implies p^2 is in this sequence. (End)

Cf. A000203, A003961, A053182, A064989, A326042, A348739, A348748, A348939 (terms of A228058 that occur here).

Cf. also A348742, A348754.

g:= prevprime: g(2):= 1:
f:= proc(n) local F,t;
  F:= ifactors(n)[2];
  mul(g(t[1])^t[2],t=F)
end proc:
select(t -> f(numtheory:-sigma(t)) > f(t), [seq(i,i=1..4000,2)]); # Robert Israel, Nov 12 2024

f[2, e_] := 1; f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[1, 4000, 2], s[DivisorSigma[1, #]] > s[#] &] (* Amiram Eldar, Nov 04 2021 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
isA348749(n) = ((n%2)&&(A064989(sigma(n)) > A064989(n)));

A348750 a(n) = A064989(A064989(sigma(A003961(A003961(n))))), where A003961 shifts the prime factorization of n one step towards larger primes, and A064989 shifts it back towards smaller primes.

Original entry on oeis.org

1, 1, 1, 23, 1, 1, 3, 7, 13, 1, 1, 23, 2, 3, 1, 305, 1, 13, 2, 23, 3, 1, 1, 7, 39, 2, 4, 69, 13, 1, 3, 69, 1, 1, 3, 299, 5, 2, 2, 7, 1, 3, 1, 23, 13, 1, 2, 305, 53, 39, 1, 46, 23, 4, 1, 21, 2, 13, 11, 23, 1, 3, 39, 19501, 2, 1, 29, 23, 1, 3, 2, 91, 3, 5, 39, 46, 3, 2, 2, 305, 2791, 1, 9, 69, 1, 1, 13, 7, 11, 13, 6

Offset: 1

Antti Karttunen, Nov 02 2021

Cf. A000203, A003961, A003973, A064989, A326042, A348751 (a(n) < n), A348752 (a(n) > n).

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A348750(n) = A064989(A064989(sigma(A003961(A003961(n)))));

a(n) = A064989(A326042(A003961(n))).

Multiplicative with a(p^e) = A064989(A064989((q^(e+1)-1)/(q-1))), where q = nextPrime(nextPrime(p)).

A266645 Permutation of natural numbers: a(n) = A064989(A250469(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 9, 8, 11, 14, 13, 22, 15, 12, 17, 26, 19, 34, 21, 20, 23, 38, 25, 18, 33, 16, 29, 46, 31, 58, 39, 28, 35, 30, 37, 62, 51, 44, 41, 74, 43, 82, 57, 24, 47, 86, 49, 50, 27, 52, 53, 94, 55, 42, 69, 68, 59, 106, 61, 118, 87, 40, 65, 66, 67, 122, 45, 76, 71, 134, 73, 142, 93, 36, 77, 70, 79, 146, 111, 32, 83, 158, 85, 78, 123

Offset: 1

Antti Karttunen, Jan 02 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A266646.
Cf. A000035, A003961, A020639, A055396, A064989, A250469.
Related permutations: A266403, A266416, A249817, A249818.

f[n_] := Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n; g[n_] := If[n == 1, 0, PrimePi@ FactorInteger[n][[1, 1]]]; Function[s, MapIndexed[ Function[{m, n}, f[Lookup[s, g[n] + 1][[m]] - Boole[n == 1]]][#1, First@ #2] &, #] &@ Map[Position[Lookup[s, g@ #], #][[1, 1]] &, Range@ 120]]@ PositionIndex@ Array[g, 10^4] (* Michael De Vlieger, Mar 09 2017, Version 10 *)

a(n) = A064989(A250469(n)).
a(n) = A064989(A249817(A003961(A249818(n)))).
As a composition of related permutations:
a(n) = A266416(A266403(n)).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]
A020639(a(n)) = A020639(n). [More generally, it preserves the smallest prime dividing n.]
A055396(a(n)) = A055396(n).

A349163 a(n) = A064989(gcd(sigma(n), A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes, while A064989 shifts it back towards smaller primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 10, 1, 2, 1, 6, 1, 2, 3, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 10, 1, 2, 1, 1, 5, 2, 1, 4, 1, 2, 1, 6, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 3, 4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 11, 5, 1, 2, 1, 2, 1

Offset: 1

Antti Karttunen, Nov 09 2021

nonn

Cf. A000203, A003961, A342671, A349161, A349162, A349165 (positions of 1's), A349166 (of terms > 1).

Cf. A349144 and A349168 [positions where a(n) is / is not relatively prime with A349164(n) = n/a(n)].

Array[Times @@ Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger@ GCD[##]] & @@ {DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &, 105] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(f=factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A349163(n) = A064989(gcd(sigma(n),A003961(n)));

a(n) = A064989(A342671(n)).

a(n) = n / A349164(n).

cp's OEIS Frontend

A349164 a(n) = A064989(A003961(n) / gcd(sigma(n), A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes, while A064989 shifts it back towards smaller primes, and sigma is the sum of divisors function.

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A244319 Self-inverse permutation of natural numbers: a(1) = 1, a(2n) = A003961(1+a(A064989(2n-1))), a(2n+1) = 1+A003961(a(A064989(2n+1)-1)).

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A329044 a(n) = A064989(A324886(n)).

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A348736 a(n) = n - A326042(n), where A326042(n) = A064989(sigma(A003961(n))).

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A348738 Numbers k for which A326042(k) < k, where A326042(n) = A064989(sigma(A003961(n))).

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A353750 a(n) = phi(sigma(n)) * A064989(sigma(n)), where A064989 shifts the prime factorization one step towards lower primes.

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A348749 Odd numbers k for which A064989(sigma(k)) > A064989(k), where A064989 shifts the prime factorization one step towards lower primes, and sigma is the sum of divisors function.

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A348750 a(n) = A064989(A064989(sigma(A003961(A003961(n))))), where A003961 shifts the prime factorization of n one step towards larger primes, and A064989 shifts it back towards smaller primes.

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