A353790 -id:A353790 - cp's OEIS frontend

A353796 Numbers k such that k divides A353790(k), where A353790(n) = phi(A003973(n)) * A064989(A003973(n)).

1, 2, 4, 8, 12, 24, 36, 44, 72, 96, 112, 128, 132, 160, 180, 220, 288, 336, 352, 360, 384, 396, 480, 528, 560, 640, 660, 880, 1044, 1056, 1152, 1232, 1344, 1404, 1440, 1680, 1760, 1920, 1980, 2088, 2352, 2376, 2464, 2496, 2640, 3168, 3600, 3696, 3920, 4032, 4400, 4736, 5220, 5280, 5376, 5760, 5824, 6075, 6144, 6160

Offset: 1

Antti Karttunen, May 12 2022

nonn

Of 5263 initial terms (terms < 2^32), only 67 are odd, and of these, only two, 1 and 1525391261 (= 503^2 * 6029) are in A007310. Of 5263 initial terms, 4653 are multiples of 3, 2331 are multiples of 81, and 3780 are multiples of 5.

Cf. A000010, A000203, A003961, A003973, A353790, A353797 (subsequence).

Cf. also A353795.

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>>valuation(n, 2))); for(i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f); };
A353790(n) = { my(s=sigma(A003961(n))); (eulerphi(s)*A064989(s)); };
isA353796(n) = !(A353790(n)%n);

A353797 Numbers k such that kA003557(A003961(k)) divides A353790(k), where A353790(n) = phi(A003973(n)) A064989(A003973(n)).

Original entry on oeis.org

1, 2, 4, 44, 132, 220, 396, 660, 1980, 3920, 4400, 8800, 11484, 13200, 13328, 22000, 26400, 30800, 39984, 57420, 66640, 74800, 92400, 119952, 149600, 199920, 224400, 269892, 277200, 448800, 523600, 599760, 673200, 771012, 1063692, 1345792, 1346400, 1570800, 3478608, 4037376, 4712400, 5664400, 6344448, 8038800, 10574080

Offset: 1

Antti Karttunen, May 12 2022

nonn

Note that A003557(A003961(n)) [= A003961(A003557(n))] is a divisor of A003972(n), therefore the set of k such that A353789(k) divides A353790(k) is a subset of this sequence.

Of 101 initial terms (terms < 2^32) all others apart from a(1) = 1 and a(2) = 2 are multiples of 4.

Subsequence of A353796.

Cf. A000010, A000203, A003557, A003961, A003972, A003973, A353789, A353790.

A003557(n) = (n/factorback(factorint(n)[, 1]));
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>>valuation(n, 2))); for(i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f); };
A353790(n) = { my(s=sigma(A003961(n))); (eulerphi(s)*A064989(s)); };
isA353797(n) = !(A353790(n)%(n*A003557(A003961(n))));

A326042 a(n) = A064989(sigma(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, 2, 11, 1, 2, 2, 3, 29, 1, 5, 22, 4, 2, 2, 49, 3, 29, 2, 11, 4, 5, 6, 6, 34, 4, 22, 22, 1, 2, 17, 55, 10, 3, 2, 319, 10, 2, 8, 3, 7, 4, 2, 55, 29, 6, 8, 98, 85, 34, 6, 44, 6, 22, 5, 6, 4, 1, 29, 22, 13, 17, 58, 1091, 4, 10, 4, 33, 12, 2, 31, 87, 3, 10, 68, 22, 10, 8, 10, 49, 469, 7, 12, 44, 3, 2, 2, 15, 25, 29, 8, 66, 34, 8

Offset: 1

Antti Karttunen, Jun 16 2019

For any other number n than those in A326182 we have a(n) < A003961(n).
Fixed points k (for which a(k) = k) satisfy A003973(k) = 2^e * A003961(k) for some exponent e >= 0. Applying A003961 to such numbers gives the odd terms in A336702, of which there are likely to be just a single instance, its initial 1. (Clarified Nov 07 2021).
Conjecture: There are no other fixed points than a(1) = 1. If true, then there are no odd perfect numbers. This condition is equivalent to the condition that if A161942 has no fixed points larger than one, then there are no odd perfect numbers. This follows as whenever k is a fixed point, that is, a(k) = k, then we should also have A003961(a(k)) = A003961(A064989(sigma(A003961(k)))) = A161942(A003961(k)) = A003961(k). Note that A003961 is an injective and surjective mapping from natural numbers to odd numbers, A064989 is its (left) inverse, and composition A003961(A064989(n)) is equivalent to A000265(n).
From Antti Karttunen, Aug 05 2020: (Start)
For any hypothetical odd perfect number x, we would have A003973(k) = 2 * A003961(k), with k = A064989(x) and x = A003961(k). Thus we would have a(k) = A064989(sigma(A003961(k))) = A064989(sigma(x)) = A064989(2*x) = A064989(x) = k. On the other hand, A003973(k) = sigma(A003961(k)) < A003961(A003961(k)) [see A286385 for the reason why], so a necessary condition for this is that x should be one of the terms of A246282. (Clarified Dec 01 2020).
(End)

Cf. A000037, A000203, A000265, A000593, A003961, A003973, A064989, A161942, A162284, A246282, A286385, A326041, A326182, A336702 (numbers whose abundancy index is a power of 2).
Cf. A348736 [n - a(n)], A348738 [a(n) < n], A348739 [a(n) > n], A348750 [= A064989(a(A003961(n)))], A348940 [gcd(n,a(n))], A348941, A348942, A351456, A353767, A353790, A353794.
Cf. also A332223 for another conjugation of sigma.

f1[p_, e_] := NextPrime[p]^e; a1[1] = 1; a1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[2, e_] := 1; f2[p_, e_] := NextPrime[p, -1]^e; a2[1] = 1; a2[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := a2[DivisorSigma[1, a1[n]]]; Array[a, 100] (* Amiram Eldar, Nov 07 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)));

a(n) = A064989(A003973(n)) = A064989(sigma(A003961(n))).
For k in A000037, a(k) = A064989(A003973(k)/2) = A064989((1/2)*sigma(A003961(k))).
Multiplicative with a(p^e) = A064989((q^(e+1)-1)/(q-1)), where q = nextPrime(p). - Antti Karttunen, Nov 05 2021
a(n) = A353790(n) / A353767(n) = A353794(n) / A351456(n). - Antti Karttunen, May 13 2022

Keyword:mult added by Antti Karttunen, Nov 05 2021

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

A353794 a(n) = A353791(sigma(A003961(n))), where A353791(n) = A003958(n) * A064989(n).

Original entry on oeis.org

1, 1, 4, 132, 1, 4, 4, 12, 870, 1, 30, 528, 16, 4, 4, 4900, 12, 870, 4, 132, 16, 30, 48, 48, 1224, 16, 528, 528, 1, 4, 306, 3960, 120, 12, 4, 114840, 120, 4, 64, 12, 70, 16, 4, 3960, 870, 48, 64, 19600, 9180, 1224, 48, 2112, 48, 528, 30, 48, 16, 1, 870, 528, 208, 306, 3480, 1191372, 16, 120, 16, 1584, 192, 4, 1116

Offset: 1

Antti Karttunen, May 11 2022

It is conjectured that a(n) is not a multiple of A353793(n) on any other n except on n=1. See also A353795.

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 computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000203, A003958, A003961, A003973, A064989, A326042, A351456, A353791, A353792, A353793, A353795 [numbers k such that k divides a(k)].

Cf. also A353790.

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
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=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353794(n) = { my(s=sigma(A003961(n))); (A003958(s)*A064989(s)); };

Multiplicative with a(p^e) = A003958(1 + q + ... + q^e) * A064989(1 + q + ... + q^e), where q is the least prime larger than p.
a(n) = A353791(A003973(n)) = A353792(A003961(n)).
a(n) = A326042(n) * A351456(n) = A064989(A003973(n)) * A003958(A003973(n)).

A353789 Multiplicative with a(p^e) = (q - 1) * q^(e-1) * p^e, where q is the least prime greater than p.

Original entry on oeis.org

1, 4, 12, 24, 30, 48, 70, 144, 180, 120, 132, 288, 208, 280, 360, 864, 306, 720, 418, 720, 840, 528, 644, 1728, 1050, 832, 2700, 1680, 870, 1440, 1116, 5184, 1584, 1224, 2100, 4320, 1480, 1672, 2496, 4320, 1722, 3360, 1978, 3168, 5400, 2576, 2444, 10368, 5390, 4200, 3672, 4992, 3074, 10800, 3960, 10080, 5016, 3480

Offset: 1

Antti Karttunen, May 10 2022

Question: Does a(n) divide A353790(n) only when n=1? Compare to A353764.

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 computed from indices in prime factorization.

Cf. A003961, A003972, A151800, A353749, A353764, A353790.

Cf. also A353793.

f[p_, e_] := ((q = NextPrime[p]) - 1) * q^(e - 1) * p^e; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 10 2022 *)

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A353789(n) = (n * eulerphi(A003961(n)));

from math import prod
from sympy import nextprime, factorint
def A353789(n): return prod((q:= nextprime(p))**(e-1)*p**e*(q-1) for p, e in factorint(n).items()) # Chai Wah Wu, May 10 2022

Multiplicative with a(p^e) = (q - 1) * q^(e-1) * p^e, where q is the least prime greater than p.
a(n) = A353749(A003961(n)) = n * A003972(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} ((p^3-p^2-p+1)/(p^3 - p*q)) = 0.836506229..., where q(p) = nextprime(p) = A151800(p). - Amiram Eldar, Dec 31 2022

A353767 a(n) = phi(sigma(A003961(n))), where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

1, 2, 2, 12, 4, 8, 4, 16, 30, 16, 6, 24, 6, 16, 16, 110, 8, 60, 8, 48, 24, 24, 8, 64, 36, 24, 48, 48, 16, 64, 18, 144, 24, 32, 32, 360, 12, 32, 36, 128, 20, 96, 16, 72, 120, 32, 18, 220, 108, 72, 32, 72, 16, 192, 48, 128, 48, 64, 30, 192, 32, 72, 120, 1092, 48, 96, 24, 96, 48, 128, 36, 480, 32, 48, 108, 96, 48, 144

Offset: 1

Antti Karttunen, May 10 2022

nonn

Cf. A000010, A000203, A003961, A003973, A062401, A326042, A353790.

f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := EulerPhi[DivisorSigma[1, s[n]]]; Array[a, 100] (* Amiram Eldar, May 10 2022 *)

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A353767(n) = eulerphi(sigma(A003961(n)));

a(n) = A000010(A003973(n)) = A062401(A003961(n)) = A000010(A000203(A003961(n))).

a(n) = A353790(n) / A326042(n).

cp's OEIS Frontend

A353796 Numbers k such that k divides A353790(k), where A353790(n) = phi(A003973(n)) * A064989(A003973(n)).

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A353797 Numbers k such that k*A003557(A003961(k)) divides A353790(k), where A353790(n) = phi(A003973(n)) * A064989(A003973(n)).

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A326042 a(n) = A064989(sigma(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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A353750 a(n) = phi(sigma(n)) * A064989(sigma(n)), where A064989 shifts the prime factorization one step towards lower primes.

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A353794 a(n) = A353791(sigma(A003961(n))), where A353791(n) = A003958(n) * A064989(n).

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A353789 Multiplicative with a(p^e) = (q - 1) * q^(e-1) * p^e, where q is the least prime greater than p.

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A353767 a(n) = phi(sigma(A003961(n))), where A003961 is fully multiplicative with a(p) = nextprime(p).

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A353797 Numbers k such that kA003557(A003961(k)) divides A353790(k), where A353790(n) = phi(A003973(n)) A064989(A003973(n)).