A336455 -id:A336455 - cp's OEIS frontend

A335915 Fully multiplicative with a(2) = 1, and a(p) = A000265(p-1)*A000265(p+1) = A000265(p^2 - 1), for odd primes p.

1, 1, 1, 1, 3, 1, 3, 1, 1, 3, 15, 1, 21, 3, 3, 1, 9, 1, 45, 3, 3, 15, 33, 1, 9, 21, 1, 3, 105, 3, 15, 1, 15, 9, 9, 1, 171, 45, 21, 3, 105, 3, 231, 15, 3, 33, 69, 1, 9, 9, 9, 21, 351, 1, 45, 3, 45, 105, 435, 3, 465, 15, 3, 1, 63, 15, 561, 9, 33, 9, 315, 1, 333, 171, 9, 45, 45, 21, 195, 3, 1, 105, 861, 3, 27, 231, 105, 15, 495, 3, 63, 33, 15, 69

Offset: 1

Antti Karttunen, Jul 09 2020

For all i, j: A324400(i) = A324400(j) => a(i) = a(j) => A336118(i) = A336118(j).

Antti Karttunen, Table of n, a(n) for n = 1..8192
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000265.

Cf. also A167344, A324400, A335904, A336118, A336455, A336456, A336466, A336467.

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]-1)*A000265(f[k,1]+1))^f[k,2])); };

Completely multiplicative with a(2) = 1, and for odd primes p, a(p) = A000265(p-1)*A000265(p+1).
For all n >= 1, A335904(a(n)) = A336118(n).
For all n >= 0, a(2^n) = a(3^n) = 1, a(5^n) = a(7^n) = 3^n.
a(n) = A336466(n) * A336467(n). - Antti Karttunen, Jan 31 2021

A336456 a(n) = A335915(sigma(sigma(n))), where A335915 is a fully multiplicative sequence with a(2) = 1 and a(p) = A000265(p^2 - 1) for odd primes p, with A000265(k) giving the odd part of k.

Original entry on oeis.org

1, 1, 3, 1, 1, 3, 3, 1, 3, 21, 3, 3, 1, 3, 3, 1, 21, 3, 3, 1, 3, 63, 3, 3, 1, 1, 3, 3, 1, 63, 3, 21, 15, 3, 15, 3, 3, 3, 3, 21, 1, 3, 3, 3, 3, 63, 15, 3, 3, 1, 63, 45, 3, 3, 63, 3, 15, 21, 3, 3, 1, 3, 9, 1, 3, 315, 3, 21, 3, 315, 63, 3, 45, 3, 3, 3, 3, 3, 15, 1, 135, 21, 3, 3, 9, 3, 3, 63, 21, 63, 15, 3, 27, 315

Offset: 1

Antti Karttunen, Jul 22 2020

nonn

Like A051027, neither this is multiplicative. For example, we have a(3) = 3, a(7) = 3, but a(21) = 3 <> 9. However, for example, a(10) = 21, and a(3*10) = 63.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000203, A051027, A335915.

Cf. also A336462, A336463, A336464.

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(sigma(sigma(n)));

a(n) = A335915(A000203(A000203(n))) = A336455(A000203(n)) = A335915(A051027(n)).

A336461 Numbers k for which A335915(k) = A335915(sigma(k)).

Original entry on oeis.org

1, 2, 3, 6, 20, 28, 40, 56, 60, 84, 120, 135, 160, 168, 176, 189, 224, 270, 304, 378, 480, 496, 528, 585, 672, 819, 912, 1170, 1372, 1488, 1638, 1836, 2156, 2744, 3025, 3672, 3724, 3780, 4116, 4312, 4572, 6050, 6076, 6468, 6525, 7448, 7560, 7956, 8128, 8232, 9075, 9144, 9225, 9261, 10224, 10880, 10976, 11172, 12152, 12936, 13050, 14144

Offset: 1

Antti Karttunen, Jul 22 2020

nonn

Numbers k such that A335915(k) = A336455(k).
If terms x and y are present and gcd(x,y) = 1, then x*y is present also. This follows because both A335915 and A336455 are multiplicative sequences.
See also comments in A336464.

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A000203, A000265, A335915, A336455.
Subsequences: A000396, A005820.
Cf. also A336462, A336463, A336464.

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]))); };
isA336461(n) = (A335915(n)==A335915(sigma(n)));

A336462 Numbers k for which A335915(sigma(k)) = A335915(sigma(sigma(k))).

Original entry on oeis.org

1, 2, 5, 12, 19, 24, 27, 28, 38, 39, 44, 54, 56, 59, 60, 65, 78, 83, 84, 87, 92, 95, 123, 124, 129, 133, 136, 143, 160, 167, 168, 178, 212, 223, 248, 258, 259, 266, 269, 276, 285, 288, 297, 308, 360, 393, 395, 413, 427, 429, 437, 446, 455, 473, 479, 501, 518, 522, 528, 555, 560, 581, 611, 612, 681, 738, 752, 755

Offset: 1

Antti Karttunen, Jul 22 2020

nonn

Numbers k for which A336455(k) = A336455(sigma(k)) = A336456(k).
Numbers k such that sigma(k) is in A336461.
See comments in A336464.

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A000203, A000265, A335915, A336455, A336456.

Cf. also A336461, A336463, A336464.

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]))); };
isA336462(n) = (A335915(sigma(n))==A335915(sigma(sigma(n))));

A336464 Numbers k for which A335915(k), A335915(sigma(k)) and A335915(sigma(sigma(k))) obtain the same value.

Original entry on oeis.org

1, 2, 28, 56, 60, 84, 160, 168, 528, 12936, 32760, 102960, 1097280, 1778400, 11740302, 19183500, 25241600, 235855620, 308308000, 317167200, 424305000, 459818240, 704700000, 787200000, 924924000, 1592025435, 2701416960, 3812244480

Offset: 1

Antti Karttunen, Jul 22 2020

Numbers k for which A335915(k) = A336455(k) = A336456(k).
Numbers k such that both k and sigma(k) are in A336461.
Note that a(26) = 1592025435 (originally found by David A. Corneth) is an odd term > 1, which factorizes as 3^5 * 7^2 * 11^2 * 5 * 13 * 17, and thus is not of the form of A228058.
It appears that if we instead list k such that both k and sigma(k) are in A336458, we will not obtain more than these three terms: 1, 2, 84.

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A000203, A000265, A228058, A335915, A336455, A336456, A336458, A336560.

Intersection of any two of these three sequences: A336461, A336462, A336463.

A336457 a(n) = A065330(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, 1, 7, 1, 1, 1, 5, 13, 1, 1, 7, 7, 1, 1, 31, 1, 13, 5, 7, 1, 1, 1, 5, 31, 7, 5, 7, 5, 1, 1, 7, 1, 1, 1, 91, 19, 5, 7, 5, 7, 1, 11, 7, 13, 1, 1, 31, 19, 31, 1, 49, 1, 5, 1, 5, 5, 5, 5, 7, 31, 1, 13, 127, 7, 1, 17, 7, 1, 1, 1, 65, 37, 19, 31, 35, 1, 7, 5, 31, 121, 7, 7, 7, 1, 11, 5, 5, 5, 13, 7, 7, 1, 1, 5, 7, 49

Offset: 1

Antti Karttunen, Jul 24 2020

Sequence removes prime factors 2 and 3 from the prime factorization of the sum of divisors of n.

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, A038502, A065330, A161942, A336458.

Cf. also A336455.

Array[Times @@ Map[#1^#2 & @@ # &, DeleteCases[FactorInteger[DivisorSigma[1, #]], ?(First@ # <= 3 &)]] &, 97] (* _Michael De Vlieger, Jul 24 2020 *)

A065330(n) = (n>>valuation(n, 2)/3^valuation(n, 3));
A336457(n) = A065330(sigma(n));

a(n) = A065330(A000203(n)) = A038502(A161942(n)).

Multiplicative with a(p^e) = A065330(1 + p + p^2 + ... + p^e).

A336463 Numbers k for which A335915(k) = A335915(sigma(sigma(k))).

Original entry on oeis.org

1, 2, 4, 7, 8, 14, 15, 16, 21, 28, 33, 42, 45, 56, 60, 64, 84, 102, 112, 128, 130, 132, 147, 152, 160, 168, 180, 240, 273, 306, 336, 406, 408, 435, 441, 448, 512, 513, 520, 528, 546, 574, 588, 615, 720, 765, 896, 960, 1023, 1026, 1224, 1323, 1344, 1470, 1568, 1740, 1764, 1785, 1989, 2052, 2112, 2184, 2296, 2352, 2432, 2625

Offset: 1

Antti Karttunen, Jul 22 2020

nonn

Numbers k such that A335915(k) = A336456(k).

See comments in A336464.

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A000203, A000265, A335915, A336455, A336456.

Cf. also A336461, A336462, A336464.

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]))); };
isA336463(n) = (A335915(n)==A335915(sigma(sigma(n))));

cp's OEIS Frontend

A335915 Fully multiplicative with a(2) = 1, and a(p) = A000265(p-1)*A000265(p+1) = A000265(p^2 - 1), for odd primes p.

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A336456 a(n) = A335915(sigma(sigma(n))), where A335915 is a fully multiplicative sequence with a(2) = 1 and a(p) = A000265(p^2 - 1) for odd primes p, with A000265(k) giving the odd part of k.

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A336461 Numbers k for which A335915(k) = A335915(sigma(k)).

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A336462 Numbers k for which A335915(sigma(k)) = A335915(sigma(sigma(k))).

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A336464 Numbers k for which A335915(k), A335915(sigma(k)) and A335915(sigma(sigma(k))) obtain the same value.

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A336457 a(n) = A065330(sigma(n)), where A065330 is fully multiplicative with a(2) = a(3) = 1, and a(p) = p for primes p > 3.

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A336463 Numbers k for which A335915(k) = A335915(sigma(sigma(k))).

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