A335915 -id:A335915 - cp's OEIS frontend

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.

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)).

A336455 a(n) = A335915(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, 1, 3, 1, 1, 1, 3, 21, 1, 1, 3, 3, 1, 1, 15, 1, 21, 3, 3, 1, 1, 1, 3, 15, 3, 3, 3, 3, 1, 1, 3, 1, 1, 1, 63, 45, 3, 3, 3, 3, 1, 15, 3, 21, 1, 1, 15, 45, 15, 1, 9, 1, 3, 1, 3, 3, 3, 3, 3, 15, 1, 21, 63, 3, 1, 9, 3, 1, 1, 1, 63, 171, 45, 15, 9, 1, 3, 3, 15, 225, 3, 3, 3, 1, 15, 3, 3, 3, 21, 3, 3, 1, 1, 3, 3, 9, 45, 21, 45

Offset: 1

Antti Karttunen, Jul 22 2020

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

Cf. A000265, A000203, A335915, 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]))); };
A336455(n) = A335915(sigma(n));
\\ Alternatively, as:
A336455(n) =  { my(f=factor(n)); prod(k=1,#f~,A335915(((f[k,1]^(1+f[k,2]))-1)/(f[k,1]-1))); };

a(n) = A335915(A000203(n)).
Multiplicative with a(p^e) = A335915(1 + p + p^2 + ... + p^e).
a(A000203(n)) = A336456(n).

A336160 Lexicographically earliest infinite sequence such that a(i) = a(j) => A335915(i) = A335915(j) and A336158(i) = A336158(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 6, 3, 7, 1, 8, 4, 9, 3, 7, 5, 10, 2, 11, 6, 12, 3, 13, 7, 5, 1, 14, 8, 15, 4, 16, 9, 17, 3, 13, 7, 18, 5, 19, 10, 20, 2, 11, 11, 15, 6, 21, 12, 22, 3, 22, 13, 23, 7, 24, 5, 19, 1, 25, 14, 26, 8, 27, 15, 28, 4, 29, 16, 30, 9, 22, 17, 31, 3, 32, 13, 33, 7, 34, 18, 35, 5, 36, 19, 25, 10, 14, 20, 37, 2, 38, 11, 39, 11, 40, 15, 41, 6, 42

Offset: 1

Antti Karttunen, Jul 11 2020

nonn

Restricted growth sequence transform of the ordered pair [A335915(n), A336158(n)].

For all i, j: A324400(i) = A324400(j) => a(i) = a(j) => A336161(i) = A336161(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000265, A046523, A324400, A335915, A336158, A336159, A336161, A336162.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A000265(n) = (n>>valuation(n,2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A336158(n) = A046523(A000265(n));
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])); };
Aux336160(n) = [A335915(n), A336158(n)];
v336160 = rgs_transform(vector(up_to, n, Aux336160(n)));
A336160(n) = v336160[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)));

A336557 Numbers k such that A336456(n) = Product_{p^e|n} A336456(p^e). Here each p^e is the maximal power of prime p that divides k, and A336456(n) = A335915(sigma(sigma(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 24, 25, 26, 27, 28, 29, 31, 32, 36, 37, 38, 39, 41, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 56, 59, 60, 61, 62, 63, 64, 67, 68, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 83, 86, 87, 89, 92, 95, 96, 97, 99, 100, 101, 103, 104, 107, 108, 109, 111, 112

Offset: 1

Antti Karttunen, Jul 25 2020

nonn

Numbers at which points A336456 appears to be multiplicative.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A335915, A336456.
Cf. A336556 (characteristic function), A336558 (complement).
Union of A000961 and A336559.
Union of A336547 and A336560.

A336559 Numbers k, not powers of primes, such that A336456(k) = Product_{p^e|k} A336456(p^e). Here each p^e is the maximal power of prime p that divides k, and A336456(k) = A335915(sigma(sigma(k))).

Original entry on oeis.org

6, 12, 14, 15, 18, 20, 24, 26, 28, 36, 38, 39, 44, 45, 48, 50, 51, 54, 56, 60, 62, 63, 68, 72, 74, 75, 76, 78, 80, 86, 87, 92, 95, 96, 99, 100, 104, 108, 111, 112, 116, 117, 122, 123, 124, 126, 134, 143, 144, 146, 147, 148, 150, 153, 158, 159, 162, 171, 172, 175, 176, 180, 183, 188, 192, 194, 196, 200, 204, 206, 207

Offset: 1

Antti Karttunen, Jul 25 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..25000

Terms of A336557 without those of A000961.
Cf. A335915, A336456, A336556, A336558, A336560.
Cf. A336549 (a subsequence).

isA336559(n) = ((n>1) && !isprimepower(n) && A336556(n)); \\ Needs also code from A336556.

A336155 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(1+i) = A007814(1+j) and A335915(i) = A335915(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 5, 2, 1, 6, 7, 2, 8, 6, 9, 2, 10, 2, 11, 6, 4, 12, 13, 2, 10, 14, 3, 6, 15, 6, 16, 2, 17, 18, 19, 2, 20, 21, 22, 6, 15, 6, 23, 12, 4, 24, 25, 2, 10, 18, 19, 14, 26, 2, 27, 6, 28, 29, 30, 6, 31, 12, 32, 2, 33, 12, 34, 18, 35, 18, 36, 2, 37, 38, 19, 21, 28, 14, 39, 6, 1, 29, 40, 6, 41, 42, 43, 12, 44, 6, 45, 24, 17, 46, 47, 2, 48, 18, 7, 18, 49, 18, 50, 14, 10

Offset: 1

Antti Karttunen, Jul 11 2020

nonn

Restricted growth sequence transform of the ordered pair [A007814(1+n), A335915(n)]. Note that A007814(1+n) gives the number of trailing 1-bits in the binary expansion of n.

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

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000265, A007814, A335915.

Cf. also A324400, A336154, A336156, A336162.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007814(n) = valuation(n,2);
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])); };
Aux336155(n) = [A007814(1+n), A335915(n)];
v336155 = rgs_transform(vector(up_to, n, Aux336155(n)));
A336155(n) = v336155[n];

A336162 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(i) = A278222(j) and A335915(i) = A335915(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, 16, 1, 17, 9, 13, 5, 18, 10, 19, 3, 20, 11, 21, 6, 22, 12, 23, 2, 13, 13, 24, 7, 25, 14, 26, 4, 27, 15, 28, 8, 29, 16, 30, 1, 31, 17, 32, 9, 33, 13, 34, 5, 35, 18, 36, 10, 37, 19, 38, 3, 39, 20, 40, 11, 41, 21, 42, 6, 43, 22, 44, 12, 45, 23, 46, 2, 47, 13, 48, 13, 49, 24, 50, 7, 36

Offset: 1

Antti Karttunen, Jul 11 2020

nonn

Restricted growth sequence transform of the ordered pair [A278222(n), A335915(n)].

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

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000265, A005940, A046523, A278222, A335915.

Cf. also A286622, A324400, A336155, A336159, A336160.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A000265(n) = (n>>valuation(n,2));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
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])); };
Aux336162(n) = [A278222(n), A335915(n)];
v336162 = rgs_transform(vector(up_to, n, Aux336162(n)));
A336162(n) = v336162[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.

cp's OEIS Frontend

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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A336455 a(n) = A335915(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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A336160 Lexicographically earliest infinite sequence such that a(i) = a(j) => A335915(i) = A335915(j) and A336158(i) = A336158(j), for all i, j >= 1.

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

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A336557 Numbers k such that A336456(n) = Product_{p^e|n} A336456(p^e). Here each p^e is the maximal power of prime p that divides k, and A336456(n) = A335915(sigma(sigma(n))).

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A336559 Numbers k, not powers of primes, such that A336456(k) = Product_{p^e|k} A336456(p^e). Here each p^e is the maximal power of prime p that divides k, and A336456(k) = A335915(sigma(sigma(k))).

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A336155 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(1+i) = A007814(1+j) and A335915(i) = A335915(j), for all i, j >= 1.

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A336162 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(i) = A278222(j) and A335915(i) = A335915(j), for all i, j >= 1.

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

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