A348931 -id:A348931 - cp's OEIS frontend

A348741 Odd numbers k for which A161942(k) < k, where A161942 is the odd part of sigma.

3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155, 157

Offset: 1

Antti Karttunen, Nov 02 2021

nonn

Differs from A088828 for the first time at n=1080, where a(1080) = 2207, while A088828(1080) = 2205 = A348743(1), the value which is missing from this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..16291
Index entries for sequences related to sigma(n)

Setwise difference A088828 \ A348743.
Cf. A161942, A348742 (complement among the odd numbers).
Cf. also A348738, A348753, A348931.

odd[n_] := n/2^IntegerExponent[n, 2]; Select[Range[1, 150, 2], odd[DivisorSigma[1, #]] < # &] (* Amiram Eldar, Nov 02 2021 *)

A000265(n) = (n >> valuation(n, 2));
isA348741(n) = ((n%2)&&A000265(sigma(n))

A348753 Numbers k congruent to 1 or 5 mod 6, for which A064989(A064989(sigma(k))) < A064989(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

5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 37, 41, 43, 47, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 125, 127, 131, 133, 137, 139, 143, 145, 149, 151, 155, 157, 161, 163, 167, 173, 179, 181, 185, 187, 191, 193, 197, 199, 203, 205, 209, 211, 215, 217, 221, 223, 227

Offset: 1

Antti Karttunen, Nov 04 2021

nonn

Sequence A003961(A003961(A348751(n))), n>=1, sorted into ascending order.
a(38) = 125 is the first term not in A276378.
Not a subsequence of A348748. The first terms that occur here but not there are: 529, 605, 2825, 6425, 7025, 8425, 10825, 15425, 16025, 16325, 16925, 17689, ...
The first squares in this sequence are: 361, 529, 961, 1369, 1849, 2209, 2809, 3721, etc., see A348935 for their square roots.
Of the natural numbers < 2^20, 347712 are in this sequence and only 1812 in A348754.

Cf. A003961, A007310, A064989, A276378, A348750, A348751, A348754.

Cf. also A348748, A348931, A348935.

f[2, e_] := 1; f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[250], MemberQ[{1, 5}, Mod[#, 6]] && s[s[DivisorSigma[1, #]]] < s[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)};
isA348753(n) = ((n%2)&&(n%3)&&(A064989(A064989(sigma(n))) < A064989(A064989(n))));

A348930 a(n) = A038502(sigma(n)), where A038502 is fully multiplicative with a(3) = 1, and a(p) = p for any other prime p.

Original entry on oeis.org

1, 1, 4, 7, 2, 4, 8, 5, 13, 2, 4, 28, 14, 8, 8, 31, 2, 13, 20, 14, 32, 4, 8, 20, 31, 14, 40, 56, 10, 8, 32, 7, 16, 2, 16, 91, 38, 20, 56, 10, 14, 32, 44, 28, 26, 8, 16, 124, 19, 31, 8, 98, 2, 40, 8, 40, 80, 10, 20, 56, 62, 32, 104, 127, 28, 16, 68, 14, 32, 16, 8, 65, 74, 38, 124, 140, 32, 56, 80, 62, 121, 14, 28

Offset: 1

Antti Karttunen, Nov 04 2021

Note that a(A005820(4)) = A005820(4) and a(A005820(6)) = A005820(6), i.e., the fourth and sixth 3-perfect numbers, 459818240 and 51001180160 are among the fixed points of this sequence, precisely because they are also terms of A323653. As the former factorizes as 459818240 = 256 * 5 * 7 * 19 * 37 * 73, it must follow that a(256)/256 * a(5)/5 * a(7)/7 * a(19)/19 * a(37)/37 * a(73)/73 = 1, because ratio a(n)/n is multiplicative. See also comments in A348738.

Index entries for sequences related to sigma(n)

Cf. A000203, A000265, A005820, A038502, A323653, A348738, A348931, A348932.

Cf. also A161942, A336457.

s[n_] := n / 3^IntegerExponent[n, 3]; Table[s[DivisorSigma[1, n]], {n, 1, 100}] (* Amiram Eldar, Nov 04 2021 *)

A038502(n) = (n/3^valuation(n, 3));
A348930(n) = A038502(sigma(n));

Multiplicative with a(p^e) = A038502(1 + p + p^2 + ... + p^e).
a(n) = A038502(A000203(n)).
For all n >= 1, A000265(a(n)) = A336457(n).

A348932 Numbers k congruent to 1 or 5 mod 6, for which A348930(k) > k.

Original entry on oeis.org

7, 13, 19, 25, 31, 37, 43, 61, 67, 73, 79, 91, 97, 103, 109, 121, 127, 133, 139, 151, 157, 163, 175, 181, 193, 199, 211, 217, 223, 229, 241, 247, 259, 271, 277, 283, 289, 301, 307, 313, 325, 331, 337, 343, 349, 367, 373, 379, 397, 403, 409, 421, 427, 433, 439, 457, 463, 469, 475, 481, 487, 499, 511, 523, 529, 541

Offset: 1

Antti Karttunen, Nov 04 2021

nonn

See comments in A348930.

Index entries for sequences related to sigma(n)

Cf. A007310, A348930, A348931.

Cf. also A348742, A348754.

s[n_] := n / 3^IntegerExponent[n, 3]; Select[Range[550], MemberQ[{1, 5}, Mod[#, 6]] && s[DivisorSigma[1, #]] > # &] (* Amiram Eldar, Nov 04 2021 *)

A038502(n) = (n/3^valuation(n, 3));
A348930(n) = A038502(sigma(n));
isA348932(n) = ((n%2)&&(n%3)&&(A348930(n)>n));

A348933 Numbers k congruent to 1 or 5 mod 6, for which A348930(k^2) < k^2.

Original entry on oeis.org

7, 13, 19, 31, 35, 37, 43, 61, 65, 67, 73, 77, 79, 91, 95, 97, 103, 109, 119, 127, 133, 139, 143, 151, 155, 157, 161, 163, 175, 181, 185, 193, 199, 203, 209, 211, 215, 217, 221, 223, 229, 241, 247, 259, 271, 277, 283, 287, 299, 301, 305, 307, 313, 323, 325, 329, 331, 335, 337, 341, 349, 365, 367, 371, 373, 377, 379

Offset: 1

Antti Karttunen, Nov 04 2021

nonn

Any hypothetical odd term y of A005820 must by necessity be a square. If y is also a nonmultiple of 3, then the square root x = A000196(y) of such a number y must satisfy the condition that for all nontrivial unitary divisor pairs d and x/d [with gcd(d,x/d) = 1, 1 < d < x], the other divisor should reside in this sequence, and the other divisor in A348934. The explanation is similar to the one given in A348738. See also comments in A348935.

Index entries for sequences related to sigma(n)

Cf. A000196, A005820, A348738, A348930, A348931, A348934, A348935.

s[n_] := n / 3^IntegerExponent[n, 3]; Select[Range[400], MemberQ[{1, 5}, Mod[#, 6]] && s[DivisorSigma[1, #^2]] < #^2 &] (* Amiram Eldar, Nov 04 2021 *)

A038502(n) = (n/3^valuation(n, 3));
A348930(n) = A038502(sigma(n));
isA348933(n) = ((n%2)&&(n%3)&&(A348930(n^2)<(n^2)));

cp's OEIS Frontend

A348741 Odd numbers k for which A161942(k) < k, where A161942 is the odd part of sigma.

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A348753 Numbers k congruent to 1 or 5 mod 6, for which A064989(A064989(sigma(k))) < A064989(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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A348930 a(n) = A038502(sigma(n)), where A038502 is fully multiplicative with a(3) = 1, and a(p) = p for any other prime p.

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A348932 Numbers k congruent to 1 or 5 mod 6, for which A348930(k) > k.

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A348933 Numbers k congruent to 1 or 5 mod 6, for which A348930(k^2) < k^2.

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