A348750 -id:A348750 - cp's OEIS frontend

A348751 Numbers k for which A348750(k) < k.

2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 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, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97

Offset: 1

Antti Karttunen, Nov 02 2021

nonn

Cf. A003961, A064989, A348750, A348751.

Cf. A348753 (corresponding 5-rough numbers, terms of A007310).

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)))));
isA348752(n) = (A348750(n)>n);

A348752 Numbers k for which A348750(k) > k.

Original entry on oeis.org

4, 9, 12, 16, 20, 25, 28, 32, 36, 48, 49, 64, 72, 80, 81, 100, 112, 116, 121, 128, 144, 162, 176, 180, 192, 196, 200, 208, 212, 225, 236, 240, 242, 243, 252, 256, 268, 272, 288, 300, 304, 320, 324, 336, 361, 384, 400, 405, 432, 441, 448, 450, 464, 468, 484, 496, 500, 512, 560, 567, 576, 588, 592, 596, 604, 625, 640

Offset: 1

Antti Karttunen, Nov 02 2021

nonn

Cf. A003961, A064989, A348750, A348751.

Cf. A348754 (corresponding 5-rough numbers, terms of A007310).

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)))));
isA348752(n) = (A348750(n)>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

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))));

A348754 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

25, 49, 121, 169, 175, 275, 289, 325, 625, 841, 925, 1225, 1445, 1525, 1675, 1681, 1825, 2401, 3025, 3125, 3481, 3757, 3925, 4075, 4225, 4375, 4825, 5041, 5275, 5929, 6125, 6875, 6925, 7075, 7225, 7825, 7921, 8125, 8275, 8281, 9025, 9925, 10201, 10525, 10625, 10693, 11425, 11875, 12005, 12025, 13075, 13225, 13475

Offset: 1

Antti Karttunen, Nov 04 2021

nonn

Sequence A003961(A003961(A348752(n))), n=1.., sorted into ascending order.

Not a subsequence of A348749. The first terms that occur here but not there are: 169, 175, 275, 1675, 3757, 4075, 5275, 7075, 8275, 10693, 12025, ...

Cf. A003961, A007310, A064989, A348750, A348752, A348753.

Cf. also A348749, A348932, A348936 (square roots of squares present).

f[2, e_] := 1; f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[15000], 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)};
isA348754(n) = ((n%2)&&(n%3)&&(A064989(A064989(sigma(n))) > A064989(A064989(n))));

A354206 a(n) = A354203(sigma(A354202(n))), where A354202 is fully multiplicative with a(p) = A354200(A000720(p)), and A354203 is its left inverse.

Original entry on oeis.org

1, 1, 1, 23, 3, 1, 1, 5, 11, 3, 2, 23, 1, 1, 3, 469, 2, 11, 1, 69, 1, 2, 1, 5, 53, 1, 4, 23, 11, 3, 7, 69, 2, 2, 3, 253, 3, 1, 1, 15, 1, 1, 1, 46, 33, 1, 2, 469, 33, 53, 2, 23, 23, 4, 6, 5, 1, 11, 13, 69, 29, 7, 11, 19507, 3, 2, 1, 46, 1, 3, 2, 55, 2, 3, 53, 23, 2, 1, 3, 1407, 2797, 1, 5, 23, 6, 1, 11, 10, 9, 33

Offset: 1

Antti Karttunen, May 23 2022

Cf. A000203, A000720, A354200, A354201, A354202, A354203, A354205, A354207.
Cf. A354361 (positions of 1's).
Cf. also A326042, A348750, A354088, A354096 for similar constructions.

A354200(n) = if(1==n,5,my(p=prime(n), m=p%4); forprime(q=1+p,,if(m==(q%4),return(q))));
A354201(n) = if(n<=3,(n+1)\2,my(m=prime(n)%4); forstep(i=n-1,0,-1,if(m==(prime(i)%4),return(prime(i)))));
A354202(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354200(primepi(f[k,1]))); factorback(f); };
A354203(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354201(primepi(f[k,1]))); factorback(f); };
A354206(n) = A354203(sigma(A354202(n)));

Multiplicative with a(p^e) = A354203((q^(e+1)-1)/(q-1)) where q = A354200(A000720(p)).
a(n) = A354203(A354205(n)) = A354203(sigma(A354202(n))).
a(n) = n - A354207(n).

A348935 Numbers k congruent to 1 or 5 mod 6, for which A064989(A064989(sigma(k^2))) < A064989(A064989(k^2)), where A064989 shifts the prime factorization one step towards lower primes, and sigma is the sum of divisors function.

Original entry on oeis.org

19, 23, 31, 37, 43, 47, 53, 61, 67, 73, 79, 83, 97, 103, 107, 109, 113, 127, 133, 137, 139, 149, 151, 157, 163, 179, 181, 191, 193, 197, 199, 211, 217, 223, 229, 233, 239, 241, 247, 251, 257, 259, 263, 269, 271, 277, 281, 283, 299, 301, 307, 311, 313, 317, 331, 335, 337, 341, 347, 349, 353, 359, 367, 371, 373, 379

Offset: 1

Antti Karttunen, Nov 04 2021

nonn

Square roots of squares present in A348753.
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 unitary divisor (d) should reside in this sequence, and the other divisor (x/d) in A348936. The explanation is similar to the one given in A348738. See also comments in A348933.
In range 1..2^20, there are 256143 numbers in this sequence and 93381 numbers in A348936.
The composites in this sequence are: 133, 217, 247, 259, 299, 301, 335, 341, 371, etc.

Cf. A000196, A007310, A348750, A348753, A348933, A348936.

f[2, e_] := 1; f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[400], MemberQ[{1, 5}, Mod[#, 6]] && s[s[DivisorSigma[1, #^2]]] < s[s[#^2]] &] (* 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)};
isA348935(n) = ((n%2)&&(n%3)&&(A064989(A064989(sigma(n^2))) < A064989(A064989(n^2))));

A348936 Numbers k congruent to 1 or 5 mod 6, for which A064989(A064989(sigma(k^2))) > A064989(A064989(k^2)), 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, 25, 29, 35, 41, 49, 55, 59, 65, 71, 77, 85, 89, 91, 95, 101, 115, 119, 121, 125, 131, 143, 145, 155, 161, 167, 169, 173, 175, 185, 187, 203, 205, 209, 215, 221, 227, 235, 245, 253, 265, 275, 287, 289, 293, 295, 305, 319, 323, 325, 329, 343, 355, 361, 365, 377, 383, 385, 391, 413, 415, 425, 445, 451

Offset: 1

Antti Karttunen, Nov 04 2021

nonn

Square roots of squares present in A348754.

See comments in A348935.

Cf. A000203, A003961, A007310, A064989, A348750, A348754, A348934, A348935.

f[2, e_] := 1; f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[450], MemberQ[{1, 5}, Mod[#, 6]] && s[s[DivisorSigma[1, #^2]]] > s[s[#^2]] &] (* 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)};
isA348936(n) = ((n%2)&&(n%3)&&(A064989(A064989(sigma(n^2))) > A064989(A064989(n^2))));

A354196 a(n) = A064989(A064989(sigma(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, 2, 1, 3, 23, 1, 9, 2, 14, 7, 2, 3, 3, 3, 1, 2, 14, 1, 3, 23, 6, 3, 9, 2, 2, 529, 1, 2, 763, 1, 7, 161, 161, 2, 2, 3, 21, 3, 483, 23, 3, 2, 7, 42, 109, 3, 2, 23, 2, 109, 3, 1, 69, 4, 6, 23, 6, 3, 3, 14, 21, 2, 3133, 23, 6, 13, 2, 201, 161, 23, 126, 3, 2, 4, 1, 21, 109, 7, 3, 461, 21, 13, 23, 21, 75, 46, 12, 1, 8

Offset: 1

Antti Karttunen, May 23 2022

nonn

Cf. A000203, A003961, A051027, A064989, A354195.

Cf. also A348750.

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); };
A354196(n) = A064989(A064989(sigma(sigma(A003961(A003961(n))))));

a(n) = A064989(A354195(A003961(n))) = A064989(A064989(A051027(A003961(A003961(n))))).

A354198 a(n) = A064989(A064989(sigma(sigma(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, 3, 1, 3, 3, 3, 2, 26, 23, 3, 3, 3, 1, 3, 21, 6, 3, 9, 14, 22, 2, 2, 7, 182, 3, 14, 313, 201, 3, 3, 3, 603, 3, 3, 3, 115, 3, 3, 2, 3, 3, 21, 2, 9, 9, 3, 2, 75, 2, 22, 3, 109, 3, 21, 46, 109, 2, 23, 7, 154, 3, 6, 22, 222, 2, 14, 2, 22, 29, 6, 1, 78, 3, 161, 69, 1407, 6, 2, 21, 44, 7, 21, 14, 201, 21, 39, 3, 529

Offset: 1

Antti Karttunen, May 24 2022

nonn

For any hypothetical odd perfect number opn that is not a multiple of 3, it holds that a(n) = A354196(n) = A348750(n) = n, where n = A064989(A064989(opn)). See also comments in A353365.

Cf. A000203, A003961, A064989, A066971, A326042, A353365, A354197.

Cf. also A348750, A354196.

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); };
A354198(n) = A064989(A064989(sigma(sigma(sigma(A003961(A003961(n)))))));

a(n) = A064989(A354197(A003961(n))) = A064989(A064989(A066971(A003961(A003961(n))))).

cp's OEIS Frontend

A348751 Numbers k for which A348750(k) < k.

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A348752 Numbers k for which A348750(k) > k.

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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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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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A348754 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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A354206 a(n) = A354203(sigma(A354202(n))), where A354202 is fully multiplicative with a(p) = A354200(A000720(p)), and A354203 is its left inverse.

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A348935 Numbers k congruent to 1 or 5 mod 6, for which A064989(A064989(sigma(k^2))) < A064989(A064989(k^2)), where A064989 shifts the prime factorization one step towards lower primes, and sigma is the sum of divisors function.

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A348936 Numbers k congruent to 1 or 5 mod 6, for which A064989(A064989(sigma(k^2))) > A064989(A064989(k^2)), where A064989 shifts the prime factorization one step towards lower primes, and sigma is the sum of divisors function.

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A354196 a(n) = A064989(A064989(sigma(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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