A351538 -id:A351538 - cp's OEIS frontend

A349745 Numbers k for which k * gcd(sigma(k), A003961(k)) is equal to sigma(k) * gcd(k, A003961(k)), where A003961 shifts the prime factorization one step towards larger primes, and sigma is the sum of divisors function.

1, 120, 216, 672, 2464, 22176, 228480, 523776, 640640, 837760, 5581440, 5765760, 7539840, 12999168, 19603584, 33860736, 38342304, 71344000, 95472000, 102136320, 197308800, 220093440, 345080736, 459818240, 807009280, 975576960, 1476304896, 1510831360, 1773584640

Offset: 1

Antti Karttunen, Nov 29 2021

nonn

Numbers k for which k * A342671(k) = A000203(k) * A322361(k).
Numbers k such that gcd(A064987(k), A191002(k)) = gcd(A064987(k), A341529(k)).
Obviously, all odd terms in this sequence must be squares.
All the terms k of A005820 that satisfy A007949(k) < A007814(k) [i.e., whose 3-adic valuation is strictly less than their 2-adic valuation] are also terms of this sequence. Incidentally, the first six known terms of A005820 satisfy this condition, while on the other hand, any hypothetical odd 3-perfect number would be excluded from this sequence. Also, as a corollary, any hypothetical 3-perfect numbers of the form 4u+2 must not be multiples of 3 if they are to appear here. Similarly for any k which occurs in A349169, for 2*k to occur in this sequence, it shouldn't be a multiple of 3 and k should also be a term of A191218. See question 2 and its partial answer in A349169.
From Antti Karttunen, Feb 13-20 2022: (Start)
Question: Are all terms/2 (A351548) abundant, from n > 1 onward?
Note that of the 65 known 5-multiperfect numbers, all others except these three 1245087725796543283200, 1940351499647188992000, 4010059765937523916800 are also included in this sequence. The three exceptions are distinguished by the fact that their 3 and 5-adic valuations are equal. In 62 others the former is larger.
If k satisfying the condition were of the form 4u+2, then it should be one of the terms of A191218 doubled as only then both k and sigma(k) are of the form 4u+2, with equal 2-adic valuations for both. More precisely, one of the terms of A351538.
(End)

Amiram Eldar, Table of n, a(n) for n = 1..52 (terms below 10^11)
Antti Karttunen, Ratio A342671(n)/A322361(n) plotted with OEIS Plot2 tool
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000203, A003961, A005101, A005820, A007814, A007949, A046060, A064987, A191002, A191218, A248150, A322361, A341529, A342671, A347870, A351534, A351536, A351538.
Cf. also A349169, A349746, A351458, A351549 for other variants.
Subsequence of A351554 and also of its subsequence A351551.
Cf. A351459 (subsequence, intersection with A351458), A351548 (terms halved).

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := NextPrime[p]^e; q[1] = True; q[n_] := n * GCD[(s = Times @@ f1 @@@ (f = FactorInteger[n])), (r = Times @@ f2 @@@ f)] == s*GCD[n, r]; Select[Range[10^6], q] (* Amiram Eldar, Nov 29 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349745(n) = { my(s=sigma(n),u=A003961(n)); (n*gcd(s,u) == (s*gcd(n,u))); };

For all n >= 1, A007814(A000203(a(n))) = A007814(a(n)). [sigma preserves the 2-adic valuation of the terms of this sequence]

A351543 Even numbers k such that there is an odd prime p that divides sigma(k), but valuation(k, p) differs from valuation(sigma(k), p), and p does not divide A003961(k), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

Original entry on oeis.org

4, 8, 12, 16, 18, 26, 32, 36, 38, 44, 48, 50, 52, 56, 58, 64, 68, 72, 74, 76, 78, 80, 82, 86, 88, 90, 92, 96, 98, 100, 104, 108, 112, 116, 118, 122, 124, 126, 128, 132, 134, 136, 144, 146, 148, 150, 152, 156, 158, 162, 164, 166, 172, 176, 178, 180, 184, 188, 192, 194, 196, 200, 202, 204, 206, 208, 212, 218, 222, 226

Offset: 1

Antti Karttunen, Feb 16 2022

nonn

Even numbers k such that sigma(k) has an odd prime factor prime(i), but prime(i-1) is not a factor of k, and A286561(k, prime(i)) <> A286561(sigma(k), prime(i)). This differs from the definition of A351542 in that prime(i) is not here required to be a factor of k itself. The condition implies also that if there is any such odd prime factor prime(i) of sigma(k), it must be >= 5.
Even numbers k for which A351555(k) > 0.
Question: Is A351538 subsequence of this sequence?

			12 = 2^2 * 3 is present as sigma(12) = 28 = 2^2 * 7, whose prime factorization contains an odd prime 7 such that neither it nor the immediately previous prime, which is 5, divide 12 itself.
196 = 2^2 * 7^2 is present as sigma(196) = 399 = 3^1 * 7^1 * 19^1, which thus has a shared prime factor 7 with 196, but occurring with smaller exponent, and with no prime 5 (which is the previous prime before 7) present in the prime factorization of 196.
364 = 2^2 * 7^1 * 13^1 is present as sigma(364) = 784 = 2^4 * 7^2, which thus has a shared prime factor 7 with 364, but occurring with larger exponent, and with no prime 5 (which is the previous prime before 7) present in the prime factorization of 364.

Cf. A000203, A003961, A286561, A351555.
Subsequences: A351541, A351542, and also conjecturally A351538.
Cf. A351553 (complement among even numbers).
No common terms with A349745.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351555(n) = { my(s=sigma(n),f=factor(s),u=A003961(n)); sum(k=1,#f~,if((f[k,1]%2) && 0!=(u%f[k,1]), (valuation(n,f[k,1])!=f[k,2]), 0)); };
isA351543(n) = (!(n%2) && A351555(n)>0);

A351548 a(n) = A349745(n) divided by 2 if it is even, and 0 if A349745(n) is odd.

Original entry on oeis.org

0, 60, 108, 336, 1232, 11088, 114240, 261888, 320320, 418880, 2790720, 2882880, 3769920, 6499584, 9801792, 16930368, 19171152, 35672000, 47736000, 51068160, 98654400, 110046720, 172540368, 229909120, 403504640, 487788480, 738152448, 755415680, 886792320, 1960686000, 2070484416, 2339064000, 2889432000

Offset: 1

Antti Karttunen, Feb 18 2022

nonn

Questions: Are all nonzero terms abundant (in A005101)? Are all terms even? Could either be proved? See also comments in A351538 and in A351549.

The terms a(2) .. a(52) are all also practical (A005153) and Zumkeller (A083207). - Antti Karttunen, Dec 05 2024

Antti Karttunen, Table of n, a(n) for n = 1..52 (prepared from the b-file of A349745 computed by _Amiram Eldar_)
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A005101, A005153, A083207, A326051 (all six known terms are present here), A329963, A349169, A349745, A351458, A351459, A351538.

Cf. also A351549.

A351548(n) = { my(u=A349745(n)); if(u%2,0,u/2); };

a(n) = 0 if A349745(n) is odd, a(n) = A349745(n)/2 otherwise.

A351537 Odd numbers k for which sigma(k) is congruent to 2 modulo 4 and is not a multiple of 3.

Original entry on oeis.org

13, 37, 61, 73, 97, 109, 117, 157, 181, 193, 229, 241, 277, 313, 325, 333, 337, 349, 373, 397, 409, 421, 433, 457, 541, 549, 577, 601, 613, 657, 661, 673, 709, 733, 757, 769, 829, 853, 873, 877, 925, 937, 981, 997, 1009, 1021, 1033, 1053, 1069, 1093, 1117, 1129, 1153, 1201, 1213, 1237, 1249, 1297, 1321, 1381, 1413

Offset: 1

Antti Karttunen, Feb 14 2022

nonn

Terms are of the form p^e*m^2 where e is 1 or 9 mod 12, p is a prime = 1 mod 12 and m is an odd number not divisible by p with sigma(m^2) not divisible by 3, i.e., q^e || m implies e is not 1 mod 3 or q = 2 mod 3. - Charles R Greathouse IV, Feb 14 2022

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Intersection of A191218 and A329963.
Contains A068228 as a subsequence.
Terms of A351538 halved.

Select[Range[1, 1440, 2], MemberQ[{2, 10}, Mod[DivisorSigma[1, #], 12]] &] (* Michael De Vlieger, Feb 14 2022 *)

isA351537(n) = if(!(n%2),0,my(s=sigma(n)); (2 == (s%4)) && (0 != (s%3)));

list(lim)=my(v=List()); forstep(m=1,sqrtint(lim\13),2, my(m2=m^2); if(sigma(m2)%3==0,next); forprimestep(p=13,lim\m2,12, m%p && listput(v,p*m2))); forstep(e=9,logint(lim\1,13),[4,8], forstep(m=1,sqrtint(lim\13^e),2, my(m2=m^2); if(sigma(m2)%3==0,next); forprimestep(p=13,lim\m2,12, m%p && listput(v,p^e*m2)))); Set(v) \\ Charles R Greathouse IV, Feb 14 2022

a(n) = A351538(n)/2.

cp's OEIS Frontend

A349745 Numbers k for which k * gcd(sigma(k), A003961(k)) is equal to sigma(k) * gcd(k, A003961(k)), where A003961 shifts the prime factorization one step towards larger primes, and sigma is the sum of divisors function.

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A351543 Even numbers k such that there is an odd prime p that divides sigma(k), but valuation(k, p) differs from valuation(sigma(k), p), and p does not divide A003961(k), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

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A351548 a(n) = A349745(n) divided by 2 if it is even, and 0 if A349745(n) is odd.

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A351537 Odd numbers k for which sigma(k) is congruent to 2 modulo 4 and is not a multiple of 3.

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A351550 Numbers of the form 4u+2 in A351551.

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