A349169 -id:A349169 - cp's OEIS frontend

A349753 Odd numbers k for which A003961(k)-2k divides A003961(k)-sigma(k), where A003961 shifts the prime factorization one step towards larger primes, and sigma is the sum of divisors function.

1, 3, 7, 25, 33, 55, 57, 69, 91, 93, 2211, 4825, 12639, 28225, 32043, 68727, 89575, 970225, 2245557, 16322559, 22799825, 48980427, 55037217, 60406599, 68258725, 325422273, 414690595, 569173299, 794579511, 10056372275, 10475647197, 10759889913, 11154517557

Offset: 1

Antti Karttunen, Dec 01 2021

nonn

Numbers k for which A326057(k) = gcd(A003961(k)-2k, A003961(k)-sigma(k)) is equal to abs(A252748(k)) = |A003961(k)-2k|.
The odd terms of A326134 form a subsequence of this sequence. Unlike in A326134, here we don't constrain the value of A252748(k) = A003961(k)-2k, thus allowing also values <= +1. Because of that, the odd terms of A048674 and A348514 are all included here, for example 57 and 68727 that occur in A348514, and 1, 3, 25, 33, 93, 970225, 325422273, 414690595 that occur in A048674.
Conjecture (1): This is a subsequence of A319630, in other words, for all terms k, gcd(k, A003961(k)) = 1.
Conjecture (2): Apart from 1, there are no common terms with A349169, which would imply that no odd perfect numbers exist.
None of the 36 initial terms is Zumkeller, in A083207, because all are deficient (in A005100). See also A337372. - Antti Karttunen, Dec 05 2024

Amiram Eldar, Table of n, a(n) for n = 1..36 (terms below 10^11)
Index entries for sequences where any odd perfect numbers must occur
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000203, A003961, A005100, A083207, A252748, A286385, A319630, A326057, A326134, A337372, A349169.
Cf. A048674, A348514.
Subsequence of A378980 (its odd terms).

f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := Divisible[(sn = s[n]) - DivisorSigma[1, n], sn - 2*n]; Select[Range[1, 10^6, 2], q] (* Amiram Eldar, Dec 04 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349753(n) = if(!(n%2), 0, my(s = A003961(n), t = (s-(2*n)), u = s-sigma(n)); !(u%t));

A364286 Composite numbers k for which A324644(k)/A324198(k) = 2.

Original entry on oeis.org

33, 51, 69, 91, 99, 135, 141, 145, 153, 159, 187, 207, 213, 217, 285, 295, 303, 321, 339, 391, 411, 423, 427, 435, 445, 477, 507, 519, 573, 637, 639, 679, 681, 699, 771, 783, 799, 843, 855, 861, 885, 895, 901, 909, 933, 951, 963, 1017, 1041, 1057, 1059, 1071, 1081, 1083, 1147, 1149, 1173, 1185, 1195, 1203, 1207

Offset: 1

Antti Karttunen, Jul 17 2023

nonn

See comments in A351458.

All terms are odd. Of the 63 initial terms of A349169, only term 13923 occurs also in this sequence. The first common term with A332458 is 161257. - Antti Karttunen, Mar 10 2024

Cf. A000203, A276086, A324198, A324644, A332458, A349169, A351458, A371082 (subsequence).

Subsequence of A082686.

f[x_] := Block[{m, i, n = x, p}, m = i = 1; While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m]; Select[Select[Range[1350], CompositeQ], GCD[#2, #3]/GCD[#1, #3] == 2 & @@ {#, DivisorSigma[1, #], f[#]} &] (* Michael De Vlieger, Mar 10 2024 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA364286(n) = if(isprime(n), 0, my(u=A276086(n)); (gcd(sigma(n),u)==2*gcd(n,u))); \\ Antti Karttunen, Mar 10 2024

A348993 a(n) = A064989(sigma(n) / gcd(sigma(n), A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes, while A064989 shifts it back towards smaller primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 1, 5, 2, 1, 1, 3, 11, 2, 2, 5, 5, 1, 2, 29, 4, 11, 3, 1, 1, 2, 2, 1, 29, 5, 1, 5, 6, 2, 1, 5, 2, 4, 2, 55, 17, 3, 5, 3, 10, 1, 7, 5, 22, 2, 2, 29, 34, 29, 4, 25, 8, 1, 4, 3, 1, 6, 6, 1, 29, 1, 11, 113, 2, 2, 13, 5, 2, 2, 4, 11, 31, 17, 29, 15, 2, 5, 3, 29, 49, 10, 10, 5, 8, 7, 2, 3, 12, 22, 5, 5, 1, 2, 6, 5

Offset: 1

Antti Karttunen, Nov 10 2021

Cf. A000203, A000265, A003961, A064989, A161942, A342671, A348992, A349162, A349169 (gives odd k for which a(k) = A319627(k)).

Array[Times @@ Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger[#1/GCD[##]]] & @@ {DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &, 96] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
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)};
A349162(n) = { my(s=sigma(n)); (s/gcd(s,A003961(n))); };
A348993(n) = A064989(A349162(n));

a(n) = A064989(A349162(n)) = A064989(A348992(n)).

A349176 Odd numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)) > 1, where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors function.

Original entry on oeis.org

135, 285, 435, 455, 855, 885, 1185, 1287, 1305, 1335, 1425, 1435, 1485, 1635, 2235, 2275, 2295, 2655, 2685, 2905, 2985, 3105, 3135, 3185, 3311, 3395, 3435, 3555, 3585, 4005, 4035, 4185, 4425, 4785, 4865, 4905, 4995, 5385, 5685, 5805, 5835, 5845, 5925, 6135, 6237, 6335, 6345, 6585, 6675, 6735, 7125, 7155, 7175, 7185

Offset: 1

Antti Karttunen, Nov 11 2021

nonn

			For n = 135 = 3^3 * 5, sigma(135) = 240 = 2^4 * 3 * 5, A003961(135) = 5^3 * 7 = 875, and gcd(135,875) = gcd(240,875) = 5, which is larger than 1, therefore 135 is included in the sequence.

Intersection of A104210 and A349174, or equally, intersection of A349166 and A349174.
Subsequence of A372567.
Cf. A000203, A003961, A319626, A348994, A349161, A349164, A349169.

Select[Range[1, 7200, 2], And[#1/#2 == #1/#3, #2 > 1] & @@ {#3, GCD[#1, #3], GCD[#2, #3]} & @@ {#, DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349176(n) = if(!(n%2),0,my(u=A003961(n),t=gcd(u,n)); (t>1)&&(gcd(u,sigma(n))==t));

A348990 a(n) = n / gcd(n, A003961(n)), where A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 7, 8, 9, 10, 11, 4, 13, 14, 3, 16, 17, 6, 19, 20, 21, 22, 23, 8, 25, 26, 27, 28, 29, 2, 31, 32, 33, 34, 5, 4, 37, 38, 39, 40, 41, 14, 43, 44, 9, 46, 47, 16, 49, 50, 51, 52, 53, 18, 55, 56, 57, 58, 59, 4, 61, 62, 63, 64, 65, 22, 67, 68, 69, 10, 71, 8, 73, 74, 15, 76, 7, 26, 79, 80, 81, 82, 83, 28

Offset: 1

Antti Karttunen, Nov 10 2021

Denominator of ratio A003961(n) / n. This ratio is fully multiplicative, and A348994(n) / a(n) = A319626(A003961(n)) / A319627(A003961(n)) gives it in its lowest terms.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A000035, A000961, A002110, A003961, A319626, A319627, A319630 (fixed points), A322361, A349169 (where equal to A348992).

Cf. A348994 (numerators).

Array[#1/GCD[##] & @@ {#, Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &, 84] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A348990(n) = (n/gcd(n, A003961(n)));

a(n) = n / A322361(n) = n / gcd(n, A003961(n)).
a(n) = A319627(A003961(n)).
For all odd numbers n, a(n) = A003961(A319627(n)).
For all n >= 1, A000035(A348990(n)) = A000035(n). [Preserves the parity]

A386430 Odd numbers k such that there are no prime factors p of sigma(k) such that p does not divide A003961(k) and the valuation(k, p) is different from valuation(sigma(k), p), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 3, 7, 15, 21, 27, 31, 33, 57, 69, 87, 91, 93, 105, 127, 141, 177, 189, 195, 217, 231, 237, 273, 285, 301, 381, 399, 447, 465, 483, 495, 513, 567, 573, 597, 609, 627, 651, 717, 775, 819, 837, 861, 889, 903, 987, 1023, 1029, 1149, 1185, 1239, 1311, 1365, 1419, 1431, 1437, 1455, 1497, 1561, 1653, 1659, 1687, 1743

Offset: 1

Antti Karttunen, Aug 22 2025

Conjecture: After the initial 1, and apart from any hypothetical odd perfect numbers, all other terms are in A248150, i.e., sigma(k) == 0 (mod 4). This would imply (with the same caveat), that this sequence has no common terms with A228058 and no squares larger than one. This is true at least for the first 709203 terms (terms in range [1..2^34]).

Terms k such that A162642(k) = 1 are rare: 3, 7, 27, 31, 127, 567, 775, 8191, 27783, 131071, 524287, 2147483647, ... (odd terms of A387160).

			a(386548) = 5919068925 = 3^4 * 5^2 * 7^2 * 11^2 * 17 * 29. sigma(5919068925) = 15355618740 = 2^2 * 3^4 * 5 * 7 * 11^2 * 19^2 * 31. The "don't care primes" is given by A003961(A007947(5919068925))) = 2947945 = 5*7*11*13*19*31, thus only odd prime factor that matters here is 3, which in case has the same exponent (4) in both n = 5919068925 and sigma(n). In a way, this number is very close to satisfying Euler's criterion for odd perfect numbers (A228058), except that it has two unitary prime factors of the form 4k+1, instead of just one, apart from the square factor. Both n/17 and n/29 are in A228058.

Odd terms of A351554.
Cf. A000203, A003961, A056169, A162642, A228058, A248150, A351555, A386424, A387160.
Cf. A349169 (subsequence).

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)); };
isA386430(n) = ((n%2) && (0==A351555(n)));

{k | k odd, A351555(k) = 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.

A349177 Odd numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)) = 1, where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 55, 59, 61, 63, 67, 69, 71, 73, 79, 81, 83, 85, 89, 91, 93, 95, 97, 101, 103, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 137, 139, 141, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 167, 169, 173

Offset: 1

Antti Karttunen, Nov 11 2021

nonn

Odd numbers k for which k and A003961(k) are relatively prime, and also sigma(k) and A003961(k) are coprime.

Subsequence of A349174 from this first differs by not having term 135 (see A349176).
Intersection of A319630 and A349174, or equally, intersection of A349165 and A349174.
Cf. A000203, A003961, A319626, A348994, A349161, A349164, A349169.

Select[Range[1, 173, 2], GCD[#1, #3] == GCD[#2, #3] == 1 & @@ {#, DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349177(n) = if(!(n%2),0,my(u=A003961(n),t=gcd(u,n)); (1==t)&&(gcd(u,sigma(n))==t));

A349746 Numbers k for which k * gcd(sigma(k), u) is equal to sigma(k) * gcd(k, u), where u is obtained by shifting the prime factorization of k two steps toward larger primes [with u = A003961(A003961(k))].

Original entry on oeis.org

1, 11466, 114660, 411264, 804384, 871416, 4999680, 46332000, 176417280, 378069120, 396168192, 485188704, 709430400, 2004912000, 3921372000, 5600534400, 6128179200, 6956471808, 7556976000, 7746979968, 9904204800, 14092001280, 14182439040, 23423662080, 31998395520

Offset: 1

Antti Karttunen, Nov 30 2021

nonn

Sigma preserves both the 2-adic and 3-adic valuation of the terms of this sequence.

All 65 known 5-multiperfect numbers (A046060) are included in this sequence, as well as the smallest 7-multiperfect number, 141310897947438348259849402738485523264343544818565120000 = A007539(7), and probably the majority of other p-multiperfect numbers as well, where p is a prime > 3. However, any term that is in A349747 is not included in this sequence.

Cf. A000203, A003961, A007539, A007814, A007949, A046060, A349747.

Cf. also A349169, A349745.

f[p_, e_] := NextPrime[p, 2]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := n * GCD[(sigma = DivisorSigma[1, n]), (u = s[n])] == sigma * GCD[n, u]; Select[Range[10^6], q] (* Amiram Eldar, Dec 01 2021 *)

A003961twice(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(1+nextprime(1+f[i, 1]))); factorback(f); };
isA349746(n) = { my(s=sigma(n),u=A003961twice(n)); (n*gcd(s,u) == (s*gcd(n,u))); };

For all n >= 1, A007814(A000203(a(n))) = A007814(a(n)) and A007949(A000203(a(n))) = A007949(a(n)). [See comment]

a(15)-a(25) from Martin Ehrenstein, Dec 17 2021

A349175 Odd numbers k for which gcd(k, A003961(k)) <> gcd(sigma(k), A003961(k)), where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors function.

Original entry on oeis.org

15, 27, 35, 45, 57, 65, 75, 77, 87, 99, 105, 143, 165, 171, 175, 177, 189, 195, 205, 221, 225, 231, 237, 245, 255, 261, 267, 297, 301, 315, 323, 325, 327, 345, 351, 375, 385, 399, 405, 415, 417, 429, 437, 447, 459, 465, 485, 495, 513, 525, 531, 537, 539, 555, 567, 585, 595, 597, 605, 609, 615, 621, 627, 629, 645

Offset: 1

Antti Karttunen, Nov 10 2021

nonn

Odd numbers for which A348994(n) <> A349161(n).

Equally, odd numbers such that A319626(n) <> A349164(n).

Cf. A000203, A003961, A319626, A348994, A349161, A349164.

Cf. A349169, A349174 (complement among the odd numbers).

Select[Range[1, 645, 2], GCD[#1, #3] != GCD[#2, #3] & @@ {#, DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349175(n) = if(!(n%2),0,my(u=A003961(n)); gcd(u,sigma(n))!=gcd(u,n));

cp's OEIS Frontend

A349753 Odd numbers k for which A003961(k)-2k divides A003961(k)-sigma(k), where A003961 shifts the prime factorization one step towards larger primes, and sigma is the sum of divisors function.

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A364286 Composite numbers k for which A324644(k)/A324198(k) = 2.

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A348993 a(n) = A064989(sigma(n) / gcd(sigma(n), A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes, while A064989 shifts it back towards smaller primes, and sigma is the sum of divisors function.

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A349176 Odd numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)) > 1, where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors function.

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A348990 a(n) = n / gcd(n, A003961(n)), where A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).

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A386430 Odd numbers k such that there are no prime factors p of sigma(k) such that p does not divide A003961(k) and the valuation(k, p) is different from valuation(sigma(k), p), 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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A349177 Odd numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)) = 1, where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors function.

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