Jaroslav Krizek - cp's OEIS frontend

A358342 Lesser of twin primes p such that sigma((p-1)/2) + tau((p-1)/2) is a prime.

3, 5, 17, 65537, 1927561217, 6015902625062501, 12370388895062501, 835920078368222501, 6448645485213008897, 50973659693056000001, 54332889713542767617, 64304984013657011717, 112112769248058062501, 147337258721536000001

Offset: 1

Jaroslav Krizek, Nov 10 2022

Lesser of twin primes p such that A000203((p-1)/2) + A000005((p-1)/2) is a prime q.
The first 4 terms are Fermat primes from A019434.
Corresponding values of primes q:  2, 5, 19, 65551, 2248681529, ...
Subsequence of A272060 and A272061.
Lesser of twin primes of the form 2*m+1 with m a term of A064205.
There are no other terms <= 10^14.
All the terms above 3 are in A145824. - Amiram Eldar, Jan 05 2023

			17 and 19 are twin primes; sigma((17-1)/2) + tau((17-1)/2) = sigma(8) + tau(8) = 15 + 4 = 19; 19 is prime, so 17 is in the sequence.

Intersection of A001359 and A272061.

Cf. A000005 (tau), A000203 (sigma), A019434, A064205, A145824, A272060.

[n: n in [3..10^7] | IsPrime(n) and IsPrime(n+2) and IsPrime(&+Divisors((n-1) div 2) + #Divisors((n-1) div 2))]

Join[{3}, Select[4*Range[25000]^2 + 1, PrimeQ[#] && PrimeQ[# + 2] && PrimeQ[DivisorSigma[1, (# - 1)/2] + DivisorSigma[0, (# - 1)/2]] &]]
(* or *)
A272061 = Cases[Import["https://oeis.org/A272061/b272061.txt", "Table"], {, }][[;; , 2]]; Select[A272061, PrimeQ[# + 2] &] (* Amiram Eldar, Jan 05 2023 *)

isok(p) = if (isprime(p) && isprime(p+2), my(f=factor((p-1)/2)); isprime(sigma(f)+numdiv(f))); \\ Michel Marcus, Nov 23 2022

A354073 Multiply-perfect numbers that are the sum of the divisors of some number.

Original entry on oeis.org

1, 6, 28, 120, 496, 672, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, 1379454720, 1476304896, 8589869056, 14182439040, 31998395520, 43861478400, 51001180160, 66433720320, 137438691328, 153003540480, 403031236608, 518666803200

Offset: 1

Jaroslav Krizek, May 16 2022

nonn

Conjecture: 8128 is only multiply-perfect number that is not in this sequence.

The distinct values of A000203(A066961(n)).

			The multiply-perfect number 28 is in the sequence because 28 = sigma(12).

Amiram Eldar, Table of n, a(n) for n = 1..50

Intersection of A007691 and A002191.

Cf. A000203, A066961, A354072, A354074.

Set(Sort([&+Divisors(m): m in [1..10^7] | IsIntegral(&+Divisors(&+Divisors(m)) / &+Divisors(m))]))

a(18)-a(27) from Amiram Eldar, May 12 2024

A354072 Perfect numbers that are the sum of the divisors of some number.

Original entry on oeis.org

6, 28, 496, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216, 13164036458569648337239753460458722910223472318386943117783728128

Offset: 1

Jaroslav Krizek, May 16 2022

nonn

The distinct values of A000203(A146542(n)).

Conjecture: 8128 is the only perfect number that is not in this sequence.

			The perfect number 28 is in the sequence because 28 = sigma(12).
sigma(727145809044307968) = sigma(1152771972099211264) = 2305843008139952128.

Amiram Eldar, Table of n, a(n) for n = 1..11

Intersection of A000396 and A002191.

Cf. A000203, A146542, A354073, A354074.

Set(Sort([&+Divisors(m): m in [1..10^7] | &+Divisors(&+Divisors(m)) eq 2 * &+Divisors(m)]))

a(8)-a(10) from Amiram Eldar, May 12 2024

A354074 Factorials that are the sum of the divisors of some m.

Original entry on oeis.org

1, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000

Offset: 1

Jaroslav Krizek, May 16 2022

nonn

Sequence of different values of A000203(A245015(n)).

Conjecture: number 2 is the only factorial that is not in this sequence.

			Number 24 is in the sequence because sigma(14) = sigma(15) = sigma(23) = 24.

Intersection of A000142 and A002191.

Cf. A000203, A338112, A055486, A055488, A245015, A354072, A354073, A338112.

More terms from Jinyuan Wang, May 17 2022

A353034 Numbers m such that tau(m) = 2 * tau(m + 1) and simultaneously sigma(m) = 2 * sigma(m + 1), where tau(k) = A000005(k) and sigma(k) = A000203(k).

Original entry on oeis.org

20118, 20712, 79338, 103410, 203898, 267630, 570342, 907710, 1093026, 1228062, 1263918, 1663752, 2322760, 3268782, 3468486, 3527250, 5483418, 6277038, 6500442, 7637980, 9181578, 9297078, 17708178, 18638646, 25274946, 25364526, 25768302, 25909254, 31118664

Offset: 1

Jaroslav Krizek, Apr 18 2022

nonn

Corresponding values of tau(m): 16, 16, 16, 32, 16, 32, 16, 32, 16, 32, 16, 32, 32, ...

Corresponding values of sigma(m): 46080, 51840, 181440, 276480, 432000, 701568, 1200960, 2211840, ...

			tau(20118) = 16 = 2 * tau(20119) = 2 * 8, sigma(20118) = 46080 = 2 * sigma(20119) = 2 * 23040.

Cf. A000005 (tau), A000203 (sigma), A054004, A347603, A353033.

Subsequence of A163193.

[m: m in [2..10^6] | #Divisors(m) eq 2 * #Divisors(m + 1) and &+Divisors(m) eq 2 * &+Divisors(m + 1)]

A353033 Numbers m such that tau(m) = 2 * tau(m - 1) and simultaneously sigma(m) = 2 * sigma(m - 1), where tau(k) = A000005(k) and sigma(k) = A000203(k).

Original entry on oeis.org

6, 47796, 111684, 123498, 224562, 228378, 384858, 773016, 1096824, 1174542, 2351240, 2529414, 3320472, 3332616, 3650376, 4605096, 4838838, 4978476, 5014842, 5788662, 6023928, 6302724, 7658024, 8298978, 9287240, 9967974, 10950024, 12677496, 14036694, 14120360, 14927990

Offset: 1

Jaroslav Krizek, Apr 18 2022

nonn

Corresponding values of tau(m): 4, 24, 24, 16, 16, 16, 16, 32, 32, 16, 32, 32, 32, ...

Corresponding values of sigma(m): 12, 127680, 268128, 274560, 483840, 483840, 855360, 1996800, 2862720, 2472960, ...

			tau(6) = 4 = 2 * tau(5) = 2 * 2, sigma(6) = 12 = 2 * sigma(5) = 2 * 6.

Cf. A000005 (tau), A000203 (sigma), A054004, A347603, A353034.

[m: m in [2..10^6] | #Divisors(m) eq 2 * #Divisors(m - 1) and &+Divisors(m) eq 2 * &+Divisors(m - 1)]

A353032 a(n) is the smallest number m with n divisors such that m+1 has n-1 divisors, or 0 if no such number exists.

Original entry on oeis.org

0, 0, 4, 8, 81, 0, 0, 0, 441, 6723, 0, 0, 0, 0, 767495140624, 2024, 665416609183179841, 0, 0, 0, 2050624, 263168, 0, 0, 670801950625, 0, 10871232294189453124, 532899, 0, 0, 0, 0, 67634176, 0, 55471075527984793933106579132930662929175947116953798971172816083061185149078369140624

Offset: 1

Jaroslav Krizek, Apr 18 2022

nonn

For n > 33, a(64) = 6890624 is the only positive term <= 10^8.
There is no number m <= 10^10 that is the first start of run of 3 consecutive integers m, m+1 and m+2 with triplet [tau(m), tau(m+1), tau(m+2)] = [tau(m), tau(m) - 1, tau(m) - 2].
If a(11) > 0 then a(11) > 10^100. - Charles R Greathouse IV, Apr 20 2022
a(36) = 1626347583, a(40) = 1173953168, a(49) = 304006671424, a(65) = 25221297570561, a(81) = 15579533124, a(96) = 68195356770303, a(100) = 1698353697680, a(136) = 28528257204224, a(256) = 334435516415. - Jon E. Schoenfield, Apr 24 2022
From Jon E. Schoenfield, May 01 2022: (Start)
a(35) is the smallest m such that m = 16*p^6 = q^16*r - 1 where p, q, and r are odd primes; a(35) <= 16*123024356097427^6 (an 86-digit number).
a(37) = a(38) = 0;
a(39) <= 1134572901070399771884918212890624;
a(41) <= 350847983^40 (a 342-digit number). (End)

			For n = 5; a(5) = 81 because 81 is the smallest number m such that tau(m) = tau(81) = 5 and tau(82) = tau(m) - 1 = 4.

Cf. A000005, A068208, A343144.

Magma
```
Ax:=func; [Ax(n): n in [1..5]]
```

a(6)-a(8) from Jon E. Schoenfield, Apr 20 2022
a(9)-a(10), a(16), a(21)-a(22), a(28), a(33) from Jaroslav Krizek, Apr 20 2022
Remaining terms through a(34) from Jon E. Schoenfield, Apr 30 2022
a(35) from Jinyuan Wang, May 21 2022

A351890 Primes p such that tau(p - 1) - 1 = tau(p - 2) = tau(p - 3), where tau(k) is the number of divisors of k (A000005).

Original entry on oeis.org

5, 17, 65537, 9632244737, 20892967937, 127831991297, 149255504897, 159667373057, 351108391937, 542497063937, 1650957730817, 2270398022657, 2322380932097, 2747956028417, 2888694547457, 3516735087617, 6029264167937, 6122338640897, 6705696695297, 11125266727937

Offset: 1

Jaroslav Krizek, Mar 03 2022

nonn

Corresponding values of tau(a(n)-1): 3, 5, 17, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, ...
Corresponding values of tau(a(n)-2) = tau(a(n)-3): 2, 4, 16, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, ...
Quadruples of [tau(a(n)-3), tau(a(n)-2), tau(a(n)-1), tau(a(n))]: [2, 2, 3, 2], [4, 4, 5, 2], [16, 16, 17, 2], [32, 32, 33, 2], [32, 32, 33, 2], [32, 32, 33, 2], [32, 32, 33, 2], [32, 32, 33, 2], [32, 32, 33, 2], ...
Quadruple [32, 32, 33, 2] holds for all 128 terms 65537 < a(n) < 10^15.
Number p-1 is a perfect square as its number of divisors is odd.
The first 3 terms are Fermat primes from A019434.
Term 103565955613697 is the smallest primes p such that tau(p - 1) - 1 = tau(p - 2) = tau(p - 3) = tau(p - 4).

			Quadruple of [tau(65534), tau(65535), tau(65536), tau(65537)]: [16, 16, 17, 2].

Subsequence of A347078.

Cf. A000005 (tau), A019434.

[m: m in [4..10^6] | IsPrime(m) and #Divisors(m - 1) eq #Divisors(m - 2) + 1 and #Divisors(m - 2) eq #Divisors(m - 3)];

A351866 Numbers m such that sigma(m) = tau(m)! where sigma(k) = A000203(k) and tau(k) = A000005(k).

Original entry on oeis.org

1, 14, 15, 20154, 21496, 22390, 25978, 26314, 26386, 26439, 27687, 28041, 28671, 28911, 29365, 29397, 29559, 29607, 31135, 32263, 32335, 32665, 32669, 32785, 33383, 33901, 34177, 34279, 34903, 35167, 35629, 35867, 36049, 36271, 36613, 36859, 205286388, 239500772

Offset: 1

Jaroslav Krizek, Feb 22 2022

nonn

Corresponding values of sigma(m): 1, 24, 24, 40320, 40320, 40320, 40320, 40320, 40320, 40320, 40320, 40320, ...

Corresponding values of tau(m): 1, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, 8, ...

			sigma(14) = 24 = tau(14)! = 4!.

Cf. A000005 (tau), A000203 (sigma), A351865.

Subsequence of A245015.

[m: m in [1..5*10^6] | &+Divisors(m) eq Factorial(#Divisors(m))];

Select[Range[40000], DivisorSigma[1, #] == DivisorSigma[0, #]! &] (* Amiram Eldar, Feb 22 2022 *)

isok(m) = my(f=factor(m)); sigma(f) == numdiv(f)!; \\ Michel Marcus, Feb 23 2022

a(37)-a(38) from Amiram Eldar, Feb 22 2022

A350936 a(n) is the smallest number m such that tau(m) = ntau(m-1) = ntau(m+1) or 0 if no such m exists, where tau(k) = A000005(k).

Original entry on oeis.org

34, 6, 12, 30, 816, 60, 192, 270, 180, 240, 56320, 420, 233472, 2112, 1620, 1320, 2162688, 2340, 786432, 3120, 4800, 15360, 62914560, 3360, 172368, 724992, 6300, 29760, 24964497408, 12240, 35433480192, 7560, 599040, 15138816, 81648, 21600, 7215545057280

Offset: 1

Jaroslav Krizek, Jan 25 2022

nonn

Corresponding values of tau(a(n)): 4, 4, 6, 8, 20, 12, 14, 16, 18, 20, 44, 24, 52, 28, 30, 32, 68, 36, 38, 40, 42, 44, 92, 48, 100, 52, 54, 56, 116, 60, 124, 64, 132, 136, 70, 72, 296, ...

Triples of [tau(a(n) - 1), tau(a(n)), tau(a(n) + 1)] = [tau(a(n)) / n, tau(a(n)), tau(a(n)) / n]: [4, 4, 4], [2, 4, 2], [2, 6, 2], [2, 8, 2], [4, 20, 4], [2, 12, 2], [2, 14, 2], [2, 16, 2], [2, 18, 2], [2, 20, 2], [4, 44, 4], ...

			a(3) = 12 because 12 is the smallest number m such that tau(m) = 3 * tau(m-1) = 3 * tau(m+1); tau(12) = 3 * tau(11) = 3 * tau(13) = 3 * 2 = 6.

Cf. A000005, A190646, A350132, A350934, A350935.

Magma
```
Ax:=func; [Ax(n): n in [1..16]]
```

a(23)-a(37) from Jon E. Schoenfield, Jan 25 2022

cp's OEIS Frontend

User: Jaroslav Krizek

A358342 Lesser of twin primes p such that sigma((p-1)/2) + tau((p-1)/2) is a prime.

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A354073 Multiply-perfect numbers that are the sum of the divisors of some number.

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A354072 Perfect numbers that are the sum of the divisors of some number.

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A354074 Factorials that are the sum of the divisors of some m.

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A353034 Numbers m such that tau(m) = 2 * tau(m + 1) and simultaneously sigma(m) = 2 * sigma(m + 1), where tau(k) = A000005(k) and sigma(k) = A000203(k).

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A353033 Numbers m such that tau(m) = 2 * tau(m - 1) and simultaneously sigma(m) = 2 * sigma(m - 1), where tau(k) = A000005(k) and sigma(k) = A000203(k).

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A353032 a(n) is the smallest number m with n divisors such that m+1 has n-1 divisors, or 0 if no such number exists.

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A351890 Primes p such that tau(p - 1) - 1 = tau(p - 2) = tau(p - 3), where tau(k) is the number of divisors of k (A000005).

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A350936 a(n) is the smallest number m such that tau(m) = ntau(m-1) = ntau(m+1) or 0 if no such m exists, where tau(k) = A000005(k).