A246281 -id:A246281 - cp's OEIS frontend

A246373 Primes p such that if 2p-1 = product_{k >= 1} A000040(k)^(c_k), then p <= product_{k >= 1} A000040(k-1)^(c_k).

2, 3, 7, 19, 29, 31, 37, 47, 67, 71, 79, 89, 97, 101, 103, 107, 109, 127, 139, 151, 157, 181, 191, 197, 199, 211, 223, 227, 229, 241, 251, 269, 271, 277, 283, 307, 317, 331, 337, 349, 359, 367, 373, 379, 397, 409, 421, 433, 439, 457, 461, 467, 487, 499, 521, 541, 547, 569, 571, 577, 601

Offset: 1

Antti Karttunen, Aug 25 2014

nonn

Primes p such that A064216(p) >= p, or equally, A064989(2p-1) >= p.

All primes of A005382 are present here, because if 2p-1 is prime q, Bertrand's postulate guarantees (after cases 2 and 3 which are in A048674) that there exists at least one prime r larger than p and less than q = 2p-1, for which A064989(q) = r.

			2 is present, as 2*2 - 1 = 3 = p_2, and p_{2-1} = p_1 = 2 >= 2.
3 is present, as 2*3 - 1 = 5 = p_3, and p_{3-1} = p_2 = 3 >= 3.
5 is not present, as 2*5 - 1 = 9 = p_2 * p_2, and p_1 * p_1 = 4, with 4 < 5.
7 is present, as 2*7 - 1 = 13 = p_6, and p_5 = 11 >= 7.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Wikipedia, Bertrand's postulate

Intersection of A000040 and A246372.
Subsequence: A005382.
A246374 gives the primes not here.
Cf. A064216, A064989, A246281.

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)};
n = 0; forprime(p=2,2^31, if((A064989((2*p)-1) >= p), n++; write("b246373.txt", n, " ", p); if(n > 9999, break)));
(Scheme, with Antti Karttunen's IntSeq-library)
(define A246373 (MATCHING-POS 1 1 (lambda (n) (and (prime? n) (>= (A064216 n) n)))))

A275717 Numbers n for which A003961(n) > A003961(n-1).

Original entry on oeis.org

2, 3, 4, 6, 8, 12, 14, 15, 16, 18, 20, 24, 26, 27, 30, 32, 35, 36, 38, 39, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 78, 80, 81, 84, 86, 87, 88, 90, 92, 95, 96, 98, 100, 102, 104, 108, 110, 112, 114, 116, 117, 119, 120, 122, 123, 124, 125, 126, 128, 130, 132, 135, 138, 140, 143, 144

Offset: 1

Antti Karttunen, Aug 07 2016

nonn

One more than the positions of ascents in permutation A048673.

Antti Karttunen, Table of n, a(n) for n = 1..10000

One more than A275721.
Cf. A003961, A048673.
Complement: A275718 (apart from 1 which is in neither sequence).
Cf. A029744 (a subsequence, apart from its initial 1).
Cf. also A275719, A275720, A246281, A246282.

f[n_] := Times @@ Map[Prime[PrimePi@ First[#] + 1]^Last[#] &, FactorInteger@ n]; Select[Range@ 145, f[# - 1] < f@ # &] (* Michael De Vlieger, Aug 07 2016 *)

A275718 Numbers n for which A003961(n) < A003961(n-1).

Original entry on oeis.org

5, 7, 9, 10, 11, 13, 17, 19, 21, 22, 23, 25, 28, 29, 31, 33, 34, 37, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 65, 67, 69, 71, 73, 76, 77, 79, 82, 83, 85, 89, 91, 93, 94, 97, 99, 101, 103, 105, 106, 107, 109, 111, 113, 115, 118, 121, 127, 129, 131, 133, 134, 136, 137, 139, 141, 142, 145, 148, 149, 151, 153, 154, 155

Offset: 1

Antti Karttunen, Aug 07 2016

nonn

One more than the positions of descents in permutation A048673.

Antti Karttunen, Table of n, a(n) for n = 1..10000

One more than A275722.
Complement: A275717 (apart from 1 which is in neither sequence).
Cf. A003961, A048673.
Cf. also A275719, A275720, A246281, A246282.

f[n_] := Times @@ Map[Prime[PrimePi@ First[#] + 1]^Last[#] &, FactorInteger@ n]; Select[Range@ 155, f[# - 1] > f@ # &] (* Michael De Vlieger, Aug 07 2016 *)

A337379 Numbers k for which A003961(k) < 2*sigma(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 101, 102, 103

Offset: 1

Antti Karttunen, Aug 27 2020

nonn

Cf. A000203, A003961, A286385.
Cf. A337378 (complement).
Positions of zeros in A337380.
Cf. also A246281, A337382 (subsequences).

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA337379(n) = (A003961(n)<2*sigma(n));

A246342 a(0) = 12, after which, if a(n-1) = product_{k >= 1} (p_k)^(c_k), then a(n) = (1/2) * (1 + product_{k >= 1} (p_{k+1})^(c_k)), where p_k indicates the k-th prime, A000040(k).

Original entry on oeis.org

12, 23, 15, 18, 38, 35, 39, 43, 24, 68, 86, 71, 37, 21, 28, 50, 74, 62, 56, 149, 76, 104, 230, 305, 235, 186, 278, 224, 1337, 1062, 2288, 8951, 4482, 16688, 67271, 33637, 16821, 66688, 571901, 338059, 181516, 258260, 455900, 1180337, 1080207, 1817863, 1157487, 984558, 1230848, 53764115

Offset: 0

Antti Karttunen, Aug 24 2014

nonn

Iterates of A048673 starting from value 12.
All numbers 1 .. 11 are in finite cycles of A048673/A064216, thus 12 is the smallest number in this cycle, regardless of whether the cycle is infinite or finite.
This sequence soon reaches much larger values than the corresponding A246343 (iterating the same cycle in the other direction). However, with the corresponding sequences starting from 16 (A246344 & A246345), there is no such pronounced difference, and with them the bias is actually the other way.

			Start with a(0) = 12; thereafter each new term is obtained by replacing each prime factor of the previous term with the next prime, to whose product 1 is added before it is halved:
12 = 2^2 * 3 = p_1^2 * p_2 -> ((p_2^2 * p_3)+1)/2 = ((9*5)+1)/2 = 23, thus a(1) = 23.
23 = p_9 -> (p_10 + 1)/2 = (29+1)/2 = 15, thus a(2) = 15.

Antti Karttunen, Table of n, a(n) for n = 0..1001

A246343 gives the terms of the same cycle when going in the opposite direction from 12.
Cf. A048673, A064216, A246344, A246345.
Cf. also A246281, A246282, A246351, A246352, A246361, A246362, A246371, A246372.

default(primelimit, 2^30);
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A048673(n) = (A003961(n)+1)/2;
k = 12; for(n=0, 1001, write("b246342.txt", n, " ", k) ; k = A048673(k));
(Scheme, with memoization-macro definec)
(definec (A246342 n) (if (zero? n) 12 (A048673 (A246342 (- n 1)))))

a(0) = 12, and for n >= 1, a(n) = A048673(a(n-1)).

A246343 a(0) = 12, after which, if (2*a(n-1)) - 1 = product_{k >= 1} (p_k)^(c_k) then a(n) = product_{k >= 1} (p_{k-1})^(c_k), where p_k indicates the k-th prime, A000040(k).

Original entry on oeis.org

12, 19, 31, 59, 44, 46, 55, 107, 134, 166, 317, 398, 282, 557, 470, 622, 763, 531, 1051, 1267, 1807, 3607, 7211, 4522, 9041, 3700, 3725, 3982, 7951, 15889, 30053, 24018, 24189, 34535, 14630, 12916, 21769, 27599, 24524, 32678, 26094, 43073, 34446, 68881, 116479, 143359, 275221, 550439, 667462, 1051489

Offset: 0

Antti Karttunen, Aug 24 2014

nonn

Iterates of A064216 starting from value 12.

All numbers from 1 to 11 are in finite cycles of A048673/A064216, thus 12 is the smallest number in this cycle, regardless of whether it is infinite or finite.

			Start with a(0) = 12; then after each new term is obtained by doubling the previous term, from which one is subtracted, after which each prime factor is replaced with the previous prime:
12 -> ((2*12)-1) = 23 = p_9, and p_8 = 19, thus a(1) = 19.
19 -> ((2*19)-1) = 37 = p_12, and p_11 = 31, thus a(2) = 31.
31 -> ((2*31)-1) = 61 = p_18, and p_17 = 59, thus a(3) = 59.
59 -> ((2*59)-1) = 117 = 3*3*13 = p_2 * p_2 * p_6, and p_1 * p_1 * p_5 = 2*2*11 = 44, thus a(4) = 44.

Antti Karttunen, Table of n, a(n) for n = 0..1001

A246342 gives the terms of the same cycle when going to the opposite direction from 12.
Cf. A048673, A064216, A246344, A246345.
Cf. also A246281, A246282, A246351, A246352, A246361, A246362, A246371, A246372.

default(primelimit, 2^30);
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)};
A064216(n) = A064989((2*n)-1);
k = 12; for(n=0, 1001, write("b246343.txt", n, " ", k); k = A064216(k));
(Scheme, with memoization-macro definec)
(definec (A246343 n) (if (zero? n) 12 (A064216 (A246343 (- n 1)))))

a(0) = 12, a(n) = A064216(a(n-1)).

A246344 a(0) = 16, after which, if a(n-1) = product_{k >= 1} (p_k)^(c_k), then a(n) = (1/2) * (1 + product_{k >= 1} (p_{k+1})^(c_k)), where p_k indicates the k-th prime, A000040(k).

Original entry on oeis.org

16, 41, 22, 20, 32, 122, 101, 52, 77, 72, 338, 434, 611, 451, 280, 1040, 4820, 7907, 3960, 30713, 15364, 22577, 12154, 9791, 4902, 8108, 9131, 5815, 4099, 2056, 3551, 2095, 1474, 1385, 984, 2903, 1455, 1768, 4361, 5869, 2940, 19058, 18845, 13227, 11053, 8707, 4357, 2182, 1640, 4064, 15917, 9432, 46238

Offset: 0

Antti Karttunen, Aug 24 2014

nonn

Iterates of A048673 starting from value 16.

Either this sequence is actually part of the cycle containing 12 (see A246342) or 16 is the smallest member of this cycle (regardless of whether this cycle is finite or infinite), which follows because all numbers 1 .. 11 are in finite cycles, and also 13 and 14 are in closed cycles and 15 is in the cycle of 12.

			Start with a(0) = 16; then after each new term is obtained by replacing each prime factor of the previous term with the next prime, to whose product is added one before it is halved:
16 = 2^4 = p_1^4 -> ((p_2^4)+1)/2 = (3^4 + 1)/2 = (81+1)/2 = 41, thus a(1) = 41.
41 = p_13 -> ((p_14)+1)/2 = (43+1)/2 = 22, thus a(2) = 22.

Antti Karttunen, Table of n, a(n) for n = 0..1001

A246345 gives the terms of the same cycle when going to the opposite direction from 16.
Cf. A048673, A064216, A246342, A246343.
Cf. also A246281, A246282, A246351, A246352, A246361, A246362, A246371, A246372.

default(primelimit, 2^30);
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i,1] = nextprime(f[i,1]+1)); factorback(f); \\ Using code of Michel Marcus
A048673(n) = (A003961(n)+1)/2;
k = 16; for(n=0, 1001, write("b246344.txt", n, " ", k) ; k = A048673(k));
(Scheme, with memoization-macro definec)
(definec (A246344 n) (if (zero? n) 16 (A048673 (A246344 (- n 1)))))

a(0) = 16, and for n >= 1, a(n) = A048673(a(n-1)).

A246345 a(0) = 16, after which, if (2*a(n-1)) - 1 = product_{k >= 1} (p_k)^(c_k) then a(n) = product_{k >= 1} (p_{k-1})^(c_k), where p_k indicates the k-th prime, A000040(k).

Original entry on oeis.org

16, 29, 34, 61, 49, 89, 106, 199, 389, 310, 617, 524, 694, 1207, 1921, 3097, 3899, 4142, 3374, 3674, 4234, 8461, 16903, 20211, 37841, 22408, 26853, 26391, 48031, 68605, 137201, 81272, 108334, 137809, 266737, 512627, 347932, 497005, 982081, 1942279, 3855031, 5292209

Offset: 0

Antti Karttunen, Aug 24 2014

nonn

Iterates of A064216 starting from value 16.

A337382 Numbers k for which A003973(k) < 2*sigma(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 10, 11, 13, 17, 19, 22, 23, 25, 26, 29, 31, 33, 34, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 116, 118, 119, 121, 122, 123, 127, 129, 131, 133, 134, 137, 139, 141

Offset: 1

Antti Karttunen, Aug 27 2020

nonn

Cf. A000203, A003961, A003973.
Cf. A337381 (complement).
Positions of zeros in A337383.
Subsequence of A337379.
Cf. also A246281.

A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
isA337382(n) = (A003973(n)<2*sigma(n));

A349573 a(n) = A048673(n) - n, where A048673(n) = (A003961(n)+1) / 2, and A003961(n) shifts the prime factorization of n one step towards larger primes.

Original entry on oeis.org

0, 0, 0, 1, -1, 2, -1, 6, 4, 1, -4, 11, -4, 3, 3, 25, -7, 20, -7, 12, 7, -2, -8, 44, 0, 0, 36, 22, -13, 23, -12, 90, 0, -5, 4, 77, -16, -3, 4, 55, -19, 41, -19, 15, 43, -2, -20, 155, 12, 24, -3, 25, -23, 134, -9, 93, 1, -11, -28, 98, -27, -6, 75, 301, -5, 32, -31, 18, 4, 46, -34, 266, -33, -12, 48, 28, -5, 50, -37

Offset: 1

Antti Karttunen, Nov 23 2021

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

Cf. A003961, A048673, A064216, A349571.

Cf. A048674 (positions of zeros), A246351 (negative terms), A246281 (nonpositive terms), A246352 (nonnegative terms), A246282 (positive terms), A269860 (even terms), A269861 (odd terms).

f[p_, e_] := NextPrime[p]^e; a[1] = 0; a[n_] := (1 + Times @@ f @@@ FactorInteger[n])/2 - n; Array[a, 100] (* Amiram Eldar, Nov 23 2021 *)

A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(1+factorback(f)); };
A349573(n) = (A048673(n)-n);

a(n) = A048673(n) - n.

a(n) = Sum_{d|n, dA349571(n/d).

cp's OEIS Frontend

A246373 Primes p such that if 2p-1 = product_{k >= 1} A000040(k)^(c_k), then p <= product_{k >= 1} A000040(k-1)^(c_k).

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A275717 Numbers n for which A003961(n) > A003961(n-1).

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Mathematica

A275718 Numbers n for which A003961(n) < A003961(n-1).

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Mathematica

A337379 Numbers k for which A003961(k) < 2*sigma(k).

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A246342 a(0) = 12, after which, if a(n-1) = product_{k >= 1} (p_k)^(c_k), then a(n) = (1/2) * (1 + product_{k >= 1} (p_{k+1})^(c_k)), where p_k indicates the k-th prime, A000040(k).

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A246343 a(0) = 12, after which, if (2*a(n-1)) - 1 = product_{k >= 1} (p_k)^(c_k) then a(n) = product_{k >= 1} (p_{k-1})^(c_k), where p_k indicates the k-th prime, A000040(k).

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A246344 a(0) = 16, after which, if a(n-1) = product_{k >= 1} (p_k)^(c_k), then a(n) = (1/2) * (1 + product_{k >= 1} (p_{k+1})^(c_k)), where p_k indicates the k-th prime, A000040(k).

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A246345 a(0) = 16, after which, if (2*a(n-1)) - 1 = product_{k >= 1} (p_k)^(c_k) then a(n) = product_{k >= 1} (p_{k-1})^(c_k), where p_k indicates the k-th prime, A000040(k).

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