A048674 -id:A048674 - cp's OEIS frontend

A246362 Numbers n such that if 2n-1 = Product_{k >= 1} (p_k)^(c_k), then n < Product_{k >= 1} (p_{k-1})^(c_k), where p_k indicates the k-th prime, A000040(k).

4, 6, 7, 9, 10, 12, 15, 16, 19, 20, 21, 22, 24, 27, 29, 30, 31, 34, 35, 36, 37, 40, 42, 44, 45, 46, 47, 48, 49, 51, 52, 54, 55, 56, 57, 60, 62, 64, 65, 66, 67, 69, 70, 71, 72, 75, 76, 78, 79, 80, 81, 82, 84, 85, 87, 89, 90, 91, 92, 96, 97, 99, 100, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 112, 114, 115

Offset: 1

Antti Karttunen, Aug 24 2014

nonn

Numbers n such that A064216(n) > n.
Numbers n such that A064989(2n-1) > n.
The sequence grows as:
      a(100) = 148
     a(1000) = 1449
    a(10000) = 14264
   a(100000) = 141259
  a(1000000) = 1418197
and the powers of 10 occur at:
        a(5) = 10
       a(63) = 100
      a(701) = 1000
     a(6973) = 10000
    a(70845) = 100000
   a(705313) = 1000000
suggesting that the ratio a(n)/n is converging to a constant and an arbitrary natural number is more than twice as likely to be here than in the complement A246361. Compare this to the ratio present in the "inverse" case A246282.

			4 is present, as 2*4 - 1 = 7 = p_4, and p_{4-1} = p_3 = 5 > 4.
5 is not present, as 2*5 - 1 = 9 = p_2 * p_2, and p_1 * p_1 = 4, with 4 < 5.
6 is present, as 2*6 - 1 = 11 = p_5, and p_{5-1} = p_4 = 7 > 6.
35 is present, as 2*35 - 1 = 69 = 3*23 = p_2 * p_9, and p_1 * p_8 = 2*19 = 38 > 35.

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

Complement: A246361.
Setwise difference of A246372 and A048674.
Cf. A000040, A064216, A064989, A246282.

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);
isA246362(n) = (A064216(n) > n);
n = 0; i = 0; while(i < 10000, n++; if(isA246362(n), i++; write("b246362.txt", i, " ", n)));

;; With Antti Karttunen's IntSeq-library.
(define A246362 (MATCHING-POS 1 1 (lambda (n) (> (A064216 n) n))))

A348514 Numbers k for which A003961(k) = 2k+1, where A003961 shifts the prime factorization of n one step towards larger primes.

Original entry on oeis.org

4, 10, 57, 1054, 2626, 68727, 12371554, 1673018314, 10475647197, 11154517557, 27594844918, 630178495917, 7239182861878

Offset: 1

Antti Karttunen, Oct 29 2021

Numbers k such that A064216(1+k) = k.
It seems that after 4, all other terms are squarefree. See conjecture in A348511.
a(9)..a(13) <= 10475647197, 11154517557, 27594844918, 630178495917, 7239182861878, which are also terms. - David A. Corneth, Oct 30 2021

Index entries for sequences computed from indices in prime factorization

Fixed points of map A108228. (Compare to A048674).
Positions of ones in A252748.
Subsequence of the following sequences: A246282, A319630, A348511, A378980 (see also A379216), A387411, A387414.
Cf. A000230, A003961, A064216, A326134.

f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^5], s[#] == 2*# + 1 &] (* Amiram Eldar, Oct 30 2021 *)

a(9)-a(11) verified by Amiram Eldar, Nov 01 2021

a(12)-a(13) verified by Martin Ehrenstein, Nov 08 2021

A245449 Fixed points of A245447 and A245448.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 13, 25, 26, 30, 33, 53, 93, 1023, 1034, 1203, 1330, 2657, 8584, 17159, 779212, 970225, 1558409, 8550146, 240902643, 244608573, 325422273, 414690595, 570131490, 1020233393, 1864797542, 2438037206

Offset: 1

Antti Karttunen, Jul 22 2014

First apply A003961(n), where the primes in the prime factorization of natural number n are shifted one step left [i.e. each p_i changes to p_{i+1}]. Then increment the resulting odd number by one to get an even number, which is divided by 2, and the same three operations are done second time to that quotient. This sequence consists of such numbers for which the final result is equal to the original n which we started from.
8550146 is the largest term <= 123456789.
Numbers which are in 1- and 2-cycles of A048673 and A064216.

			For n = 30 = 2*3*5 = p_1 * p_2 * p_3, the first shift operation results p_2 * p_3 * p_4 = 3*5*7 = 105, and (105+1)/2 = 53, which is the 16th prime, p_16. Shifting this once left results p_17 = 59, and (59+1)/2 = 30 again. Thus 30 is included in the sequence. For the same reason 53 is also included in the sequence.

A048674 is a subsequence.

Cf. A048673, A064216, A245447, A245448.

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;
isA245449(n) = ((A048673(A048673(n)) == n))
i=0; for(n=1, 123456789, if(isA245449(n), i++; write("b245449.txt", i, " ", n)))

;; With Antti Karttunen's IntSeq-library.
(define A245449 (FIXED-POINTS 1 1 A245447))

a(25)-a(32) added by Antti Karttunen, Sep 13 2014

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.

Original entry on oeis.org

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));

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).

Original entry on oeis.org

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)))))

A269860 Numbers n such that n and A048673(n) are of the same parity.

Original entry on oeis.org

1, 2, 3, 6, 8, 9, 11, 13, 18, 20, 22, 23, 24, 25, 26, 27, 28, 31, 32, 33, 35, 37, 39, 46, 47, 49, 50, 54, 59, 60, 62, 66, 68, 69, 70, 71, 72, 74, 75, 76, 78, 80, 81, 83, 84, 85, 88, 89, 93, 94, 95, 96, 97, 98, 99, 104, 105, 107, 109, 111, 112, 116, 117, 118, 119, 121, 128, 131, 133, 138, 139, 141, 142, 143, 145, 147, 150

Offset: 1

Antti Karttunen, Mar 16 2016

nonn

Union of odd terms of A246261 and even terms of A246263.

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

Complement: A269861.
Cf. A048674 (a subsequence).
Cf. A000035, A048673, A246261, A246263.
Cf. also A270430.

f[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n; Select[Range@ 150, Xor[EvenQ@ f@ #, OddQ@ #] &] (* Michael De Vlieger, Mar 17 2016 *)

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).

A285701 a(n) = number of iterations x -> A064216(x) needed to reach a nonprime number when starting from n, a(2) = a(3) = 1.

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Antti Karttunen, Apr 26 2017

nonn

			For n=2, A064216(2) = 2, thus there is exactly one distinct prime that can be reached when iterating A064216 starting from 2, thus a(2) = 1.
For n=19, A064216(19) = 31 (a prime), A064216(31) = 59 (a prime) and A064216(59) = 44 (not a prime), thus there are exactly three distinct primes that are encountered when iterating A064216 starting from 19 before a nonprime is reached, thus a(19) = 3 (the count includes also the starting prime 19).

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

Cf. A000040, A000720, A010051, A048674, A064216, A064989, A245449, A246373, A285700, A285706.

Cf. A005382 (gives positions of terms > 1 from its third term 7 onward).

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);
A285701(n) = if(!isprime(n),0,if((2==n)||(3==n),1,1+A285701(A064216(n))));

(definec (A285701 n) (cond ((zero? (A010051 n)) 0) ((or (= 2 n) (= 3 n)) 1) (else (+ 1 (A285701 (A064216 n))))))
;; Another version not requiring A064216 and A064989:
(definec (A285701 n) (cond ((zero? (A010051 n)) 0) ((or (= 2 n) (= 3 n)) 1) ((zero? (A010051 (+ n n -1))) 1) (else (+ 1 (A285701 (A000040 (+ -1 (A000720 (+ n n -1)))))))))

If A010051(n) = 0 [when n is a nonprime], a(n) = 0, otherwise a(n) = 1 + a(A064216(n)), with a(2) = a(3) = 1.

A378983 Numbers k such that (A003961(k)-2*k) divides (A003961(k)-(1+sigma(k))), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 10, 11, 15, 16, 17, 25, 26, 29, 32, 33, 35, 39, 41, 57, 59, 64, 71, 93, 101, 107, 125, 128, 137, 149, 161, 179, 191, 197, 227, 239, 256, 269, 281, 311, 347, 419, 431, 461, 512, 521, 569, 599, 617, 641, 659, 782, 809, 821, 827, 857, 881, 1019, 1024, 1030, 1031, 1034, 1049, 1054, 1061, 1091, 1151

Offset: 1

Antti Karttunen, Dec 13 2024

nonn

Conjecture: A202274 gives all terms of A028982 that occur in this sequence.

			For k=16 we have A003961(16) = 81, A003961(k)-2*k = 49, and 49 divides (A003961(k)-(1+sigma(k))) = 81-32 = 49, therefore 16 is included in this sequence.
For k=25 we have A003961(25) = 49, A003961(k)-2*k = -1, and -1 divides (A003961(k)-(1+sigma(k))) regardless of what the latter is, therefore 25 is included.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Positions of 0's in A378982.
Subsequences: A048674, A348514, A202274.
Cf. also A378980.

PARI
```
isA378983(n) = !A378982(n);
```

A379216 Difference 2*k - A003961(k) computed for k for which this difference divides difference (A003961(k)-sigma(k)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

Original entry on oeis.org

1, 1, 1, -1, -3, 3, -1, 1, 1, -43, 1, 5, 19, -1, -7, -5, 1, -2005, 1, -1, 149, -193, -1, -3, -79243, 1243, 1253, -7, 51, 581, -1, 3093, 1, 155491, 919, 1, -1, 15833, -877, -4295498497, 5129369, 31, 5779339, -69187, -29, 6745, 1, 181, 1, 69197, -397, -117433, -101, -1, 1, 2759, 1, -29479, 1, -5626288431709, 29669, -1, -132239, -1, -1, 14591, -2267959, -3187, 787250461

Offset: 1

Antti Karttunen, Dec 20 2024

sign

Among the initial 69 terms, there are eleven +1's and eleven -1's. The former correspond in A378980 with those of its terms that are in A048674 (1, 2, 3, 25, 26, 33, 93, 1034, ...), while the latter here correspond in A378980 with those of its terms that are in A348514 (4, 10, 57, 1054, 2626, ...).

Cf. A003961, A048674, A252748, A348514, A378980, A378981, A379217.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A378981(n) = ((A003961(n)-sigma(n))%((2*n)-A003961(n)));
is_A378980(n) = !A378981(n);
for(n=1,2^25,if(is_A378980(n),print1((2*n)-A003961(n),", ")));

a(n) = -A252748(A378980(n)).

cp's OEIS Frontend

A246362 Numbers n such that if 2n-1 = Product_{k >= 1} (p_k)^(c_k), then n < Product_{k >= 1} (p_{k-1})^(c_k), where p_k indicates the k-th prime, A000040(k).

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A348514 Numbers k for which A003961(k) = 2k+1, where A003961 shifts the prime factorization of n one step towards larger primes.

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A245449 Fixed points of A245447 and A245448.

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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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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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A269860 Numbers n such that n and A048673(n) are of the same parity.

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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.

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A285701 a(n) = number of iterations x -> A064216(x) needed to reach a nonprime number when starting from n, a(2) = a(3) = 1.

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