A324108 -id:A324108 - cp's OEIS frontend

A324109 Numbers n such that A324108(n) = A324054(n-1).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 40, 41, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 56, 58, 59, 61, 62, 64, 67, 68, 71, 72, 73, 74, 76, 79, 80, 81, 82, 83, 86, 87, 88, 89, 92, 94, 96, 97, 98, 100, 101, 103, 104, 106, 107, 108, 109, 112, 113, 116, 118, 121

Offset: 1

Antti Karttunen and David A. Corneth, Feb 15 2019

nonn

Numbers n such that A324054(n-1) is equal to A324108(n), which is a multiplicative function with A324108(p^e) = A324054((p^e)-1).
Prime powers (A000961) is a subsequence by definition.
Also A070776 is a subsequence. This follows because for every n of the form 2^i * p^j (where p is an odd prime, and i >= 0, j >= 0), we have A324108(2^i * p^j) = A324054(2^i - 1)*A324054(p^j - 1) = sigma(A005940(2^i)) * sigma(A005940(p^j)). Because A005940(1) = 1, and A005940(2n) = 2*A005940(n), the powers of two are among the fixed points of A005940 (cf. A029747), thus the left half of product is sigma(2^i), while on the other hand, we know that A005940(p^j) is odd (because A005940 also preserves parity), and thus the whole product is equal to sigma(2^i * A005940(p^j)) = sigma(A005940(2^i * p^j)) = A324054((2^i * p^j)-1).
See subsequence A324111 for less regular solutions.

Antti Karttunen, Table of n, a(n) for n = 1..10001

Union of A070776 and A324111.

Cf. A000961 (a subsequence), A029747, A324054, A324107, A324108, A324110 (complement).

A324054(n) = { my(p=2,mp=p*p,m=1); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, if(3==(n%4),mp *= p,m *= (mp-1)/(p-1))); n>>=1); (m); };
A324108(n) = { my(f=factor(n)); prod(i=1, #f~, A324054((f[i,1]^f[i,2])-1)); };
isA324109(n) = (A324054(n-1)==A324108(n));
for(n=1,121,if(isA324109(n), print1(n,", ")));

A324111 Numbers n for which A324108(n) = A324054(n-1) and which are neither prime powers nor of the form 2^i * p^j, where p is an odd prime, with either exponent i or j > 0.

Original entry on oeis.org

1, 87, 174, 348, 696, 1392, 2091, 2784, 4182, 5568, 8364, 11136, 16683, 16728, 22272, 33215, 33366, 33456, 44544, 66430, 66732, 66912, 89088, 132860, 133464, 133824, 178176, 265720, 266928, 267179, 267648, 356352, 531440, 533856, 534358, 535296, 712704, 1062880, 1066877, 1067712, 1068716, 1070592, 1319235, 1425408

Offset: 1

Antti Karttunen, Feb 15 2019

nonn

Setwise difference of A324109 and A070776.
Setwise difference of A070537 and A324110.
If an odd number n > 1 is present, then all 2^k * n are present also. Odd terms > 1 are given in A324112.

			87 is a term, as 87 = 3*29, A324054(3-1) = 4, A324054(29-1) = 156, and A324108(87) = 4*156 = 624 = A324054(87-1).

Antti Karttunen, Table of n, a(n) for n = 1..173

Cf. A070537, A070776, A324109, A324110, A324112.

A000265(n) = (n/2^valuation(n, 2));
A324054(n) = { my(p=2,mp=p*p,m=1); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, if(3==(n%4),mp *= p,m *= (mp-1)/(p-1))); n>>=1); (m); };
A324108(n) = { my(f=factor(n)); prod(i=1, #f~, A324054((f[i,1]^f[i,2])-1)); };
isA324111(n) = ((1!=omega(n))&&(1!=omega(A000265(n)))&&(A324054(n-1)==A324108(n)));
for(n=1,2^20,if(isA324111(n), print1(n,", ")))

A324110 Numbers k such that A324108(k) != A324054(k-1).

Original entry on oeis.org

15, 21, 30, 33, 35, 39, 42, 45, 51, 55, 57, 60, 63, 65, 66, 69, 70, 75, 77, 78, 84, 85, 90, 91, 93, 95, 99, 102, 105, 110, 111, 114, 115, 117, 119, 120, 123, 126, 129, 130, 132, 133, 135, 138, 140, 141, 143, 145, 147, 150, 153, 154, 155, 156, 159, 161, 165, 168, 170, 171, 175, 177, 180, 182, 183, 185, 186, 187, 189, 190, 195, 198, 201

Offset: 1

Antti Karttunen and David A. Corneth, Feb 15 2019

nonn

This is a subsequence of A070537. The missing terms 1, 87, 174, 348, 696, 1392, 2091, ..., are at A324111.

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

Cf. A070537, A324054, A324108, A324109 (complement), A324111.

A324054(n) = { my(p=2,mp=p*p,m=1); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, if(3==(n%4),mp *= p,m *= (mp-1)/(p-1))); n>>=1); (m); };
A324108(n) = { my(f=factor(n)); prod(i=1, #f~, A324054((f[i,1]^f[i,2])-1)); };
isA324110(n) = (A324054(n-1)!=A324108(n));
for(n=1,201,if(isA324110(n), print1(n,", ")))

A324112 Odd numbers n for which A324108(n) = A324054(n-1), and which themselves are not powers of primes (in A000961).

Original entry on oeis.org

87, 2091, 16683, 33215, 267179, 1066877, 1319235, 4228521, 4330579, 8668351, 9769751, 34662043, 35924003, 50892875, 68239949, 83920375, 143201615, 151730823, 311513495, 419564887, 537386921, 538253299, 539511051, 605140375

Offset: 1

Antti Karttunen, Feb 16 2019

Odd terms > 1 in A324111.

			(Here f is A324054 and g is A324108).
87 is a term, as 87 = 3 * 29, f(3-1) = 4, f(29-1) = 156, and g(87) = 4 * 156 = 624 = f(87-1).
1066877 is a term, as 1066877 = 7^2 * 21773, f(49-1) = 133, f(21773-1) = 8035200, and g(1066877) = 133 * 8035200 = 1068681600 = f(1066877-1).
537386921 is a term, as 537386921 = 2083 * 257987, f(2083-1) = 1440, f(257987-1) = 31554095246856, and g(537386921) = 1440 * 31554095246856 = 45437897155472640 = f(537386921-1).

Cf. A324111.

A324054 a(n) = A000203(A005940(1+n)).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 13, 15, 8, 18, 24, 28, 31, 39, 40, 31, 12, 24, 32, 42, 48, 72, 78, 60, 57, 93, 124, 91, 156, 120, 121, 63, 14, 36, 48, 56, 72, 96, 104, 90, 96, 144, 192, 168, 248, 234, 240, 124, 133, 171, 228, 217, 342, 372, 403, 195, 400, 468, 624, 280, 781, 363, 364, 127, 18, 42, 56, 84, 84, 144, 156, 120, 112, 216, 288, 224

Offset: 0

Antti Karttunen, Feb 14 2019

nonn

As noted by David A. Corneth, the function f(n) = a(n-1) [that is, the offset-1 version of this sequence] seems to be "almost multiplicative". Sequence A324109 gives the positions n where f(n) satisfies the multiplicativity in a sense that f(n) = f(p(1)^e(1)) * ... * f(p(k)^e(k)), when n = p(1)^e(1) * ... * p(k)^e(k), and A324110 the positions where this does not hold.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..70327
Index entries for sequences related to binary expansion of n
Michael De Vlieger, Annotated fan-style binary tree diagram showing 16 levels, where blue represents A336834(n) = 1 and yellow A336834(n) = 0. Labels are a(n) for the 8 smallest levels.

Cf. A000203, A005940, A038712, A324055, A324056, A324057, A324058, A324108, A324109, A324110, A324111, A324112, A356321.

Cf. also A106737, A290077 (tau and phi similarly permuted).

nn = 76; a[0] = 1; Do[Set[a[n], Prime[1 + DigitCount[n, 2, 0]]*a[n - 2^Floor@ Log2@ n]], {n, nn}]; Array[DivisorSigma[1, a[#]] &, nn, 0] (* Michael De Vlieger, Aug 03 2022 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A324054(n) = sigma(A005940(1+n));

A324054(n) = { my(p=2,mp=p*p,m=1); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, if(3==(n%4),mp *= p,m *= (mp-1)/(p-1))); n>>=1); (m); };

from math import prod
from itertools import accumulate
from collections import Counter
from sympy import prime
def A324054(n): return prod(((p:=prime(len(a)+1))**(b+1)-1)//(p-1) for a, b in Counter(accumulate(bin(n)[2:].split('1')[:0:-1])).items()) # Chai Wah Wu, Mar 10 2023

a(n) = A000203(A005940(1+n)).

a(n) = A324056(n) * A038712(1+n).

A324106 Multiplicative with a(p^e) = A005940(p^e).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 15, 12, 25, 18, 15, 16, 11, 14, 21, 20, 27, 30, 45, 24, 49, 50, 75, 36, 125, 30, 81, 32, 45, 22, 45, 28, 55, 42, 75, 40, 77, 54, 105, 60, 35, 90, 135, 48, 121, 98, 33, 100, 245, 150, 75, 72, 63, 250, 375, 60, 625, 162, 63, 64, 125, 90, 39, 44, 135, 90, 99, 56, 91, 110, 147, 84, 135, 150, 189, 80, 143, 154, 231, 108, 55

Offset: 1

Antti Karttunen, Feb 15 2019

Question: are there any other numbers n besides 1 and those in A070776, for which a(n) = A005940(n)? At least not below 2^25. This is probably easy to prove.

			For n = 85 = 5*17, a(85) = A005940(5) * A005940(17) = 5*11 = 55. Note that A005940(5) is obtained from the binary expansion of 5-1 = 4, which is "100", and A005940(17) is obtained from the binary expansion of 17-1 = 16, which is "1000".

Antti Karttunen, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Fan style binary tree of a(n), n = 1..2^12, color coded to show the smallest values in the range r = (2^r - 1)..2^(r+1) in blue and highlighting the largest with red.
Index entries for sequences related to binary expansion of n

Cf. A005940, A070776, A324107 (fixed points), A324108, A324109.

nn = 128; Array[Set[a[#], #] &, 2]; Do[If[EvenQ[n], Set[a[n], 2 a[n/2]], Set[a[n], Times @@ Power @@@ Map[{Prime[PrimePi[#1] + 1], #2} & @@ # &, FactorInteger[a[(n + 1)/2]]]]], {n, 3, nn}]; Array[Times @@ Map[a, Power @@@ FactorInteger[#]] &, nn] (* Michael De Vlieger, Sep 18 2022 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A324106(n) = { my(f=factor(n)); prod(i=1, #f~, A005940(f[i,1]^f[i,2])); };

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A324109 Numbers n such that A324108(n) = A324054(n-1).

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A324111 Numbers n for which A324108(n) = A324054(n-1) and which are neither prime powers nor of the form 2^i * p^j, where p is an odd prime, with either exponent i or j > 0.

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A324110 Numbers k such that A324108(k) != A324054(k-1).

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A324112 Odd numbers n for which A324108(n) = A324054(n-1), and which themselves are not powers of primes (in A000961).

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A324054 a(n) = A000203(A005940(1+n)).

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A324106 Multiplicative with a(p^e) = A005940(p^e).

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