A324109 -id:A324109 - cp's OEIS frontend

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

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).

A070776 Numbers k such that number of terms in the k-th cyclotomic polynomial is equal to the largest prime factor of k.

Original entry on oeis.org

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, 88, 89, 92, 94, 96, 97, 98, 100

Offset: 1

Labos Elemer, May 07 2002

Numbers k such that A051664(k) = A006530(k).
This is also numbers in the form of 2^i*p^j, i >= 0 and j >= 0, p is an odd prime number. - Lei Zhou, Feb 18 2012
From Zhou's formulation (where the exponents i and j should actually have been specified as i > 0 OR j > 0, to exclude 1) it follows that this is a subsequence of A324109. It also follows that A005940(a(n)) = A324106(a(n)) for all n >= 1. - Antti Karttunen, Feb 15 2019
Also from Zhou's formulation, the union (disjoint) of A000079\{1} and A336101. - Peter Munn, Jul 16 2020
Numbers k>=2 such that A078701(k) = A299766(k). - Juri-Stepan Gerasimov, Jun 02 2021

			n=10: Cyclotomic[10,x]=1-x+x^2-x^3+x^4 with 5 terms [including 1] which equals largest prime factor (5) of 10=n.

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

Positions of zeros in A070536.
Cf. A005940, A006530, A051664, A061345, A070537 (complement), A324106, A324111.
Subsequence of A324109.
Subsequences: A000079\{1}, A336101.

Select[Range[1000],(a=FactorInteger[#];b=Length[a];(b==1)||((b==2)&&(a[[1]][[1]]==2)))&] (* Lei Zhou, Feb 18 2012 *)

A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1); \\ From A006530.
A051664(n) = length(select(x->x!=0, Vec(polcyclo(n)))); \\ After program in A051664
A070536(n) = (A051664(n) - A006530(n));
isA070776(n) = (!A070536(n)); \\ Antti Karttunen, Feb 15 2019
k=0; n=0; while(k<10000, n++; if(isA070776(n), k++; write("b070776.txt", k, " ", 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])); };

A324108 Multiplicative with a(p^e) = A324054((p^e)-1).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 13, 15, 8, 18, 24, 28, 31, 39, 24, 31, 12, 24, 32, 42, 52, 72, 78, 60, 57, 93, 124, 91, 156, 72, 121, 63, 96, 36, 78, 56, 72, 96, 124, 90, 96, 156, 192, 168, 48, 234, 240, 124, 133, 171, 48, 217, 342, 372, 144, 195, 128, 468, 624, 168, 781, 363, 104, 127, 186, 288, 56, 84, 312, 234, 156, 120, 112

Offset: 1

Antti Karttunen and David A. Corneth, Feb 15 2019

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000203, A005940, A324054, A324106, A324109, A324110.

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)); };

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,", ")))

A324107 Fixed points of A324106, where A324106 is a multiplicative function with A324106(p^e) = A005940(p^e).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 63, 64, 80, 96, 120, 126, 128, 160, 192, 240, 252, 256, 315, 320, 384, 480, 504, 512, 630, 640, 768, 960, 1008, 1024, 1260, 1280, 1536, 1920, 2016, 2048, 2520, 2560, 3072, 3840, 4032, 4096, 5040, 5120, 6144, 7680, 8064, 8192, 10080, 10240, 12288, 15360, 16128, 16384

Offset: 1

Antti Karttunen, Feb 15 2019

nonn

Numbers n such that A324106(n) = n.

			For n = 63 = 3^2 * 7^1, we find that A005940(9) = 7 and A005940(7) = 9, thus A324106(63) = 7*9 = 63, and 63 is a member of this sequence.

Cf. A005940, A070776, A324106, A324109.

Cf. A029747 (a subsequence).

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])); };
isA324107(n) = (n==A324106(n));
for(n=1,16384,if(isA324107(n), print1(n,", ")))

cp's OEIS Frontend

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

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A070776 Numbers k such that number of terms in the k-th cyclotomic polynomial is equal to the largest prime factor of k.

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

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A324108 Multiplicative with a(p^e) = A324054((p^e)-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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A324107 Fixed points of A324106, where A324106 is a multiplicative function with A324106(p^e) = A005940(p^e).

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