A293442 -id:A293442 - cp's OEIS frontend

A293514 a(n) = Product_{d|n, d>1} prime(A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing n.

1, 2, 2, 6, 2, 8, 2, 20, 6, 8, 2, 48, 2, 8, 8, 84, 2, 48, 2, 48, 8, 8, 2, 320, 6, 8, 20, 48, 2, 128, 2, 264, 8, 8, 8, 864, 2, 8, 8, 320, 2, 128, 2, 48, 48, 8, 2, 2688, 6, 48, 8, 48, 2, 320, 8, 320, 8, 8, 2, 3072, 2, 8, 48, 1560, 8, 128, 2, 48, 8, 128, 2, 11520, 2, 8, 48, 48, 8, 128, 2, 2688, 84, 8, 2, 3072, 8, 8, 8, 320

Offset: 1

Antti Karttunen, Nov 11 2017

nonn

			For n = 24, its divisors larger than one are: 2, 3, 4, 6, 8, 12, 24. Only 2 has valuation > 1, namely A286561(24,2) = 3 (as 2^3 divides 24), while the other six have valuation 1. Thus a(24) = prime(1)^6 * prime(3) = 64*5 = 320.
For n = 64, its divisors larger than one are: 2, 4, 8, 16, 32, 64. We see that 2^6 = 4^3 = 8^2 = 64, while valuation of the last three 16, 32 and 64 is 1. Thus a(64) = prime(1)^3 * prime(2) * prime(3) * prime(6) = 2^3 * 3 * 5 * 13 = 1560.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A000040, A286561.

Cf. also A293515, A293524, A294876, A293442.

A293514(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(valuation(n,d)))); m; };

a(n) = Product_{d|n, d>1} A000040(A286561(n,d)).
Other identities. For all n >= 1:
A001222(a(n)) = A032741(n).
A007814(a(n)) = A056595(n) [See A046951.]
1+A056239(a(n)) = A169594(n).
A064989(a(n)) = A293515(n).

A331592 a(n) is the smaller of the number of terms in the factorizations of n into (1) powers of distinct primes and (2) powers of squarefree numbers with distinct exponents that are powers of 2.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen and Peter Munn, Jan 21 2020

nonn

See A329332 for a description of the relationship between the two factorizations. From this relationship we get the formula a(n) = min(A001221(n), A001221(A225546(n))).
The result depends only on the prime signature of n.
k first appears at A191555(k).

			The factorization of 6 into powers of distinct primes is 6 = 2^1 * 3^1 = 2 * 3, which has 2 terms. Its factorization into powers of squarefree numbers with distinct exponents that are powers of 2 is 6 = 6^(2^0) = 6^1, which has 1 term. So a(6) is min(2,1) = 1.
The factorization of 40 into powers of distinct primes is 40 = 2^3 * 5^1 = 8 * 5, which has 2 terms. Its factorization into powers of squarefree numbers with distinct exponents that are powers of 2 is 40 = 10^(2^0) * 2^(2^1) = 10^1 * 2^2 = 10 * 4, which has 2 terms. So a(40) is min(2,2) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A000961, A001221, A191555, A293442, A329332.
Sequences with related definitions: A331308, A331591, A331593.
A003961, A225546 are used to express relationship between terms of this sequence.
Differs from = A071625 for the first time at n=216, where a(216) = 2, while A071625(216) = 1.

A331592(n) = min(omega(n), A331591(n)); \\ Uses also code from A331591.

a(n) = min(A001221(n), A331591(n)) = min(A001221(n), A001221(A293442(n))).
a(A225546(n)) = a(n).
a(A003961(n)) = a(n).
a(n^2) = a(n).

A293443 Multiplicative with a(p^e) = A019565(A193231(e)).

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 2, 3, 6, 4, 2, 12, 2, 4, 4, 10, 2, 12, 2, 12, 4, 4, 2, 6, 6, 4, 3, 12, 2, 8, 2, 5, 4, 4, 4, 36, 2, 4, 4, 6, 2, 8, 2, 12, 12, 4, 2, 20, 6, 12, 4, 12, 2, 6, 4, 6, 4, 4, 2, 24, 2, 4, 12, 15, 4, 8, 2, 12, 4, 8, 2, 18, 2, 4, 12, 12, 4, 8, 2, 20, 10, 4, 2, 24, 4, 4, 4, 6, 2, 24, 4, 12, 4, 4, 4, 10, 2, 12, 12, 36, 2, 8, 2, 6, 8

Offset: 1

Antti Karttunen, Oct 31 2017

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A019565, A028234, A067029, A193231.

Cf. also A270428, A293442, A293231, A293439.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) }; \\ And this from Franklin T. Adams-Watters
vecproduct(v) = { my(m=1); for(i=1,#v,m *= v[i]); m; };
A293443(n) = vecproduct(apply(e -> A019565(A193231(e)), factorint(n)[, 2]));

;; With memoization-macro definec.
(definec (A293443 n) (if (= 1 n) n (* (A019565 (A193231 (A067029 n))) (A293443 (A028234 n)))))

a(1) = 1; for n > 1, a(n) = A019565(A193231(A067029(n))) * a(A028234(n)).
For all n >= 1, A007814(a(n)) = A293439(n).
For all k in A270428, A007814(a(k)) = A001221(k).

A294876 a(n) = Product_{d|n, d>1} prime(gcd(d,n/d)).

Original entry on oeis.org

1, 2, 2, 6, 2, 8, 2, 18, 10, 8, 2, 72, 2, 8, 8, 126, 2, 200, 2, 72, 8, 8, 2, 648, 22, 8, 50, 72, 2, 128, 2, 882, 8, 8, 8, 23400, 2, 8, 8, 648, 2, 128, 2, 72, 200, 8, 2, 31752, 34, 968, 8, 72, 2, 5000, 8, 648, 8, 8, 2, 10368, 2, 8, 200, 16758, 8, 128, 2, 72, 8, 128, 2, 2737800, 2, 8, 968, 72, 8, 128, 2, 31752, 1150, 8, 2, 10368, 8, 8, 8, 648, 2, 80000, 8, 72

Offset: 1

Antti Karttunen, Nov 11 2017

nonn

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

Cf. A294877 (rgs-version of this filter).
Cf. A000040, A000188, A034444, A055155.
Cf. also A293442, A293514, A293524.

A294876[n_] := Product[Prime[GCD[d, n/d]], {d, Rest[Divisors[n]]}];
Array[A294876, 100] (* Paolo Xausa, Feb 22 2024 *)

A294876(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(gcd(d,n/d)))); m; };

a(n) = Product_{d|n, d>1} A000040(gcd(d,n/d)).
Other identities. For all n >= 1:
1+A007814(a(n)) = A034444(n).
1+A056239(a(n)) = A055155(n).
For n > 1, A061395(a(n)) = A000188(n).

A302791 A filter sequence for Fermi-Dirac factorization: restricted growth sequence transform of A046523(A302024(n)).

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 4, 2, 4, 2, 3, 2, 4, 4, 2, 2, 4, 2, 3, 4, 4, 2, 5, 2, 4, 4, 4, 2, 6, 2, 4, 4, 4, 3, 4, 2, 4, 4, 6, 2, 6, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 6, 4, 7, 4, 4, 2, 5, 2, 4, 3, 4, 4, 6, 2, 4, 4, 6, 2, 7, 2, 4, 4, 4, 4, 6, 2, 4, 2, 4, 2, 6, 4, 4, 4, 7, 2, 7, 4, 4, 4, 4, 4, 6, 2, 4, 3, 4, 2, 6, 2, 7, 6

Offset: 1

Antti Karttunen, Apr 15 2018

nonn

For all i, j: a(i) = a(j) => A064547(i) = A064547(j).
For all i, j: a(i) = a(j) => A302790(i) = A302790(j).
See also comments in A302024.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A037445, A046523, A050376 (gives the positions of 2's), A052331, A064547, A293442, A302024, A302787, A302790.

allocatemem(2^30);
up_to = 65537;
v050376 = vector(up_to);
A050376(n) = v050376[n];
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to,break));
A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
A302024(n) = A005940(1+A052331(n));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
Aux302791(n) = A046523(A302024(n));
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
write_to_bfile(1,rgs_transform(vector(up_to,n,Aux302791(n))),"b302791.txt");

A293447 Fully additive with a(p^e) = e * A000225(PrimePi(p)), where PrimePi(n) = A000720(n) and A000225(n) = (2^n)-1.

Original entry on oeis.org

0, 1, 3, 2, 7, 4, 15, 3, 6, 8, 31, 5, 63, 16, 10, 4, 127, 7, 255, 9, 18, 32, 511, 6, 14, 64, 9, 17, 1023, 11, 2047, 5, 34, 128, 22, 8, 4095, 256, 66, 10, 8191, 19, 16383, 33, 13, 512, 32767, 7, 30, 15, 130, 65, 65535, 10, 38, 18, 258, 1024, 131071, 12, 262143, 2048, 21, 6, 70, 35, 524287, 129, 514, 23, 1048575, 9, 2097151, 4096, 17, 257, 46, 67, 4194303, 11, 12

Offset: 1

Antti Karttunen, Nov 09 2017

nonn

Original, equal definition: totally additive with a(p^e) = e * A005187(2^(PrimePi(p)-1)), where PrimePi(n) = A000720(n).

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences computed from indices in prime factorization

Cf. A000225, A000720, A003557, A005187, A087207.

Cf. also A046645, A048675, A293442, A331591, A331740.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; }; \\ This function from Charles R Greathouse IV
A293447(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2] * A005187(2^(primepi(f[k, 1])-1))); }

(define (A293447 n) (cond ((= 1 n) 0) (else (+ (A005187 (A087207 n)) (A293447 (A003557 n))))))
;; Alternatively:
(define (A293447 n) (if (= 1 n) 0 (+ (A005187 (A000079 (+ -1 (A061395 n)))) (A293447 (/ n (A006530 n))))))

Totally additive with a(p^e) = e * A005187(2^(PrimePi(p)-1)), where PrimePi(n) = A000720(n).
a(1) = 0, and for n > 1, a(n) = A005187(A087207(n)) + a(A003557(n)).
Other identities:
For all n >= 1, a(A293442(n)) = A046645(n).
For all n >= 2 and all k >= 0, a(n^k) = k*a(n).
For all n >= 1, a(n) >= A048675(n) >= A331740(n) >= A331591(n).

Definition simplified by Antti Karttunen, Feb 05 2020

A293524 a(n) = Product_{d|n, d>1} prime(A052409(d)).

Original entry on oeis.org

1, 2, 2, 6, 2, 8, 2, 30, 6, 8, 2, 48, 2, 8, 8, 210, 2, 48, 2, 48, 8, 8, 2, 480, 6, 8, 30, 48, 2, 128, 2, 2310, 8, 8, 8, 864, 2, 8, 8, 480, 2, 128, 2, 48, 48, 8, 2, 6720, 6, 48, 8, 48, 2, 480, 8, 480, 8, 8, 2, 3072, 2, 8, 48, 30030, 8, 128, 2, 48, 8, 128, 2, 17280, 2, 8, 48, 48, 8, 128, 2, 6720, 210, 8, 2, 3072, 8, 8, 8, 480

Offset: 1

Antti Karttunen, Nov 11 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A000040, A052409, A294873, A294874, A294875.

Cf. also A293514, A293442.

A293524(n) = { my(m=1,e); fordiv(n,d, if(d>1, e = ispower(d); if(!e, m += m, m *= prime(e)))); m; };

a(n) = Product_{d|n, d>1} A000040(A052409(d)).
Other identities. For all n >= 1:
A001222(a(n)) = A032741(n).
A007814(a(n)) = A183096(n).
A064989(a(n)) = A294875(n).

A318469 Multiplicative with a(p^e) = A019565(A003714(e)).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 10, 2, 6, 2, 6, 4, 4, 2, 10, 3, 4, 5, 6, 2, 8, 2, 7, 4, 4, 4, 9, 2, 4, 4, 10, 2, 8, 2, 6, 6, 4, 2, 20, 3, 6, 4, 6, 2, 10, 4, 10, 4, 4, 2, 12, 2, 4, 6, 14, 4, 8, 2, 6, 4, 8, 2, 15, 2, 4, 6, 6, 4, 8, 2, 20, 10, 4, 2, 12, 4, 4, 4, 10, 2, 12, 4, 6, 4, 4, 4, 14, 2, 6, 6, 9, 2, 8, 2, 10, 8

Offset: 1

Antti Karttunen, Aug 30 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A003714, A019565, A072649, A318464, A318465.

Cf. also A293442, A293443, A300834, A318470.

A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A318469(n) = factorback(apply(e -> A019565(A003714(e)),factor(n)[,2]));

For all n >= 1, A001222(a(n)) = A318464(n).

A340323 Multiplicative with a(p^e) = (p + 1) * (p - 1)^(e - 1).

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 3, 8, 18, 12, 12, 14, 24, 24, 3, 18, 24, 20, 18, 32, 36, 24, 12, 24, 42, 16, 24, 30, 72, 32, 3, 48, 54, 48, 24, 38, 60, 56, 18, 42, 96, 44, 36, 48, 72, 48, 12, 48, 72, 72, 42, 54, 48, 72, 24, 80, 90, 60, 72, 62, 96, 64, 3, 84, 144, 68, 54

Offset: 1

José María Grau Ribas, Jan 04 2021

Starting with any integer and repeatedly applying the map x -> a(x) reaches the fixed point 12 or the loop {3, 4}.

			a(2^s) = 3 for all s>0.

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

Cf. A003958, A003963, A048250, A068468, A064989, A167344, A327938, A293442, A064988, A108548, A124859, A279513, A324106, A322360, A326297, A340324, A340325, A340368.

f:= proc(n) local  t;
  mul((t[1]+1)*(t[1]-1)^(t[2]-1),t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Jan 07 2021

fa[n_]:=fa[n]=FactorInteger[n];
phi[1]=1; phi[p_, s_]:= (p + 1)*( p - 1)^(s - 1)
phi[n_]:=Product[phi[fa[n][[i, 1]], fa[n][[i, 2]]], {i,Length[fa[n]]}];
Array[phi, 245]

A340323(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,(f[i,1]+1)*((f[i,1]-1)^(f[i,2]-1)))); \\ Antti Karttunen, Jan 06 2021

a(n) = A167344(n) / A340368(n) = A048250(n) * A326297(n). - Antti Karttunen, Jan 06 2021

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(6)/(2*zeta(2)*zeta(3))) * Product_{p prime} (1 + 2/p^2) = 0.56361239505... . - Amiram Eldar, Nov 12 2022

A294932 Multiplicative with a(p^e) = A019565(A289814(e)).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1

Offset: 1

Antti Karttunen, Nov 11 2017

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A019565, A289814, A293442, A294931, A294933.

(define (A294932 n) (if (= 1 n) n (* (A019565 (A289814 (A067029 n))) (A294932 (A028234 n)))))

cp's OEIS Frontend

A293514 a(n) = Product_{d|n, d>1} prime(A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing n.

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A331592 a(n) is the smaller of the number of terms in the factorizations of n into (1) powers of distinct primes and (2) powers of squarefree numbers with distinct exponents that are powers of 2.

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A293443 Multiplicative with a(p^e) = A019565(A193231(e)).

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A294876 a(n) = Product_{d|n, d>1} prime(gcd(d,n/d)).

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A302791 A filter sequence for Fermi-Dirac factorization: restricted growth sequence transform of A046523(A302024(n)).

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A293447 Fully additive with a(p^e) = e * A000225(PrimePi(p)), where PrimePi(n) = A000720(n) and A000225(n) = (2^n)-1.

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A293524 a(n) = Product_{d|n, d>1} prime(A052409(d)).

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A318469 Multiplicative with a(p^e) = A019565(A003714(e)).

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