A344593 -id:A344593 - cp's OEIS frontend

A344592 a(n) = A003557(A276086(A108951(n))).

1, 1, 1, 3, 1, 5, 1, 1, 1, 7, 1, 125, 1, 11, 16807, 15, 1, 35, 1, 343, 161051, 13, 1, 25, 9317, 17, 1, 1331, 1, 2401, 1, 1, 371293, 19, 253333223, 42875, 1, 23, 1419857, 1, 1, 1, 1, 2197, 14641, 29, 1, 49, 371293, 6684099653, 2476099, 4913, 1, 55, 37349, 19487171, 6436343, 31, 1, 5929, 1, 37, 20449, 21, 582622237229761, 1792160394037

Offset: 1

Antti Karttunen, May 26 2021

nonn

Cf. A003557, A085731, A108951, A276086, A324886, A328572, A329047, A342002, A342920, A346091, A346092 [= A276085(a(n))].

Cf. A344591 (positions of ones), A344593 (rgs-transform).

Block[{b = MixedRadix[Reverse@ Prime@ Range@ 20]}, Array[#/(Times @@ FactorInteger[#][[All, 1]]) &@ Apply[Times, Power @@@ #] &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #} &@ IntegerDigits[#, b] &@ Apply[Times, Map[(Times @@ Prime@ Range@ PrimePi@ #1)^#2 & @@ # &, FactorInteger[#]]] &, 66]] (* Michael De Vlieger, Jul 14 2021 *)

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) }; \\ From A108951
A328572(n) = { my(m=1, p=2); while(n, if(n%p, m *= p^((n%p)-1)); n = n\p; p = nextprime(1+p)); (m); };
A344592(n) = A328572(A108951(n));

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A344592(n) = A003557(A276086(A108951(n)));

a(n) = A328572(A108951(n)) = A003557(A324886(n)) = A003557(A276086(A108951(n))).
a(n) = A329047(n) / A342920(n).
a(n) = A085731(A324886(n)) = gcd(A324886(n), A329047(n)) = A324886(n) / A346091(n). - Antti Karttunen, Jul 09 2021

A344591 Numbers k such that the primorial inflation of k is a sum of distinct primorial numbers.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 11, 13, 17, 19, 23, 27, 29, 31, 32, 37, 40, 41, 42, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 115, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 228, 229, 233, 239, 241, 251, 252, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

Antti Karttunen, May 26 2021

nonn

Numbers k such that A108951(k) is in A276156.

			A108951(40) = 240 and 240 is in A276156 because 240 = A002110(4) + A002110(3) = 210 + 30, therefore 40 is included in this sequence.

Positions of ones in A329344, in A344592 and in A344593.
Positions of squarefree terms in A324886.
Union of A008578 and A351959.
Cf. A002110, A108951, A276156, A351957 (characteristic function).
Cf. also A351958.

Name changed by Antti Karttunen, Apr 04 2022

A344594 Lexicographically earliest infinite sequence such that a(i) = a(j) => A342920(i) = A342920(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 4, 2, 1, 5, 1, 2, 6, 7, 1, 8, 1, 5, 6, 2, 1, 9, 10, 2, 7, 5, 1, 11, 1, 12, 6, 2, 13, 14, 1, 2, 6, 15, 1, 8, 1, 5, 16, 2, 1, 17, 18, 19, 6, 5, 1, 20, 21, 3, 6, 2, 1, 22, 1, 2, 23, 24, 25, 4, 1, 5, 6, 26, 1, 27, 1, 2, 28, 5, 29, 4, 1, 30, 31, 2, 1, 14, 32, 2, 6, 3, 1, 33, 34, 5, 6, 2, 35, 36, 1, 37, 38, 39, 1, 4, 1, 3, 40

Offset: 1

Antti Karttunen, May 26 2021

nonn

Restricted growth sequence transform of A342920, where A342920(n) = A342002(A108951(n)) = A329047(n) / A344592(n).

Cf. A108951, A324886, A329047, A342002, A342920, A344592.

Cf. also A344593.

up_to = 65537;
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; };
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A342002(n) = { my(u=A276086(n)); (A003415(u) / A003557(u)); };
A342920(n) = A342002(A108951(n));
v344594 = rgs_transform(vector(up_to, n, A342920(n)));
A344594(n) = v344594[n];

cp's OEIS Frontend

A344592 a(n) = A003557(A276086(A108951(n))).

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A344591 Numbers k such that the primorial inflation of k is a sum of distinct primorial numbers.

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A344594 Lexicographically earliest infinite sequence such that a(i) = a(j) => A342920(i) = A342920(j), for all i, j >= 1.

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