A348999 -id:A348999 - cp's OEIS frontend

A348949 a(n) = A003959(A276086(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e, and A276086 gives the prime product form of primorial base expansion of n.

1, 3, 4, 12, 16, 48, 6, 18, 24, 72, 96, 288, 36, 108, 144, 432, 576, 1728, 216, 648, 864, 2592, 3456, 10368, 1296, 3888, 5184, 15552, 20736, 62208, 8, 24, 32, 96, 128, 384, 48, 144, 192, 576, 768, 2304, 288, 864, 1152, 3456, 4608, 13824, 1728, 5184, 6912, 20736, 27648, 82944, 10368, 31104, 41472, 124416, 165888, 497664, 64

Offset: 0

Antti Karttunen, Nov 07 2021

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A003959, A276086, A348950, A348997, A348999.

Cf. also A324653, A346470, A348996.

A348949(n) = { my(m=1, p=2); while(n, m *= ((1+p)^(n%p)); n = n\p; p = nextprime(1+p)); (m); };

a(n) = A003959(A276086(n)).

a(n) = A276086(n) + A348950(n).

A348929 a(n) = gcd(n, A003959(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 12, 1, 2, 3, 1, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 1, 4, 1, 6, 1, 1, 3, 2, 1, 36, 1, 2, 1, 2, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 2, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 72, 1, 2, 3, 4, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 3, 4, 1, 18, 7, 4, 1, 2, 5, 12, 1, 2, 3, 4, 1, 6, 1, 2, 3

Offset: 1

Antti Karttunen, Nov 07 2021

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

Cf. A003959, A085731, A126865, A323166, A348507, A348928, A348999 [= a(A276086(n))].

Differs from similar A126795 for the first time at n=36, where a(36) = 36, while A126795(36) = 12.

f[p_, e_] := (p + 1)^e; a[n_] := GCD[n, Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Nov 07 2021 *)

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348929(n) = gcd(n, A003959(n));

a(n) = gcd(n, A003959(n)) = gcd(n, A348507(n)) = gcd(A003959(n), A348507(n)).

A348950 a(n) = A348507(A276086(n)), where A348507(n) = A003959(n) - n, A003959 is multiplicative with a(p^e) = (p+1)^e, and A276086 gives the prime product form of primorial base expansion of n.

Original entry on oeis.org

0, 1, 1, 6, 7, 30, 1, 8, 9, 42, 51, 198, 11, 58, 69, 282, 351, 1278, 91, 398, 489, 1842, 2331, 8118, 671, 2638, 3309, 11802, 15111, 50958, 1, 10, 11, 54, 65, 258, 13, 74, 87, 366, 453, 1674, 113, 514, 627, 2406, 3033, 10674, 853, 3434, 4287, 15486, 19773, 67194, 5993, 22354, 28347, 98166, 126513, 418914, 15, 94, 109

Offset: 0

Antti Karttunen, Nov 06 2021

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A003959, A276086, A348507, A348949, A348999.

Cf. also A327860.

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A348507(n) = (A003959(n) - n);
A348950(n) = A348507(A276086(n));

A348950(n) = { my(m1=1, m2=1, p=2); while(n, m1 *= (p^(n%p)); m2 *= ((1+p)^(n%p)); n = n\p; p = nextprime(1+p)); (m2-m1); };

a(n) = A348949(n) - A276086(n) = A348507(A276086(n)).

A348998 a(n) = A348928(A276086(n)), where A348928(n) = gcd(n, A003958(n)), and A003958 is multiplicative with a(p^e) = (p-1)^e, and A276086 gives the prime product form of primorial base expansion of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 6, 3, 6, 1, 2, 3, 6, 3, 6, 1, 2, 3, 6, 3, 6, 1, 2, 3, 6, 3, 6, 1, 2, 3, 6, 3, 6, 1, 2, 3, 6, 9, 18, 1, 2, 3, 6, 9, 18, 1, 2, 3, 6, 9, 18, 1, 2, 3, 6, 9, 18, 1, 2, 3, 6, 9, 18, 1, 2, 3, 6, 9, 18, 1, 2, 3, 6, 9, 18

Offset: 0

Antti Karttunen, Nov 07 2021

After each primorial number (A002110), the apparent periodicity grows more complex.

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A002110, A003958, A276086, A348928, A348999.

A348998(n) = { my(m1=1, m2=1, p=2); while(n, m1 *= (p^(n%p)); m2 *= ((p-1)^(n%p)); n = n\p; p = nextprime(1+p)); gcd(m1,m2); };

A349000 a(n) = A323166(A276086(n)), where A323166(n) = gcd(n, usigma(n)), usigma (A034448) is multiplicative with a(p^e) = (p^e)+1, and A276086 gives the prime product form of primorial base expansion of n.

Original entry on oeis.org

1, 1, 1, 6, 1, 6, 1, 2, 3, 6, 15, 90, 1, 2, 1, 6, 5, 30, 1, 2, 3, 6, 45, 90, 1, 2, 1, 6, 5, 30, 1, 2, 1, 6, 1, 6, 1, 2, 3, 6, 15, 90, 1, 2, 1, 6, 5, 30, 7, 14, 21, 42, 315, 630, 1, 2, 1, 6, 5, 30, 1, 2, 1, 6, 1, 6, 5, 10, 15, 30, 15, 90, 25, 50, 25, 150, 25, 150, 175, 350, 525, 1050, 7875, 15750, 25, 50, 25, 150, 125, 750

Offset: 0

Antti Karttunen, Nov 07 2021

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A034448, A276086, A323166, A348996, A348997, A348999.

A349000(n) = { my(m1=1, m2=1, p=2, u); while(n, if(n%p, u = p^(n%p); m1 *= u; m2 *= (1+u)); n = n\p; p = nextprime(1+p)); gcd(m1,m2); };

a(n) = A323166(A276086(n)) = gcd(A276086(n), A348996(n)).

cp's OEIS Frontend

A348949 a(n) = A003959(A276086(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e, and A276086 gives the prime product form of primorial base expansion of n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A348929 a(n) = gcd(n, A003959(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A348950 a(n) = A348507(A276086(n)), where A348507(n) = A003959(n) - n, A003959 is multiplicative with a(p^e) = (p+1)^e, and A276086 gives the prime product form of primorial base expansion of n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A348998 a(n) = A348928(A276086(n)), where A348928(n) = gcd(n, A003958(n)), and A003958 is multiplicative with a(p^e) = (p-1)^e, and A276086 gives the prime product form of primorial base expansion of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A349000 a(n) = A323166(A276086(n)), where A323166(n) = gcd(n, usigma(n)), usigma (A034448) is multiplicative with a(p^e) = (p^e)+1, and A276086 gives the prime product form of primorial base expansion of n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula