A392863 a(n) = Sum_{i=1..n} (A143293(n) mod prime(i))*A002110(i-1), where A002110 are the primorial numbers, and A143293 give their partial sums.

0, 1, 1, 25, 145, 1615, 27025, 27025, 1558555, 147053905, 1485611125, 72652236655, 1877696647825, 31560649187065, 12505821453802675, 64836866780482795, 16051971214081259455, 928548408575402511895, 108603128017211310299815, 577756653454839194232895, 424927120411789260211703755, 13257566213325865362462257725

Offset

0,4

Links

Antti Karttunen, Table of n, a(n) for n = 0..349

Index entries for sequences related to primorial base.

Index entries for sequences related to primorial numbers.

Formula

a(n) = Sum_{i=1..n} (A143293(i) mod prime(i))*A002110(i-1). [An equivalent formula, valid because of the self-embedding property]

a(n) = Sum_{i=0..n-1} A357270(i)*A002110(i).

For n > 0, a(n) = A391930(n, A143293(n-1)).

Example

n                     a(n)                           A049345(a(n))
---+---------------------------------------------------------------
0  |                    0,                                      0,
1  |                    1,                                      1,
2  |                    1,                                      1,
3  |                   25,                                  4:0:1,
4  |                  145,                                4:4:0:1,
5  |                 1615,                              7:4:4:0:1,
6  |                27025,                           11:7:4:4:0:1,
7  |                27025,                           11:7:4:4:0:1,
8  |              1558555,                       3:0:11:7:4:4:0:1,
9  |            147053905,                    15:3:0:11:7:4:4:0:1,
10 |           1485611125,                  6:15:3:0:11:7:4:4:0:1,
11 |          72652236655,               11:6:15:3:0:11:7:4:4:0:1,
12 |        1877696647825,             9:11:6:15:3:0:11:7:4:4:0:1,
13 |       31560649187065,           4:9:11:6:15:3:0:11:7:4:4:0:1,
14 |    12505821453802675,        41:4:9:11:6:15:3:0:11:7:4:4:0:1,
15 |    64836866780482795,      4:41:4:9:11:6:15:3:0:11:7:4:4:0:1,
16 | 16051971214081259455,   26:4:41:4:9:11:6:15:3:0:11:7:4:4:0:1,
etc.
Note that the digits of the growing primorial base expansion are given (when read from right to left) by A357270.
Compare to the pattern in A392864.

Prog

(PARI) A392863(n) = if(!n, n, my(P=1, s=1, z=0); forprime(p=2, prime(n), s+=(P*=p); z += (s%p)*(P/p)); (z));
(PARI) A392863(n) = sum(i=0, n-1, A357270(i)*A002110(i));

Crossrefs

Cf. A002110, A049345, A143293, A357270, A391930, A392615, A392864.

Cf. also A062347, A391936.

Sequence in context: A193438 A139152 A123014 * A366167 A095971 A147421

Adjacent sequences: A392860 A392861 A392862 * A392864 A392865 A392866

Keyword

nonn,base,easy

Author

Antti Karttunen, Jan 26 2026

Status

approved