Sums of all Squares

[davar55]

While proving a few such important sequences infinite might help us
triangulate on the keys - the k-tuples, and the mersennes.

Is that blind faith, or do you have some reason for this belief?

I'll just say here it's not blind and it's not faith.

Forgive me my keys.

[davar55]

Originally Posted by davar55

From above:

...

If we prove that enough sequences' infinitude (like this one) depends on that k-tuple conjecture, then proving even one of them infinite would
be evidence that all are - including the mersennes.

I suppose this might be considered more evidence, huh?

[davar55]

Originally Posted by davar55

Another possibly interesting variation:

2^2 + 3^3 + 5^5 + ... + p^p = 10^mK

What is the smallest prime p such that
the sum of squares of all primes up to p
is a multiple of 10 (or 100 or 1000 and so on).

(I think somewhere in these series we'll get
numeric pointers to empirical evidence
connecting some number theoretical conjectures.)

Just checking.

[davar55]

From this thread:

Originally Posted by davar55

2^2 + 3^2 + 5^2 + ... + p^2 = 10^mK

What is the smallest prime p such that
the sum of squares of all primes up to p
is a multiple of 10 (or 100 or 1000).

Originally Posted by CRGreathouse

907, 977, 977, 36643, 1067749, 17777197, 71622461, 2389799983, ...

The next term (if one exists) is more than 4 trillion.

Originally Posted by cheesehead

Not yet in the OEIS.

http://www.research.att.com/~njas/sequences/

I think it qualifies. Also, I'm fond of OEIS entries with relatively large initial terms -- especially when the next few terms are so closely spaced as in this one. (Might it set some record in that regard -- highest ratio of initial term to average spacing of next n terms, for n = 3?)

I'd be glad to submit it, but I think it should be one of you guys.

How about generalizing to other bases?

Originally Posted by davar55

This is fine work by all of you. If you wish to submit the sequence to
oeis, please go ahead. I couldn't do justice to the calculations, which
I'm really impressed by. Joint discovery (attribution) is fine.

Originally Posted by bsquared

Now in OEIS: A174862

Originally Posted by davar55

With all the work done on the OP, it shouldn't be too hard
to generalize the problem a bit.

I think cubes.

2^3 + 3^3 + 5^3 + ... + p^3 = 10^mK

What is the smallest prime p such that
the sum of cubes of all primes up to p
is a multiple of 10 (or 100 or 1000 or 10000 or ...).

I'm also curious about how these (squares and cubes) results
compare to first powers (sum of primes themselves).

Since these series depend on the properties of a number
in base ten, they could be considered recreational --
interesting but not necessarily useful. Still, perhaps the
sequence of sequences can someday be used to derive some
important number theoretic fact. That's one of the purposes
of the oeis.

Yes indeed.

[yae9911]

Hardly almost 9 years have passed, and already the topic comes out again. When I searched for entries of b² in the OEIS, I came across the sequence A174862, which contained a link to this discussion.

But with the two variants p³ and p^p not much more happened, and I have now taken the liberty to close these gaps in OEIS with new entries A330308 and A330309. It is remarkable how quickly Giovanni Resta found two more terms in the p^p sequence A330309, but Giovanni is notorious for such amazing results.

[bsquared]

Originally Posted by yae9911

Hardly almost 9 years have passed, and already the topic comes out again. When I searched for entries of b² in the OEIS, I came across the sequence A174862, which contained a link to this discussion.

But with the two variants p³ and p^p not much more happened, and I have now taken the liberty to close these gaps in OEIS with new entries A330308 and A330309. It is remarkable how quickly Giovanni Resta found two more terms in the p^p sequence A330309, but Giovanni is notorious for such amazing results.

Can't believe it has been that long since this happened. Thanks for submitting the p³ sequence, that must have been overlooked before. Also glad you are having fun with variants. Looking at the timing of things, it can't have been more than a day or so for additional terms to have been added which makes Giovanni's additions very impressive indeed.

[sweety439]

Smallest prime p such that the sum of all powers of primes 2^2 + 3^3 + 5^5 + 7^7 + 11^11 + ... + p^p up to p is a multiple of n:

n,p
1,2
2,2
3,5
4,2
5,11
6,5
7,13
8,41
9,11
10,11
11,137
12,5
13,19
14,23
15,11
16,211
17,131
18,11
19,23
20,17
21,23
22,137
23,73
24,41
25,227
26,167
27,11
28,233
29,71
30,11
31,3
32,211
33,439
34,257
35,727
36,41
37,367
38,23
39,167
40,191
41,67
42,23
43,359
44,283
45,11
46,73
47,389
48,211
49,61
50,227
51,1151
52,167
53,181
54,11
55,137
56,233
57,23
58,233
59,173
60,17
61,227
62,47
63,613
64,823
65,29
66,439
67,151
68,257
69,73
70,727
71,227
72,41
73,151
74,367
75,227
76,211
77,1697
78,167
79,241
80,211
81,11
82,67
83,773
84,233
85,257
86,389
87,233
88,283
89,67
90,11
91,977
92,919
93,47
94,389
95,211
96,211
97,23
98,727
99,647
100,751
101,907
102,1433
103,743
104,1913
105,977
106,1489
107,653
108,41
109,383
110,137
111,911
112,829
113,1039
114,23
115,1453
116,233
117,167
118,353
119,1553
120,211
121,647
122,227
123,67
124,83
125,1129
126,1367
127,617
128,823
129,359
130,431
131,191
132,991
133,23
134,1039
135,11
136,709
137,443
138,73
139,503
140,829
141,941
142,227
143,2663
144,211
145,929
146,367
147,1213
148,367
149,269
150,227
151,71
152,211
153,1151
154,1697
155,83
156,167
157,1609
158,241
159,1873
160,211
161,349
162,11
163,383
164,191
165,2539
166,773
167,2113
168,233
169,167
170,257
171,211
172,389
173,5009
174,233
175,883
176,283
177,1741
178,67
179,239
180,83
181,1187
182,977
183,227
184,919
185,773
186,47
187,601
188,389
189,2633
190,211
191,79
192,4007
193,167
194,23
195,619
196,3253
197,269
198,991
199,3221
200,797
201,1097
202,907
203,233
204,1553
205,67
206,1471
207,2539
208,1913
209,601
210,977
211,5147
212,3623
213,227
214,653
215,829
216,41

A supsequence of A330309.

[yae9911]

If someone has such an idea, it is best if he himself submits this as a proposal for a new sequence. I'm not going to take this job off your hands for now. At best, I can help with the procedure of how to submit new entries. Opinions about whether something is accepted or rejected differ widely. I do not presume to decide whether this deserves a new entry in the OEIS database. This decision is made jointly by the team of editors.