[quote]While proving a few such important sequences infinite might help us
triangulate on the keys  the ktuples, and the mersennes. [/quote][quote] Is that blind faith, or do you have some reason for this belief? [/quote]I'll just say here it's not blind and it's not faith. Forgive me my keys. 
[QUOTE=davar55;243589]From above:
... If we prove that enough sequences' infinitude (like this one) depends on that ktuple conjecture, then proving even one of them infinite would be evidence that all are  including the mersennes.[/QUOTE] I suppose this might be considered more evidence, huh? 
[QUOTE=davar55;241455]Another possibly interesting variation:
2^2 + 3^3 + 5^5 + ... + p^p = 10[sup]m[/sup]K 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.)[/QUOTE] Just checking. 
From this thread:
[QUOTE=davar55;208752]2^2 + 3^2 + 5^2 + ... + p^2 = 10[sup]m[/sup]K 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).[/QUOTE] [QUOTE=CRGreathouse;208783]907, 977, 977, 36643, 1067749, 17777197, 71622461, 2389799983, ... The next term (if one exists) is more than 4 trillion.[/QUOTE] [QUOTE=cheesehead;208799]Not yet in the OEIS. [URL="http://www.research.att.com/%7Enjas/sequences/"]http://www.research.att.com/~njas/sequences/[/URL] 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?[/QUOTE] [QUOTE=davar55;210100]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.[/QUOTE] [QUOTE=bsquared;210433]Now in OEIS: [URL="http://www.research.att.com/%7Enjas/sequences/A174862"]A174862[/URL][/QUOTE] [QUOTE=davar55;210785]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[sup]m[/sup]K 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.[/QUOTE] Yes indeed. 
Hardly almost 9 years have passed, and already the topic comes out again. When I searched for entries of b[SUP]2[/SUP] in the OEIS, I came across the sequence A[OEIS]174862[/OEIS], which contained a link to this discussion.
But with the two variants p[SUP]3[/SUP] and p[SUP]p[/SUP] not much more happened, and I have now taken the liberty to close these gaps in OEIS with new entries A[OEIS]330308[/OEIS] and A[OEIS]330309[/OEIS]. It is remarkable how quickly Giovanni Resta found two more terms in the p[SUP]p[/SUP] sequence A330309, but Giovanni is notorious for such amazing results. 
[QUOTE=yae9911;532746]Hardly almost 9 years have passed, and already the topic comes out again. When I searched for entries of b[SUP]2[/SUP] in the OEIS, I came across the sequence A[OEIS]174862[/OEIS], which contained a link to this discussion.
But with the two variants p[SUP]3[/SUP] and p[SUP]p[/SUP] not much more happened, and I have now taken the liberty to close these gaps in OEIS with new entries A[OEIS]330308[/OEIS] and A[OEIS]330309[/OEIS]. It is remarkable how quickly Giovanni Resta found two more terms in the p[SUP]p[/SUP] sequence A330309, but Giovanni is notorious for such amazing results.[/QUOTE] Can't believe it has been that long since this happened. Thanks for submitting the p[SUP]3[/SUP] 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. 
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:
[CODE] 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 [/CODE] A supsequence of A330309. 
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.

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