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 A353502 Numbers with all prime indices and exponents > 2. 5
 1, 125, 343, 625, 1331, 2197, 2401, 3125, 4913, 6859, 12167, 14641, 15625, 16807, 24389, 28561, 29791, 42875, 50653, 68921, 78125, 79507, 83521, 103823, 117649, 130321, 148877, 161051, 166375, 205379, 214375, 226981, 274625, 279841, 300125, 300763, 357911 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. LINKS David A. Corneth, Table of n, a(n) for n = 1..10000 (first 3000 terms from Amiram Eldar) FORMULA Sum_{n>=1} 1/a(n) = Product_{p prime > 3} (1 + 1/(p^2*(p-1))) = (72/95)*A065483 = 1.0154153584... . - Amiram Eldar, May 28 2022 EXAMPLE The initial terms together with their prime indices: 1: {} 125: {3,3,3} 343: {4,4,4} 625: {3,3,3,3} 1331: {5,5,5} 2197: {6,6,6} 2401: {4,4,4,4} 3125: {3,3,3,3,3} 4913: {7,7,7} 6859: {8,8,8} 12167: {9,9,9} 14641: {5,5,5,5} 15625: {3,3,3,3,3,3} 16807: {4,4,4,4,4} 24389: {10,10,10} 28561: {6,6,6,6} 29791: {11,11,11} 42875: {3,3,3,4,4,4} MATHEMATICA Select[Range[10000], #==1||!MemberQ[FactorInteger[#], {_?(#<5&), _}|{_, _?(#<3&)}]&] CROSSREFS The version for only parts is A007310, counted by A008483. The version for <= 2 instead of > 2 is A018256, # of compositions A137200. The version for only multiplicities is A036966, counted by A100405. The version for indices and exponents prime (instead of > 2) is: - listed by A346068 - counted by A351982 - only exponents: A056166, counted by A055923 - only parts: A076610, counted by A000607 The version for > 1 instead of > 2 is A062739, counted by A339222. The version for compositions is counted by A353428, see A078012, A353400. The partitions with these Heinz numbers are counted by A353501. A000726 counts partitions with multiplicities <= 2, compositions A128695. A001222 counts prime factors with multiplicity, distinct A001221. A004250 counts partitions with some part > 2, compositions A008466. A056239 adds up prime indices, row sums of A112798 and A296150. A124010 gives prime signature, sorted A118914. A295341 counts partitions with some multiplicity > 2, compositions A335464. Cf. A000720, A003963, A065483, A114901, A181819, A353503, A353508. Sequence in context: A175486 A250579 A161855 * A252057 A045184 A059470 Adjacent sequences: A353499 A353500 A353501 * A353503 A353504 A353505 KEYWORD nonn AUTHOR Gus Wiseman, May 16 2022 STATUS approved

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Last modified September 21 17:21 EDT 2023. Contains 365503 sequences. (Running on oeis4.)