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A383535 Heinz number of the positive first differences of the 0-prepended prime indices of n.

%I A383535 #6 May 21 2025 16:41:48
%S A383535 1,2,3,2,5,4,7,2,3,6,11,4,13,10,6,2,17,4,19,6,9,14,23,4,5,22,3,10,29,
%T A383535 8,31,2,15,26,10,4,37,34,21,6,41,12,43,14,6,38,47,4,7,6,33,22,53,4,15,
%U A383535 10,39,46,59,8,61,58,9,2,25,20,67,26,51,12,71,4,73
%N A383535 Heinz number of the positive first differences of the 0-prepended prime indices of n.
%C A383535 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%C A383535 Also Heinz number of the first differences of the distinct 0-prepended prime indices of n.
%F A383535 A001222(a(n)) = A001221(n).
%F A383535 A056239(a(n)) = A061395(n).
%F A383535 A055396(a(n)) = A055396(n).
%F A383535 A061395(a(n)) = A241919(n).
%e A383535 The terms together with their prime indices begin:
%e A383535      1: {}        2: {1}        31: {11}       38: {1,8}
%e A383535      2: {1}      17: {7}         2: {1}        47: {15}
%e A383535      3: {2}       4: {1,1}      15: {2,3}       4: {1,1}
%e A383535      2: {1}      19: {8}        26: {1,6}       7: {4}
%e A383535      5: {3}       6: {1,2}      10: {1,3}       6: {1,2}
%e A383535      4: {1,1}     9: {2,2}       4: {1,1}      33: {2,5}
%e A383535      7: {4}      14: {1,4}      37: {12}       22: {1,5}
%e A383535      2: {1}      23: {9}        34: {1,7}      53: {16}
%e A383535      3: {2}       4: {1,1}      21: {2,4}       4: {1,1}
%e A383535      6: {1,2}     5: {3}         6: {1,2}      15: {2,3}
%e A383535     11: {5}      22: {1,5}      41: {13}       10: {1,3}
%e A383535      4: {1,1}     3: {2}        12: {1,1,2}    39: {2,6}
%e A383535     13: {6}      10: {1,3}      43: {14}       46: {1,9}
%e A383535     10: {1,3}    29: {10}       14: {1,4}      59: {17}
%e A383535      6: {1,2}     8: {1,1,1}     6: {1,2}       8: {1,1,1}
%t A383535 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A383535 Table[Times@@Prime/@DeleteCases[Differences[Prepend[prix[n],0]],0],{n,100}]
%Y A383535 For multiplicities instead of differences we have A181819.
%Y A383535 Positions of first appearances are A358137.
%Y A383535 Positions of squarefree numbers are A383512, counted by A098859.
%Y A383535 Positions of nonsquarefree numbers are A383513, counted by A336866.
%Y A383535 These are Heinz numbers of rows of A383534.
%Y A383535 A000040 lists the primes, differences A001223.
%Y A383535 A048767 is the Look-and-Say transform, union A351294, complement A351295.
%Y A383535 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A383535 A320348 counts strict partitions with distinct 0-appended differences, ranks A325388.
%Y A383535 A325324 counts partitions with distinct 0-appended differences, ranks A325367.
%Y A383535 Cf. A122111, A124010 (prime signature), A130091, A287352, A325351, A325366, A325368, A355536, A381431, A384008, A384009.
%K A383535 nonn
%O A383535 1,2
%A A383535 _Gus Wiseman_, May 21 2025